Entanglement Theory and its Applications in Gaussian Quantum Information
Spyros Tserkis B.Sc., M.Sc.
A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland in 2019
School of Mathematics and Physics ii Abstract
Entanglement is a physical property, emerging from the superposition of composite quantum systems, and manifested through non-classical correlations of quantum observables. In the field of Quantum Information, entanglement is considered the “resource” for various quantum protocols, so the ability to quantify it would allow us to associate it with the success of those protocols. Entanglement of formation is a proper way to quantify entanglement, but an analytical expression for this measure exists only for special cases. In this thesis, we focus on two-mode Gaussian states, and we derive narrow upper and lower bounds for this measure that get tight for several special cases. We further study how quantum teleportation and entanglement distillation can be employed in order to error-correct information encoded on a Gaussian state that suffers from Gaussian noise. In particular, we derive every physical state able to simulate a given phase-insensitive Gaussian channel through teleportation with finite-energy resources, and show how error-correction of a state is related to the simulation ofa less decohering channel that the state has to pass through. We also discuss how the premise that the whole environment is under control of the adversary in quantum key distribution can be eliminated if we consider a teleportation-based eavesdropping attack. More specifically, we propose an all-optical teleportation attack that under collective measurements can reach optimality in the limit of infinite amount of entanglement, while for finite entanglement resources it outperforms the corresponding optimal individual attack. Finally, we show how using a class of finite-energy resource states we can increasingly approximate the infinite-energy bounds for decreasing purity, so that they provide tight upper bounds to the secret-key capacity of single-mode phase-insensitive Gaussian channels.
Declaration by author
This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis. I have clearly stated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analysis, significant technical procedures, professional editorial advice, financial support and any other original research work used or reported in my thesis. Thecontentof my thesis is the result of work I have carried out since the commencement of my higher degree by research candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award. I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the policy and procedures of The University of Queensland, the thesis be made available for research and study in accordance with the Copyright Act 1968 unless a period of embargo has been approved by the Dean of the Graduate School. I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material. Where appropriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis and have sought permission from co-authors for any jointly authored works included in the thesis.
Spyros Tserkis
Publications included in this thesis
1. Spyros Tserkis and Timothy C. Ralph, Quantifying entanglement in two-mode Gaussian states, Physical Review A 96, 062338 (2017).
2. Spyros Tserkis, Josephine Dias, and Timothy C. Ralph, Simulation of Gaussian channels via teleportation and error correction of Gaussian states, Physical Review A 98, 052335 (2018).
3. Spyros Tserkis, Sho Onoe, and Timothy C. Ralph, Quantifying entanglement of formation for two-mode Gaussian states: Analytical expressions for upper and lower bounds and numerical estimation of its exact value, Physical Review A 99, 052337 (2019).
4. Riccardo Laurenza, Spyros Tserkis, Leonardo Banchi, Samuel L. Braunstein, Timothy C. Ralph, and Stefano Pirandola, Tight bounds for private communication over bosonic Gaussian channels based on teleportation simulation with optimal finite resources, Physical Review A 100, 042301 (2019).
5. Spyros Tserkis, Neda Hosseinidehaj, Nathan Walk, and Timothy C. Ralph, Teleportation-based collective attacks in Gaussian quantum key distribution, Physical Review Research 2, 013208 (2020).
Other publications and preprints during candidature
1. Hao Jeng, Spyros Tserkis, Jing Yan Haw, Helen M. Chrzanowski, Jiri Janousek, Timothy C. Ralph, Ping Koy Lam, and Syed M. Assad, Entanglement properties of a measurement-based entanglement distillation experiment, Physical Review A 99, 042304 (2019).
2. Cheng Jiang, Spyros Tserkis, Kevin Collins, Sho Onoe, Yong Li, Lin Tian, Switchable bipartite and genuine tripartite entanglement via an optoelectromechanical interface, arXiv:1910.13173.
Contributions by others to the thesis
Prof. Timothy C. Ralph has supervised all the projects/papers discussed in this thesis (presented in Chapters 6-9), and, along with Dr. Austin P. Lund, provided critical feedback for improvements and revisions of the whole thesis. I also acknowledge the contributions by Josephine Dias in Ref. [1], Sho Onoe in Ref. [2], Neda Hosseinidehaj and Nathan Walk in Ref. [3], who helped in the preparation of those manuscripts by giving feedback and editing parts of the manuscripts. Contributions by Stefano Pirandola, Leonardo Banchi, Riccardo Laurenza, and Samuel L. Braunstein regarding Ref. [4], are discussed in the beginning of Chapter 9.
Statement of parts of the thesis submitted to qualify for the award of another degree
No works submitted towards another degree have been included in this thesis.
Research involving human or animal subjects
No animal or human subjects were involved in this research.
Acknowledgements
I would like to acknowledge, and express my gratitude to my principal advisor, Tim Ralph, for the patient guidance, encouragement, and advice he has provided me throughout my PhD candidature. I would also like to thank my associate advisor, Austin Lund, for his support and the fruitful discussions during my studies. My sincere acknowledgement goes to my colleagues and co-authors, Josephine Dias, Nedasadat Hosseibnidehaj and Sho Onoe. I am also grateful for meeting and collaborating with Stefano Pirandola, Riccardo Laurenza, Samuel Braunstein, Syed Assad, Hao Jeng, Ping Koy Lam, Nathan Walk, and Leonardo Banchi. I would like to thank our group administrator, Kaerin Gardner, and the postgrad administrator, Murray Kane, for being so helpful to me during my studies. Special thanks to Farid Shahandeh and Anatoly Kulikov for our interesting discussions. I am also thankful for meeting all the members of the Quantum Optics and Quantum Information group.
Financial support
This research was supported by:
• the University of Queensland International Scholarship (tuition fee award and living allowance stipend), awarded by The University of Queensland.
• the Research Higher Degree Scholarship (living allowance top-up), awarded by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technol- ogy (CE110001027 and CE170100012) and the Australian Department of Defence, Innovation Hub.
Keywords
Keywords: quantum information, quantum communication, quantum entanglement, quantum correla- tions, quantum teleportation, quantum error correction, quantum key distribution, quantum channel capacities
Australian and New Zealand Standard Research Classifications (ANZSRC)
• ANZSRC code: 020603 Quantum Information, Computation and Communication 80%
• ANZSRC code: 020604 Quantum Optics, 20%
Fields of Research (FoR) Classification
• FoR code: 0206 Quantum Physics, 90%
• FoR code: 0105 Mathematical Physics, 10%
Table of contents
List of figures xxxi
I Introduction1
1 Introductory Remarks3 1.1 Historical Note ...... 3 1.2 Goal of the Thesis ...... 4 1.3 Structure of the Thesis ...... 5
II Literature Review9
2 Mathematical Preliminaries 11 2.1 Vector Spaces ...... 12 2.1.1 Hilbert Space ...... 13 2.1.2 Fock Space ...... 15 2.2 Linear Operators ...... 15 2.2.1 Normal Operators ...... 18 2.2.2 Convexity and Concavity ...... 20 2.2.3 Composite operators ...... 21 2.2.4 Superoperators ...... 22
3 Quantum Mechanics and Quantum Information 23 3.1 Quantum State ...... 24 3.1.1 Pure and Mixed States ...... 24 3.1.2 Interpreting the Quantum State ...... 26 3.1.3 Quantum Entropy of a State ...... 26 3.2 Quantum Operation ...... 28 3.2.1 Evolution of a Closed Quantum System ...... 29 3.2.2 Evolution of an Open Quantum System ...... 29 3.2.3 Measurement of a Quantum State ...... 32 3.3 Distinguishing Quantum States ...... 34 3.3.1 Trace Distance ...... 34 xxviii Table of contents
3.3.2 Fidelity ...... 35 3.3.3 Relative Entropy ...... 36 3.4 Examples of Quantum Systems ...... 36 3.4.1 Finite-dimensional System ...... 36 3.4.2 Infinite-dimensional System ...... 38 3.5 Quantum Mechanics in Phase Space ...... 40
4 Gaussian Systems 43 4.1 Gaussian States and Operations ...... 43 4.1.1 Vacuum States ...... 45 4.1.2 Coherent States and the Displacement Operator ...... 46 4.1.3 Squeezed States and the Squeezing Operator ...... 49 4.1.4 Thermal States ...... 51 4.1.5 Phase Rotation Operator ...... 52 4.1.6 Beam Splitter Operator ...... 52 4.1.7 Two-mode Squeezing ...... 53 4.2 Symplectic Analysis of Gaussian States ...... 54 4.2.1 Decomposition of a Gaussian State ...... 54 4.2.2 Purification of a Gaussian State ...... 55 4.2.3 Two-mode Gaussian States ...... 56 4.3 Gaussian Measurements ...... 58 4.3.1 Homodyne Detection ...... 59 4.3.2 Heterodyne Detection ...... 60 4.3.3 Partial Measurements ...... 60 4.4 Distinguishing Gaussian States ...... 61 4.4.1 Fidelity ...... 61 4.4.2 Relative Entropy ...... 62 4.5 Gaussian Channels ...... 62 4.5.1 One-mode Gaussian Channels ...... 62 4.5.2 Phase-Insensitive Gaussian Channels ...... 63
5 Entanglement Theory 67 5.1 Separability and Entanglement ...... 68 5.2 Separability Criteria ...... 69 5.2.1 Pure States ...... 69 5.2.2 Mixed States ...... 70 5.2.3 Separability in Gaussian Systems ...... 71 5.3 Quantifying Entanglement ...... 73 5.3.1 Local Operations and Classical Communication ...... 73 5.3.2 Majorization ...... 74 5.3.3 Quantifying Entanglement in Pure States (Entanglement Entropy) ...... 74 Table of contents xxix
5.3.4 Quantifying Entanglement in Mixed States ...... 75 5.4 Survey of Entanglement Measures ...... 78 5.4.1 Distillable Entanglement and Entanglement Cost ...... 78 5.4.2 Entanglement of Formation ...... 79 5.4.3 Relative Entropy of Entanglement ...... 80 5.4.4 Squashed Entanglement ...... 81 5.4.5 Negativity and Logarithmic Negativity ...... 81 5.4.6 Rains Bound ...... 82
III Results 83
6 Entanglement of Formation in Gaussian Systems 85 6.1 Introduction ...... 85 6.2 Two-mode Gaussian States ...... 86 6.3 Entanglement of Formation ...... 88 6.4 Lower bound for Entanglement of Formation ...... 89 6.5 Upper bound for Entanglement of Formation ...... 94 6.6 Numerical estimation of EOF ...... 96 6.7 EoF of the Choi-state ...... 96 6.8 Comparison with Logarithmic Negativity ...... 97 6.9 Conclusion ...... 99
7 Continuous-Variable Teleportation and Error Correction 101 7.1 Introduction ...... 101 7.2 CV Teleportation ...... 102 7.3 Channel Simulation ...... 104 7.4 Error Correction of Gaussian States ...... 106 7.5 Error correction protocol ...... 109 7.5.1 Pure Lossy and Amplifier Channels ...... 109 7.5.2 Thermal Lossy and Amplifier Channels ...... 110 7.6 Error Correction with NLA ...... 111 7.7 Conclusion ...... 114
8 All-optical Teleportation and Quantum Key Distribution 115 8.1 Introduction ...... 115 8.2 Cryptography ...... 116 8.3 Quantum Key Distribution ...... 117 8.4 Gaussian CV-QKD ...... 118 8.5 Entangling Cloner Attack ...... 119 8.6 All-optical Teleportation Protocol ...... 120 8.7 All-optical Teleportation Attack ...... 122 xxx Table of contents
8.8 Conclusion ...... 127
9 Secret Key Capacity of Gaussian Channels 129 9.1 Introduction ...... 129 9.2 Secret-key capacity and Bounds ...... 130 9.2.1 Upper Bound of Secret-ket Capacity ...... 130 9.2.2 Gaussian Relative Entropy of Entanglement ...... 131 9.3 Upper Bounds for Bosonic Gaussian Channels ...... 133 9.3.1 Thermal Lossy Channel ...... 134 9.3.2 Thermal Amplifier Channel ...... 134 9.3.3 Additive Noise Channel ...... 136 9.3.4 Finite uses of the channel ...... 136 9.4 Conclusion ...... 138
IV Conclusion 139
10 Concluding Remarks and Future Outlook 141 10.1 Quantification of Entanglement in Gaussian States ...... 141 10.2 Error Correction of Gaussian States under Gaussian Noise ...... 142 10.3 Teleportation-Based Eavesdropping Attacks ...... 142 10.4 Bounds for Secret-Key Capacities of Gaussian Channels ...... 143
References 145
V Appendix 161
Appendix A Rigged Hilbert Space 163
Appendix B Noiseless Linear Amplification 167 List of figures
2.1 Convex and concave sets. On the left we have a convex set and on the right a concave one. As we see, for two arbitrary elements of the sets, A and B, we can always find a straight line which lies inside the set, that connects them when the shape is convex, while this is not true for a concave set...... 20
3.1 Pure and mixed states as a convex set [25]. With red dots we represent the pure states |ψ⟩,|φ⟩,|u⟩,|v⟩ and with the yellow dot the mixed state ρˆ. The mixed state ρˆ can be si- 1 1 multaneously represented as ρˆ = x|ψ⟩⟨ψ| + y|φ⟩⟨φ| or ρˆ = 2 |u⟩⟨u| + 2 |v⟩⟨v|, where |u⟩ = √ √ √ √ x|ψ⟩ + y|φ⟩ and |v⟩ = x|ψ⟩ − y|φ⟩...... 25 3.2 Quantum state interpretations. Each of the six distinct interpretations (see numbers in the diagram) involves the selection of one of the two possibilities from each dichotomy. Note that an ontic interpretation is necessarily objective and a subjective interpretation is necessarily epistemic...... 26 3.3 Venn diagram for the von Neumann entropies of a bipartite state ρˆAB. Note that the area of the diagram does not necessarily represent positive value of the entropies, since the conditional von Neumann entropies S(A|B) and S(B|A) can also take negative values for entangled states. 27 3.4 Stinespring dilation of a quantum channel. The input state ρˆ interacts with the environmental
state |e0⟩ through the unitary operator U, and that gives as the primary output the state M(ˆρ) and as the complementary output the state Mf(ˆρ). If the channel M is degradable then the relationship between the primary and the complementary channels is given by M ◦ D = Mf, if it is anti-degradable we have A ◦ Mf = M...... 30 3.5 Qubit in a Bloch sphere. We graphically represent a two-dimensional quantum state in a 3- dimensional sphere. Based on the Eq. (3.72) the values θ and φ correspond to polar coordinates
of a qubit. We depict a generic pure state ⃗rψ, which lies on the surface of the sphere, and a
mixed state ⃗rρ, which has a shorter length. The maximally mixed state, i.e., ⃗rmm = 1/2 is the
origin of the sphere. Also note that orthogonal states, e.g., |0⟩ ⊥ |1⟩, represented by vectors ⃗r0
and ⃗r1, respectively, are antipodal in this representation, meaning that they are on the opposite side of the sphere...... 37 3.6 1st and 2nd quantisation. The quantum state described by the ket notation |1⟩ ⊗ |2⟩ ⊗ |3⟩ is represented for each type of quantization. The lines represent modes and the dots represent particles...... 40 xxxii List of figures
3.7 Wigner function for Fock states. We plot the the function W(x,p) for the case of a Fock state given in Eq. (3.97) for: (a) n = 0, (b) n = 2, and (c) n = 4 photons, respectively. All quantities plotted are dimensionless...... 41
4.1 Contour of the Wigner function for: (a) a vacuum state in light blue, (b) a thermal zero- displaced thermal state in dark blue, (c) a coherent state in yellow, and (d) a displaced squeezed state in red...... 47 4.2 Probability distributions. In figure (a) we have the probability distribution of a coherent state, where the different colors denote different values of amplitude |α|. It worths noting that the probability distribution of the coherent state is a Poisson distribution. In figure (b) we have the probability distribution of photons of a thermal state, where the different colors denote different values of mean number of photons per mode n¯. All quantities plotted are dimensionless. 48 4.3 Wigner function representation of Gaussian states. In figure (a) we have the W function of a vacuum state. Coherent states are just displaced vacuum states so their Wigner function is just shifted respectively. In figure (b) we plot squeezed vacuum state, that is squeezed on X quadrature and anti-squeezed in P quadrature. All quantities plotted are dimensionless. ... 50 4.4 Bloch-Messiah decomposition. In this figure an arbitrary symplectic transformation Σ is graphically represented in the Bloch-Messiah (or Euler) reduction, by its main components.
The operations ΣK and ΣL are symplectic and orthogonal transformations that represent the
passive elements, while the array of ΣS(ri) are squeezing operations that represent the active elements of a symplectic transformation Σ...... 56 4.5 Gaussian measurements. In figure (a) we sketch the main components of a homodyne detection
measurement, i.e., the input mode associated with a bosonic operator A1 that we mix in a beam
splitter of transmissivity τ with the reference mode associated with a bosonic operator A2, the
output mode A3 on which we perform the photo-detection, and the secondary output mode A4 that we are not concerned about. In figure (b) we depict the heterodyne measurement detection. As we see it is a dual homodyne detection scheme applied on the output modes of the initial
mode A1 that is mixed on a beam splitter with an ancillary vacuum state |0⟩...... 60 4.6 Gaussian phase-insensitive channels. The different classes of phase-insensitive Gaussian channels are presented in this graph. With blue we have the lossy channels, L, and the dark blue line indicates the specific case of pure lossy channels. With brown we have the amplifier channels, A, and, respectively, the dark brown line represents the pure amplifier channels. The central vertical grey line corresponds to the classical additive noise channels, N , and the green dot indicates the identity channel, 1. Channels above the dashed line are entanglement-breaking channels, i.e., v ⩾ 1 + |τ|, and channels below the dark blue and brown lines are non-physical. All quantities plotted are dimensionless...... 64 List of figures xxxiii
4.7 Stinespring dilation for phase-insensitive Gaussian channels. On the panel (a) we have a lossy channel L that is modeled through a beamsplitter B with transmissivity 1 > τ = cosθ2. The complementary channel Le is also a lossy channel with transmissivity (1 − τ). On the
panel (b) we have an amplifier channel A that is modeled through a two-mode squeezer S2 with gain τ = coshr > 1. The complementary channel Ae is a contravariant channel with negative transmissivity. If we substitute the thermal states ρˆth with the vacuum |0⟩ we get a pure lossy and a pure amplifier/cotnravariant channel, respectively. The identity channel 1 is the case where τ = 1 (for either lossy or amplifier channel), so there is no interaction with the environment and no noise is induced. Finally, the classical additive-noise channel N is a limited case with τ ≈ 1 and a highly thermal state as an input...... 65 4.8 Thermal channel decomposition. Every thermal channel G can be decomposed as a
sequence of a pure loss channel Lp followed by a pure amplifier Ap, i.e., G ≡ Ap ◦ Lp, or reversely (if it is non-entangling breaking), as a pure amplifier Ap followed by a pure loss channel Lp, i.e., G ≡ Lp ◦ Ap...... 66
5.1 Entanglement witness. The line tr(W ρˆ) = 0 signifies a line (hyperplane) corresponding to the witness W . All states located to the left of the hyperplane (or belonging to it), provide
a non-negative value for the witness, i.e., tr(W ρˆs) ⩾ 0, and states located to the right are
entangled states detected by this witness, i.e., tr(W ρˆe) < 0. Note that the extremal points of the separable states, i.e., pure product states, are also extremal points of the set of all states (located on the border of the total set of states)...... 71 5.2 Relative entropy of entanglement is given by the distance (measured by the quantum relative
entropy) between a given entangled state ρˆe (orange dot) and the closest separable one ρˆs (yellow dot)...... 80
6.1 Decomposition of a two-mode Gaussian state. In figure (a) and (b) we present two symplectic
transformations Σ→ and Σ←, given in Eq. (6.24) and Eq. (6.33), respectively. Both of them are decomposed into a sequence (direct and reverse) of a two-mode squeezing transformation
S2 and two single-mode squeezing transformation S. Every state in the standard form can ′ prepared by applying Σ→ or Σ← onto a classical state, i.e., ρc and ρc ...... 91 6.2 Lower bounds for entanglement of formation. Entanglement of formation is in general an
unbounded function, but it depends only on the optimum two-mode squeezing parameter ro.
Γ −2ro We plot with (black) dots above the optimum symplectic eigenvalue νo− = e for randomly −2r created states against the corresponding value based on r−, i.e., e − . The symplectic − sf sf eigenvalue is a bounded value ∈ (0,1], which shows that: (i) EF (ρ ) ⩽ EF (ρ ), and (ii) that the bound is tight for separable and infinitely entangled states. We also depict with (blue) squares ■ [199] and (red) triangles ▲ [200] the corresponding values we get from the previously known lower bounds. The closer the dots are to the diagonal the smaller the deviation from the real value of entanglement. It is clear that our bound is, on average, tighter than the previous bounds. All the quantities plotted are dimensionless...... 95 xxxiv List of figures
6.3 Lower and upper bound for randomly created states. In this figure we plot the percentile relative difference between both the upper δ+ (blue dots) and lower δ− (red crosses) bound and the actual value of the entanglement of formation, given in Eq. (6.52), against the purity of randomly created entangled states. It is apparent that the less the purity the larger the difference sf ± sf between EF (ρ ) and EF (ρ ). We also observe that on average the upper bound is closer to the real value than the lower bound. All quantities plotted are dimensionless...... 97 6.4 Comparison between entanglement of formation (solid blue line) and logarithmic negativity
(dashed red line). Assuming a pure state ρp(r) with squeezing parameter r = 1 is sent through a pure lossy channel with transmissivity 0 ⩽ τ ⩽ 1, we compare the two measures. The deterministic upper bounds (upper lines), i.e., the amount of entanglement assuming an infinitely squeezed state is sent through the same channel, are also depicted, since theyprovide further insight regarding the qualitative differences between entanglement of formation and logarithmic negativity. The deterministic bound for logarithmic negativity can be found in ref. [213]. Specifically, for logarithmic negativity, the deterministic bound of a state with
transmissivity value τa, can also be reached by sending the squeezed state (r = 1) through a
channel of transmissivity τb, with τb > τa. However, in contrast, entanglement of formation predicts that we cannot reach the deterministic bound with a squeezed state (r = 1) regardless of how much we raise the transmissivity. This is a critical difference, since the two quantifiers disagree on whether a physical upper bound has been reached or not. All the quantities plotted are dimensionless...... 98
7.1 Teleportation and channel simulation. The teleportation protocol [225] is represented in figure (a) via its basic components: i) the dual homodyne detection, HD, between the resource state, ρ,
and the initial state σin, ii) the classical channel, CC, iii) the displacement, D, and iv) the output
state, σout. In figure (b) we have the corresponding channel that the teleportation protocol simulates...... 104 7.2 Channel simulation through entanglement distillation. Figure (a) represents the channel, G, that we want to error-correct. Figure (b) shows the resource state ρ, which is sent through the same channel, G, and then through an entanglement distillation process before it is used in a teleportation protocol (see also Fig. 7.1). In figure (c) we have the effective transformation
for a successful distillation process, i.e., ρe, and finally in figure (d) the simulated channel Gc, ′ which leads to a state σout...... 107 7.3 Problems with fidelity. Let us assume that a pure state with squeezing parameter equalto
ζ = 0.8 is going through both a thermal amplifier channel A(τ2,ε2 = 2.5) and a thermal lossy
channel L(τ1,ε1 = 1.01). For different values of τ1 and τ2 we calculate the fidelity of the
input/output state and we get two sets of states: (i) the set with F1 < F2, colored with blue (on the left of the solid line, sections I and IV) and (ii) the ones with no entanglement left, colored with brown (on the top of the dashed line, sections I and II). As we can see there is an overlap between those two sets, i.e., section I, where we have both an entanglement-breaking situation
and F1 < F2. All quantities plotted are dimensionless...... 108 List of figures xxxv
7.4 Error correction with ideal NLA. An initial thermal lossy channel with τ = 0.5 and ε = 1.05
induces noise into one mode of a pure two-mode squeezed state σin with squeezing parameter ζ = 0.5. We apply the protocol using a resource state ρ with squeezing parameter χ = 0.5. In figure (a) we present both the (optimized over teleportation gain) entanglement of theoutput ′ state, maxλ{EF (σout)}, (red solid line), and the entanglement of the distilled resource state,
EF (ρe), (red dashed line), against the NLA gain, g. With solid blue and dashed blue lines we
have the entanglement of the output state without the protocol, EF (σout), and the deterministic Choi upper bound of entanglement for this channel, EF (L ), respectively. We observe that for
g = 1/χ the entanglement of the Choi-state is reached and from then on and until we reach gmax ′ we are into the error correction area (light blue shaded), i.e., maxλ{EF (σout)} ⩾ EF (σout). In figure (b) the contour lines indicate equally decohering channels (with parameters τ and v), i.e., channels that decohere the entanglement by the same amount. The yellow triangle represents the initial channel. Applying the protocol without distilling the resource state, i.e., g = 1, we get the channel shown with the red dot. Increasing the NLA gain, and specifically for g = 1/χ, we simulate a channel (yellow dot in the graph), that decoheres the state by the same amount as the initial channel (both channels lie on the same dashed contour line). The best channel
we can simulate is achieved for gmax, and is represented by the blue dot. With red/yellow/blue diamonds we indicate the corresponding simulated channels of an error correcting protocol based on a less entangled resource state, i.e., χ′ = 0.45. Thus, we can visually interpret error correction as the process of simulating a channel “closer” to the identity (represented by the green dot) than the initial one. All quantities plotted are dimensionless...... 112 7.5 Error correction range. We plot the range of all the possible channels with parameters 0 ⩽ τ ⩽ 1 and 1 ⩽ v ⩽ 2 that can be error corrected with the protocol, based on both the NLA condition, gmax > 1/χ, and the entanglement-breaking condition, v ⩾ 1 + τ. It is apparent that for increasing values of squeezing parameter χ, the set of channels is increased as well. All quantities plotted are dimensionless...... 113 7.6 Error correction with a single quantum scissor. The thermal lossy channel that we want to error correct has transmissivity τ = 0.01 and noise ε = 1.0002. Both the resource and the initial state ′ have squeezing parameter equal to χ = ζ = 0.5. The red solid line depicts maxλ{EF (σout)} QS ′ for the ideal NLA, and the brown dashed one depicts the corresponding maxλ{EF (σout)} for the realistic NLA with one quantum scissor. As we see the NLA gain needed to cross the value
EF (σout) is greater for the realistic NLA compared with the ideal one. All quantities plotted are dimensionless...... 114 xxxvi List of figures
8.1 One-time pad scheme. In this figure we graphically represent the one-time pad scheme that consists of two parties, Alice and Bob that want to secretly transmit a message though a public channel. The public channel is assumed to be under an eavesdropper control, called Eve. In this example, we assume that the message that Alice has to transmit is the bit sequence “101101010”, on which she applies (by modular addition) a pre-established (random) secret key, i.e., “001101011”. She sends the encrypted-text, i.e., “100000001”, through the public channel to Bob, who applies the same secret key (by modular addition) to the encrypted-text in order to retrieve the initial message...... 117 8.2 Entangling cloner attack. In this figure we present the entangling cloner attack, where Evehas
full control of the environment and simulates the channel G through a beam-splitter Bτ (with the same transmissivity as the channel). One input of the beam-splitter is Alice’s signal and ′ the other is one arm of Eve’s state φ . With M1−2 we represent the measurements that Eve performs later on her quantum memory on her state µ′, and 1 the identity channel...... 120 8.3 All-optical teleportation protocol. In this protocol, the basic components are: (i) a two-mode
squeezer Sg, (ii) a beam-splitter Bt, (iii) the decoherence that the signal has to go through modeled with a quantum channel G, and (iv) the entangled resource state ρ...... 121 8.4 All-optical teleportation attack. In this attack, Eve has no access to the quantum channel G and she performs an all-optical teleportation over the signal sent from Alice. With ρ we denote
Eve’s distilled resource state. The all-optical teleportation consists of a two-mode squeezer Sg
with gain g, which is in Eve’s first station close to Alice’s laboratory, and a beam-splitter Bt with transmissivity t = 1/g, which is in Eve’s second station close to Bob’s laboratory. One
mode of Eve’s resource state ρ is sent to the first station as an input of Sg. The other mode of
ρ is sent to the second station and is mixed on a beam-splitter Bη with another state φ, before
it becomes an input to Bt. Finally Eve performs the measurements M1−3 on her state µ that was stored in the quantum memory. For a pure channel G, both the entangling cloner and the all-optical teleportation attack need to substitute the state φ with a single-mode vacuum state |0⟩.124 8.5 Eve’s information and key rate. In figure (a) with the solid blue line we plot the amount of information S(b:E) Eve can extract in the reverse reconciliation scenario against the entanglement of her pure resource state E(ρ), parametrized over the squeezing parameter γ
[see Eq. (8.19)]. The minimum value of entanglement E(γmin) corresponds to about 4.2dB of squeezing. The horizontal red dashed line represents the Holevo bound χ(b:E), that is the maximum amount of information an eavesdropper can physically extract. In the limit of infinite entanglement, i.e., E(γ → 1) → ∞, we see that this bound is reached. With the green dot-dashed line we indicate the maximum amount of information Eve can extract through an optimal individual attack [251, 252]. In figure (b) we have the corresponding key rate that Alice and Bob measure given Eve’s collective attack. Again, the red dashed line is the minimum possible key rate that they can extract, and the green dot-dashed represents the key rate based on the optimal individual attack. In both plots we indicate the non-physical areas with grey color. All quantities plotted are dimensionless...... 125 List of figures xxxvii
8.6 Hybrid teleportation protocol. In this protocol, the basic components are: (i) the beamsplitters that split and later recombine the signal, and (ii) the conventional DV teleportations, with their corresponding DV resource states, Bell measurements BM, and final corrections through unitary operations U...... 127
9.1 Adaptive QKD protocol. In the first step, Alice and Bob prepare the initial separable state ρˆab
of their local registers a and b by applying an adaptive LOCC Λ0. After the preparation of these registers, there is the first transmission through the quantum channel G. Alice picks a
quantum system from her local register a1 ∈ a, which is therefore depleted as a → aa1; then,
system a1 is sent through the channel G, with Bob getting the output b1. After transmission,
Bob includes the output system b1 in his local register, which is augmented as b1b → b. This
is followed by Alice and Bob applying another adaptive LOCC Λ1 to their registers a and b.
In the second transmission, Alice picks and sends another system a2 ∈ a through the quantum
channel G with output b2 received by Bob. The remote parties apply another adaptive LOCC n Λ2 to their registers and so on. This procedure is repeated n times, with the output state ρˆab being finally generated for Alice’s and Bob’s local registers...... 131 9.2 Finite-resource simulation of bosonic Gaussian channels. In panel (a), we depict a phase-
insensitive Gaussian channel G transforming the input state σˆin into the output state σˆout. In panel (b), we show its simulation by means of a teleportation LOCC Λ. Its basic components
are: (i) a dual Homodyne Detection (HD) between the input state σˆin and the resource state
ρˆ =ρ ˆτ,v as in Eqs. (7.5a)-(7.5c); (ii) the classical communication (CC) of the Homodyne Detection outcomes; and (iii) a conditional phase-space displacement D with suitable gain
[225] which provides the output teleported state σˆout...... 132 9.3 Upper bounds to the secret-key rate capacity of lossy and amplifier channels (secret bits per channel use versus transmissivity 0 ⩽ τ ⩽ 1 or gain τ ⩾ 1). In panels (a) and (c) we show the results for pure lossy and pure amplifier channels, while panels (b) and (d) show the corresponding results for thermal lossy and thermal amplifier channels with n¯ = 1. In the
panels the lower blue line indicates the infinite-energy bound B2¯n+1,∞ of Ref. [114] while the green dashed line is the approximate finite-energy bound B˜ of Ref. [285], which is computed over the class of states of Ref. [220]. The black dashed line corresponds to our finite-energy
bound B1,1 computed with a pure resource state (ν± = 1). Note that, for pure lossy channels,
this bound B1,1 coincides with the previous finite-energy bound given in285 [ ]. Then, the red
dashed line is our finite-energy bound Bν−,ν+ , computed with ν− = 1, ν+ = 100 for pure lossy
and pure amplifier channels, and with ν− = 2¯n + 1, ν+ = 500 for thermal lossy and thermal
amplifier channels. As we see for increasing values of ν+, and thus increasing simulation
energy, we can approximate B2¯n+1,∞ as closely as we want, while keeping the energy of the resource state finite (although large). All quantities plotted are dimensionless...... 135 xxxviii List of figures
9.4 Upper bounds to the secret-key capacity of the additive-noise Gaussian channel (secret bits per channel use versus added noise v). The lower blue line indicates the infinite-energy bound
B∞,∞ of Ref. [114]. Then, we show our improved finite-energy bound Bν,ν which is plotted for pure resource state, i.e., ν = 1 (black dashed line) and for ν = 10 (red dashed line). Note
that the previous bound given in [285] coincides with our finite-bound B1,1. For increasing
values of ν we can approximate B∞,∞ as closely as we want, while keeping the energy of the resource state finite (despite being large). All quantities plotted are dimensionless...... 136 9.5 Secret-key bits versus number n of uses of a thermal lossy channel L with transmissivity τ = 0.01 corresponding to 100km of standard optical fiber and thermal numbern¯ = 0.0011 corresponding to δ ≃ 0.1 excess noise. We assume a security parameter ϵ = 10−2. We plot
the optimized finite-size bound Φn,ϵ(L) computed from Eq. (9.26b) (solid red line) which
approaches the asymptotic value B2¯n+1,∞(L) of Eq. (9.16) for large n (red dashed line). The optimal resource state ρˆ may have low energy at finite n. For instance, at n = 2 × 103, this
state has spectrum ν− ≃ 0.5005 and ν+ ≃ 1.66832. For comparison, we also plot the bound
(solid blue line) that we would obtain with a resource state of high energy, namely ν− ≃ 0.5011 7 and ν+ ≃ 1.99561 × 10 ...... 137
A.1 Discretization of a real continuous function. A continuous function ψ(x) corresponds to an Pn infinitely dimensional vector |ψ⟩ = limn→∞ i=0 ψi|i⟩, if we discretize the interval L into equal n parts of length ∆x = L/n and then associate the value of the function at this point
with an element ψi...... 164
B.1 Noiseless linear amplification. Each quantum scissor operation (QS) consists of twobeam splitters. The input signal is mixed on a balanced beam splitter with an ancilla signal and
both outputs are measured using photon detectors D1 and D2. The ancilla signal is one of the two outputs of a single photon passing through a tunable beam splitter with ratio ξ, while the other signal is the overall output of the quantum scissor. Successful quantum scissor operation
is heralded when a single photon is detected at D1 and none at D2 or vice versa. Using N quantum scissors and two N splitters (one to divide and another to recombine the signal), we can approximate the ideal NLA in the limit of N → ∞...... 168 List of symbols and abbreviations
V Vector Space T (·,·) Trace Distance
H Hilbert Space F(·,·) Fidelity
F (H ) Fock Space ⃗rψ Bloch Vector
N Nuclear Space A Annihilation Operator A† Creation Operator N Normal Operator (Ch. 2) Q Quadrature Operator N Number Operator (Ch. 3) Q(φ) Generalized Quadrature Operator U Unitary Operator X Position Operator H Hermitian Operator P Momentum Operator P Positive Operator W(·) Wigner Function Π Projection Operator Q(·) Husimi Function 1 Identity Operator P(·) P function P Probability Distribution R Vectorial Bosonic Operator ρˆ Density Matrix A⃗ Vectorial Annihilation Operator H(·) Shannon Entropy A⃗† Vectorial Creation Operator H(·|·) Classical Conditional Entropy Ω Commutation Matrix or Symplectic Form H(·:·) Classical Mutual Entropy H(2) Second-order Hamiltonian
S(·) von Neumann Entropy Σ Symplectic Transformation
S(·,·) Quantum Joint Entropy |0⟩ Vacuum State
S(·|·) Quantum Mutual Entropy |a⟩ Coherent State |n⟩ Fock or Number State S(·:·) Quantum Mutual Entropy |r⟩ Squeezed State S(·∥·) Relative Entropy D(α) Displacement Operator Sr(·) Rényi Entropy S(r) Squeezing Operator St(·) Tsalis Entropy S2(r) Two-mode Squeezing Operator M(·) Quantum Operation SN (r) N-mode Squeezing Operator C(·) Classical Capacity R(φ) Phase Rotation Operator Q(·) Quantum Capacity B(θ) Beam Splitter Operator V (·) Covariance Matrix (Ch. 3-5) νi Symplectic Eigenvalues σσ,ρρ Covariance Matrix (Ch. 6-9) G Gibbs matrix Var(·) Variance E Entanglement measure and Ent. of Entanglement
σ Standard Deviation EF Entanglement of formation xl List of figures
± EC Entanglement cost |Φ/Ψ ⟩ Bell States
ED Distillable entanglement W Entanglement Witness
ER Relative entropy of entanglement Λ LOCC operation
ES Squashed entanglement K Key Rate
EN Negativity G Gaussian channel
EL Logarithmic Negativity L Loss channel
EB Rains Bound A Amplifier channel
ID Direct Coherent Information Ae Contravariant channel
IR Reverse Coherent Information N Additive noise channel
General note on the nomenclature. Scalar values are denoted by regular small letters, while matrices/operators by bold small letters or capital letters. Vectors are denoted by either letters accompanied with the vector sign, e.g., ⃗r, or through the Dirac notation, e.g., |ψ⟩. Caligpaphical capital letters are used for either quantum operations such as channels, e.g., L, or for significant figures of merit such as fidelity, e.g., F. Finally, curly capital letters are used to represent mathematical spaces, e.g. Hilbert space H . Part I
Introduction
Chapter 1
Introductory Remarks
“Some authors state that the last stage in this chain of measurements involves ‘consciousness’, or the ‘intellectual inner life’ of the observer, by virtue of the ‘principle of psycho-physical parallelism’. Other authors introduce a wave function for the entire universe. In this book, I shall refrain from using concepts that I do not understand."
— Asher Peres
Quantum Computation and Quantum Information (QCQI) is multidisciplinary scientific field that studies the connection between Quantum Physics and Information Theory. As a subject QCQI can be approached via various perspectives, e.g., physical, mathematical, computational, experimental, commercial. In this work, we mainly focus on the mathematical and physical implications of the theory, aiming to provide new mathematical tools for the analysis of the quantum systems, and propose new protocols that enhance our current knowledge on manipulating quantum information. Before we introduce the purpose and the contents of this thesis, let us briefly give a review of some historical milestones that shaped the field as we know it today.
1.1 Historical Note
Historically, the first work in the field of QCQI is due to A. Holevo, who in 1973 discussedthe limitations of extracting classical information from a quantum mechanical systems [5] (a concept that is now called the “Holevo bound”). In 1975, R. P. Poplavskii published a work [6] showing that classical computers are unable to simulate quantum systems, and a year later, R. S. Ingarden attempted to extend Shannon’s classical information theory to quantum systems [7]. 4 Introductory Remarks
A work that attracted a lot of attention in the early years of QCQI was due to P. Benioff [8], who in 1980 described a quantum mechanical model of Turing Machines1, followed by R. Feynman in 1982 [9] who also discussed how quantum mechanical systems can be utilized from a computational point of view. More formally, a model of a universal quantum computer, was proposed by D. Deutsch in 1985 [10], who later, in 1992, alongside with R. Jozsa proposed a quantum algorithm [11] that can efficiently solve a computational problem on a quantum computer (called the “Deutsch-Jozsa algorithm”). Apart from the quantum computational proposals, C. Bennett and G. Brassard in 1984, showed that quantum properties can be also harnessed for cryptographic purposes [12]. In 1993, another seminal paper appeared by C. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres and W. Wootters [13], proposing a teleportation protocol able to disassemble in one place and later reconstruct in another place a quantum state, using classical information in conjunction with a quantum property called “entanglement”. The work that probably had the highest impact in the scientific community, and practically established the field was due to P. Shor, who in 1994 came up with a quantum algorithm [14] (known as the “Shor’s algorithm”) that is able to factorize a large integer exponentially faster than any known classical algorithm. The reason this result sparked a tremendous interest (especially in the computer science field) is the fact that current cryptographic schemes, e.g. RSA (Rivest-Shamir- Adleman), are based on the computational hardness of this mathematical problem, i.e., factorizing large integers, and thus if a universal quantum computer exists it can potentially break many of the current cryptographic systems. Currently (2019), QCQI is a matured field, with results that gradually making their way out ofthe laboratories and even meeting commercial needs, e.g., in cryptographic schemes, quantum random number generators etc. A universal quantum computer has yet to be constructed, but the progress that has been done in the recent years gives promise that a fully functional quantum computer will be soon a reality.
1.2 Goal of the Thesis
Quantum Mechanics offers a wide platform of physical systems that can be used as the building blocks for QCQI. In this thesis, we mainly focus on quantum systems that lie within a special class, called “Gaussian systems”. Gaussian quantum states can be relatively easily constructed and manipulated experimentally, e.g., a laser is an example of a quantum Gaussian system, and at the same time allow a convenient mathematical description. The aim of this thesis is to:
• Mathematically investigate and analyze certain properties of Gaussian systems, i.e., (i) entan- glement in Gaussian states, and (ii) transmission limitations of quantum information through Gaussian channels. 1A Turing machine is an abstract mathematical model of computation that defines a hypothetical machine able to manipulate symbols according to a list of rules. 1.3 Structure of the Thesis 5
• Theoretically propose Gaussian quantum protocols for communication purposes, i.e., (i) error correction of quantum states due to signal degradation, and (ii) new cryptographic schemes and models of eavesdropping attacks.
1.3 Structure of the Thesis
The thesis is divided into four parts:
Part I - Introduction. This part consists of a single chapter, i.e.,
Chapter 1: Introductory Remarks, where a brief historical review of QCQI is presented, accompanied with the goals and the structure of this thesis.
Part II - Literature Review. This part is intended to provide a self-consistent presentation of the mathematical and physical background needed for the comprehension of the results presented in the part III. In particular, we have:
Chapter 2: Mathematical Preliminaries introduces the notions of Hilbert Space and Fock Space, which are the mathematical spaces that the quantum systems are typically associated with, e.g., a quantum state is an element of a Hilbert space. We also define mappings within those spaces called Linear Operators, which are the mathematical objects that describe the evolution and measurement of quantum states. Basic knowledge of linear algebra and analysis is assumed throughout the chapter. Chapter 3: Quantum Mechanics and Quantum Information presents the basics of the Quantum Mechanics theory in the context of Quantum Information. We formally present the two postulates that Quantum Mechanics is based on, related to the notion of the quantum state and the quantum operations applied to it, respectively. We discuss the distinction between pure and mixed quantum states, and we also introduce the concept of quantum entropy. We present the different ways a quantum state can be evolved or measured, and we also discuss how we operationally distinguish two quantum states. Finally, we show how to represent a quantum state depending on the dimensions of the system, and give physical examples for each case. We particularly focus on infinite dimensional states, that describe bosonic modes that lay the basis for the rest of the thesis. The representation of quantum states in a phase space is also discussed. Chapter 4: Gaussian Systems introduces the special class of quantum systems that we focus on in this thesis, i.e., continuous-variable states with Gaussian statistics. Different types of Gaussian states and Gaussian operations are discussed, e.g., coherent states, squeezed states, thermal states etc., along with the mathematical tools, i.e., symplectic analysis, that offer a nice description of those systems. We also mention how Gaussian states can be measured and distinguished from one another. Finally, we discuss the formalism that describes the evolution of Gaussian states, i.e., Gaussian channels. 6 Introductory Remarks
Chapter 5: Entanglement Theory is devoted to a quantum property that this thesis mainly focuses on, i.e., entanglement. We formally define when a state is called entangled or separable, and provide the corresponding mathematical criteria that distinguish between the two cases (both in general and in Gaussian systems). We also discuss the notions of local operations and classical communication (LOCC), and briefly mention how an entangled state can be transformed into another under LOCC (majorisation theory). We proceed by postulating several criteria that an entanglement measure should satisfy, and finally present various entanglement measures that have been defined in the literature, aiming to quantitatively describe entanglement.
Part III - Results. In this part we present the results of this thesis. More specifically,
Chapter 6: Entanglement of Formation in Gaussian Systems. In this chapter, we include the results regarding the quantification of entanglement in two-mode Gaussian systems through an entanglement measure called “entanglement of formation”. Analytical lower and upper bounds are derived for entanglement of formation, accompanied by a numerical method for the estimation of the measure. We also provide a comparison with another entanglement quantifier, i.e., logarithmic negativity, that is the most common wayto measure entanglement in the literature. This chapter includes results from Refs. [15, 1, 2]. Chapter 7: Continuous-Variable Teleportation and Error Correction. In this chapter, we propose a quantum communication protocol that aims to reduce the Gaussian noise induced to a Gaussian state through its transmission via optical fibers or free space. We also provide an introduction to the standard quantum teleportation protocol for continuous- variable states, that the error correction protocol is based on. This chapter includes results from Ref. [1]. Chapter 8: All-optical Teleportation and Quantum Key Distribution. In this chapter, we focus on Gaussian quantum key distribution, and we affirmatively answer the question of whether Eve can optimally attack a system under collective measurements without having access to the shared quantum channel (also known as the purification assumption). More specifically, we propose a teleportation-based eavesdropping attack that serves as an alternative scheme to the well-known entangling cloner. A brief introduction to the all-optical teleportation attack is also presented. This chapter includes results from Ref. [3]. Chapter 9: Secret Key Capacity of Gaussian Channels. In this chapter, we investi- gate whether the upper bounds for the secret-key quantum capacity of one-mode phase- insensitive Gaussian channels can be achieved under finite-energy resource states. We conclude that the upper bound can be closely approximated for increasing energy and decreasing purity of the states. This chapter includes results from Ref. [4].
Part IV - Conclusion. This part consists of the final chapter of the thesis. 1.3 Structure of the Thesis 7
Chapter 10: Concluding Remarks is the chapter of the thesis, where we provide an overall conclusion to the research done during this PhD candidature. New directions and open problems that worth looking into in the future are also discussed.
Part II
Literature Review
Chapter 2
Mathematical Preliminaries
“In mathematics we get to make the rules, pick the axioms, we can even decide what kinds of reasoning are allowed or not, and after doing so we get to play around and see what happens. Mathematics is the plane of games. Science is finding out what game you are playing, and we still don’t know whatgamewe have found ourselves in.”
— Michael Stevens
It was obvious since the very early years of the development of Quantum Physics, that the peculiar nature of Quantum Physics and the weird phenomena that emerge in the atomic/subatomic world needed a radically different mathematical structure compared to the well-established one from Classical Physics. W. Heisenberg was the first who attempted to mathematically formulate the new theory[16] through Matrix Mechanics, followed by E. Schrödinger [17], who approached this task through Wave Mechanics. It was P. Dirac some years later, who in his seminal monograph [18] proved the mathematical equivalence of those two seemingly different structures. A more mature treatment of Quantum Physics, was due to J. von Neumann [19] who mathematically formulated Quantum Mechanics under Hilbert Spaces, that later supplemented by I. Gelfand and N. Y. Vilenkin [20] introducing the rigged Hilbert Spaces, resolving some technical issues which were still unclear at the time. In this chapter, we introduce the mathematical formalism that is necessary for the development and understanding of the rest of this thesis. In particular, we define the Hilbert space, which isthe mathematical structure that the quantum systems (that will be introduced in chapter3) “live in”. As we will see in the next chapter, the elements of this space are associated with the “states” of the quantum system and the operations within it correspond to their evolution. Basic knowledge on linear algebra and analysis is assumed throughout this chapter. 12 Mathematical Preliminaries
Note on references. The mathematical foundations of Quantum Mechanics and Quantum Informa- tion can be found in several textbooks, introductory [21–23] and more advanced [24–29].
2.1 Vector Spaces
A linear vector space V is a set of elements, called vectors, which is closed under addition and multiplication of scalars. Let V be a vector space over a scalar field C (that could also be a real P field R), a list of vectors {⃗ei} in V is said to be linearly independent if the relation i ψi⃗ei = 0 has ψ1 = ··· = ψn = 0 as the only solution (otherwise they are called linearly dependent). The least upper bound of linearly independent vectors in V is called the dimension of the space, denoted as dimV . We say that a set of vectors {⃗ei} spans the space V if every vector ψ⃗ ∈ V in this space can be written as a linear combination X ψ⃗ = ψi⃗ei , (2.1) i where ψi ∈ C. If {⃗ei} is, in addition, a set of linearly independent vectors it is called the basis of V , and the vector ψ⃗ can be represented as the (finite-dimensional for our purposes) column
ψ1 ψ2 ψ⃗ = . (2.2) . . ψn
In this chapter, we focus on finite-dimensional vector spaces, e.g., the set {ψi} in the vector above has finite amount of elements, but a discussion about infinite-dimensional vector spaces is provided in the AppendixA. The linear vector space V is called an inner product space if it is equipped with an inner product ⟨·,·⟩, defined as the map ⟨·,·⟩ : V × V → C, satisfying the following properties for any ψ,⃗ ⃗φ,⃗χ ∈ V and λ ∈ C:
⃗ ⃗ ⃗ ⃗ ⃗ • ⟨ψ,ψ⟩ ⩾ 0, and ⟨ψ,ψ⟩ = 0 ⇐⇒ ψ = 0 ,
• ⟨ψ,⃗ ⃗φ⟩ = ⟨⃗φ,ψ⃗⟩∗ ,
• ⟨ψ,λ⃗φ⃗ + ⃗χ⟩ = λ⟨ψ,⃗ ⃗φ⟩ + ⟨ψ,⃗ ⃗χ⟩ .
A norm is a function that assigns a strictly positive “length” or “size” to each vector ψ⃗ ∈ V . Let us ⃗ define the p-norm of a vector ψ, with p ⩾ 1, as
!1/p X p ∥ψ⃗∥p ≡ |ψi| , (2.3) i which can be interpreted as the generalized mean value, and satisfies the following properties:
⃗ ⃗ ⃗ • ∥ψ∥p ⩾ 0, ∥ψ∥p = 0 ⇐⇒ ψ = 0 , 2.1 Vector Spaces 13
• ∥λψ⃗∥p = |λ| · ∥ψ⃗∥p , ⃗ ⃗ • ∥ψ + ⃗φ∥p ⩽ ∥ψ∥p + ∥⃗φ∥p .
For any ψ,⃗ ⃗φ ∈ V we have the so-called Hölder’s inequality [30], which relates the inner-product of two vectors with their p-norms as follows
1 1 ⟨ψ,⃗ ⃗φ⟩ ∥ψ⃗∥ · ∥⃗φ∥ , with + = 1, (2.4) ⩽ p q p q where the equality holds if and only if ψ⃗ and ⃗φ are linearly dependent. Note that for p = q = 2 it reduces to the well-known Cauchy–Schwarz inequality [31]. To any linear vector space V there exists an associated space, called the dual space V ′, of linear ′ functionals on V . For any vectors ψ⃗1,ψ⃗2 ∈ V , the functional F ∈ V , assigns the scalars F (ψ⃗1),F (ψ⃗2), such that
F (λψ⃗1 + µψ⃗2) = λF (ψ⃗1) + µF (ψ⃗2), (2.5)
′ with λ,µ ∈ C, and for the functionals F1,F2 ∈ V we have
(F1 + F2)(ψ⃗) = F1(ψ⃗) + F2(ψ⃗). (2.6)
The Riesz theorem states that there is a one-to-one correspondence between linear functionals F in V ′ and vectors ψ,⃗ ⃗φ in V , such that F (ψ⃗) = ⟨⃗φ,ψ⃗⟩. (2.7)
This correspondence means that the spaces V and V ′ are isomorphic.
2.1.1 Hilbert Space
When an inner product vector space V is equipped with a norm it is called a normed space. A sequence {ψ⃗n} in a normed space is called a Cauchy sequence if
∥ψ⃗n − ψ⃗m∥p → 0 as n,m → ∞. (2.8)
A normed space is called complete if all Cauchy sequences converge in V , i.e., there exists a vector ψ⃗ such that ∥ψ⃗n − ψ⃗∥p → 0, and a complete normed space is called a Banach space.
Definition (Hilbert space). Hilbert space H is a normed inner-product space that is complete with respect to the 2-norm (also called the Euclidean norm), i.e., for an element ψ⃗ ∈ H we have
q ∥ψ⃗∥2 = ⟨ψ,⃗ ψ⃗⟩. (2.9)
If ⟨ψ,⃗ ⃗φ⟩ = 0, then ψ⃗ and ⃗φ are called orthogonal, denoted as ψ⃗ ⊥ ⃗φ. A vector ψ⃗ is said to be normalized if ∥ψ⃗∥2 = 1, and a family of vectors {⃗ei} is called orthonormal if
⟨⃗ei,⃗ej⟩ = δij , (2.10) 14 Mathematical Preliminaries
where δij is the so-called Kronecker delta, defined as
1 for i = j δij ≡ . (2.11) 0 for i ̸= j
Given an orthonormal basis {⃗ei} ∈ H the following properties hold:
• Any vector ψ⃗ ∈ H can be written as
X ψ⃗ = ⟨⃗ei,ψ⃗⟩⃗ei , (2.12) i
where ⟨⃗ei,ψ⃗⟩ is also called the Fourier coefficient of ψ⃗.
• The inner product of two vectors ψ,⃗ ⃗φ ∈ H satisfy
X ⟨ψ,⃗ ⃗φ⟩ = ⟨ψ,⃗e⃗ i⟩⟨⃗ei, ⃗φ⟩, (2.13) i
so, the norm of a vector ψ⃗ ∈ H is given by the Parseval’s identity, i.e.,
s X 2 ∥ψ⃗∥2 = |⟨⃗ei,ψ⃗⟩| . (2.14) i
⃗ P P The inner product for complex-valued vectors ψ = i ψi⃗ei ∈ H and ⃗φ = i φi⃗ei ∈ H is defined as ⃗ X ∗ ⟨ψ, ⃗φ⟩ ≡ ψi φi , (2.15) i with ψi,φi ∈ C.
Composite Hilbert Spaces
Let us have two Hilbert spaces, i.e., H A with elements {ψ⃗}, and H B with elements {⃗φ}. We can construct a linear composite space by defining two operations, i.e., the tensor sum and the tensor product. For the tensor sum operation, the set of elements of the space H A ⊕ H B is {ψ⃗ ⊕ ⃗φ}, with the following properties: • ψ⃗1 ⊕ ⃗φ1 + ψ⃗2 ⊕ ⃗φ2 = ψ⃗1 + ψ⃗2 ⊕ (⃗φ1 + ⃗φ2) , • λ ψ⃗ ⊕ ⃗φ = λψ⃗ ⊕ λ⃗φ , with λ ∈ C. The dimension of the new space is
dim(H A ⊕ H B) = dimH A + dimH B . (2.16)
For the tensor product operation, the set of elements of the space H A ⊗ H B is {ψ⃗ ⊗ ⃗φ}, where:
• (ψ⃗1 + ψ⃗2) ⊗ ⃗φ = ψ⃗1 ⊗ ⃗φ + ψ⃗2 ⊗ ⃗φ , 2.2 Linear Operators 15
• λ(ψ⃗ ⊗ ⃗φ) = (λψ⃗) ⊗ ⃗φ = ψ⃗ ⊗ (λ⃗φ) , with λ ∈ C. The dimension of the new space is equal to
dim(H A ⊗ H B) = dimH A × dimH B . (2.17)
The inner product between two vectors ψ⃗1 ⊗ ⃗φ1 and ψ⃗2 ⊗ ⃗φ2, is defined as
⟨ψ⃗1 ⊗ ⃗φ1,ψ⃗2 ⊗ ⃗φ2⟩ ≡ ⟨ψ⃗1,ψ⃗2⟩⟨⃗φ1, ⃗φ2⟩. (2.18)
For a subspace K ⊂ H , every element ψ⃗ ∈ H can be uniquely decomposed as
ψ⃗ = ⃗φ + ⃗χ, (2.19) for every ⃗φ ∈ K and ⃗χ ∈ K ⊥ if and only if ⟨ψ,⃗ ⃗φ⟩ = 0. This is also called the Projection theorem. The set K ⊥ is then called the orthogonal complement of K and H can be expressed as the direct sum of H = K ⊕ K ⊥.
2.1.2 Fock Space
Definition (Fock space). Fock space is a tensor algebra over a Hilbert space H . We denote (N) (N) NN (0) by H the N-fold tensor product H = i H , we set H = C, and define the (total) Fock space as ∞ h (1) (2) i M (i) F [H ] ≡ Sx C ⊕ H ⊕ H ⊕ ··· = SxH , (2.20) i=0
where Sx is an operator which symmetrizes or antisymmetrizes a tensor. h i A vector Ψ⃗ ∈ F [H ] is written as Ψ⃗ = ψ⃗(0),ψ⃗(1),...,ψ⃗(N) with ψ⃗(i) ∈ H (i), and depending on whether the Hilbert space describes particles obeying bosonic (x = +) or fermionic (x = −) statistics we have two important subspaces of the Fock space, which are called the Bosonic (or symmetric) Fock space and the Fermionic (or antisymmetric) Fock space, respectively. In this work we will focus on the first one.
2.2 Linear Operators
Definition (Linear operator). Let D be a subspace of H . A map A on D is called a linear operator if it preserves linear combinations, i.e.
A(λψ⃗ + µ⃗φ) = λAψ⃗ + µA⃗φ, (2.21)
with ψ,⃗ ⃗φ ∈ D and λ,µ ∈ C. 16 Mathematical Preliminaries
Domain of A is the space D, denoted as domA, and range of A is the set {Aψ⃗ | ∀ ψ⃗ ∈ D}, denoted as ranA. An operator is called bounded if domA ≡ H and there exists a constant m > 0 satisfying
⃗ ⃗ ∥Aψ∥2 ⩽ m∥ψ∥2 , (2.22) for any ψ⃗ ∈ H , otherwise the operator is called unbounded. From now on we will adopt the so-called Dirac notation. In this notation, an element ψ⃗ ∈ H is represented by |ψ⟩ and is called “ket”. A linear functional F in the dual space H ′ corresponds to a vector ⟨φ|, called “bra”, and the numerical value of the functional F (|ψ⟩) is given by
F (|ψ⟩) ≡ ⟨φ|ψ⟩ ≡ ⟨⃗φ,ψ⃗⟩, (2.23) which is called “bra-ket”, and it is essentially a reformulation of the inner product. It is worth noting that due to the isomorphism between a space and its dual space there is an anti-linear correspondence between bras and kets, i.e., λ∗⟨φ| ←→ λ|φ⟩. (2.24)
A vector under this notation can be written as
|ψ⟩ = X ⟨i|ψ⟩|i⟩, (2.25) i
P where {|i⟩} is an orthonormal basis, and the inner product of two vectors |ψ⟩ = i ψi|i⟩ and |φ⟩ = P i φi|i⟩ takes the form: φ1 φ2 ⟨ψ,φ⟩ ≡ ⟨ψ|φ⟩ ≡ ψ∗,ψ∗,··· ,ψ∗ . (2.26) 1 2 n . . φn We can also define the outer product of two vectors |ψ⟩ and ⟨φ| as
∗ ∗ ψ1 ψ1φ1 ··· ψ1φn ∗ ∗ ψ2 ψ2φ1 ··· ψ2φn |ψ⟩⟨φ| ≡ φ∗,φ∗,··· ,φ∗ = . (2.27) . 1 2 n . .. . . . . . ∗ ∗ ψn ψnφ1 ··· ψnφn
Based on the above definitions any operator A ∈ H can be written as
X X X A = |i⟩⟨i|A|j⟩⟨j| = ⟨i|A|j⟩|i⟩⟨j| = aij|i⟩⟨j|, (2.28) i,j i,j i,j where {|i⟩} and {|j⟩} are orthonormal bases, and aij are the elements of the corresponding matrix of the linear transformation A. Using a third orthonormal basis, i.e., {|k⟩}, the composition AB is given 2.2 Linear Operators 17 by X X X AB = ⟨i|AB|j⟩|i⟩⟨j| = ⟨i|A|k⟩⟨k|B|j⟩|i⟩⟨j| = aikbkj|i⟩⟨j|. (2.29) i,j i,j,k i,j,k P The transpose and the conjugate transpose (also called adjoint) of an operator A ≡ i,j aij|i⟩⟨j| are defined as T X A ≡ aij|j⟩⟨i|, (2.30) i,j and † X ∗ A ≡ aij|j⟩⟨i|, (2.31) i,j respectively, where {|i⟩} and {|j⟩} are orthonormal bases. Transpose and conjugate transpose opera- tions satisfy the following properties (for λ ∈ C):
• (AT )T = A , • (A†)† = A ,
• (A + B)T = AT + BT , • (A + B)† = A† + B† ,
• (AB)T = BT AT , • (AB)† = B†A† ,
• (λA)T = λAT , • (λA)† = λ∗A† ,
• (AT )−1 = (A−1)T , • (A†)−1 = (A−1)† ,
• det(AT ) = det(A) , • det(A†) = det(A)∗ .
The trace of an operator is defined as
X X tr(A) ≡ ⟨i|A|i⟩ ≡ aii , (2.32) i i where {|i⟩} is an orthonormal basis. The trace has the following properties:
• tr(AB) = tr(BA) ,
P P • tr( i Ai) = i tr(Ai) ,
• tr(|ψ⟩⟨ψ|A) = ⟨ψ|A|ψ⟩ , • tr A† = [tr(A)]∗ .
Analogously to vectors, we can define norms for operators/matrices as well. The Schatten p-norm of an operator, with p ⩾ 1, is defined as
n h † p/2io1/p ∥A∥p ≡ tr (A A) , (2.33) and satisfies the following properties:
• ∥A∥p ⩾ 0, ∥A∥p = 0 ⇐⇒ A = 0 , 18 Mathematical Preliminaries
• ∥λA∥p = |λ| · ∥A∥p ,
• ∥A + B∥p ⩽ ∥A∥p + ∥B∥p .
A space of matrices becomes a Hilbert space H , under the inner product of matrices, defined as
⟨A|B⟩ ≡ tr A†B , (2.34) where we have used the Schatten 2-norm, also called Hilbert-Schmidt norm or Frobenius norm, given by r † ∥A∥2 = tr A A , (2.35) and it is analogous to the Euclidean norm for vectors. However, another interesting norm is the Schatten 1-norm, also called trace norm, given by √ † ∥A∥1 = tr A A . (2.36)
If the application of an operator A on a certain vector |ψ⟩ gives as an output the scalar multiple λ of the same vector, i.e., A|ψ⟩ = λ|ψ⟩, (2.37) then we call the vector |ψ⟩ an eigenvector and the scalar λ an eigenvalue of the operator A.
2.2.1 Normal Operators
A significant type of linear operators are the so-called normal operators N, for which it holds
N †N = NN † . (2.38)
Spectral decomposition is a fundamental theorem, which states that any normal operator N ∈ H is diagonal with respect to some orthonormal basis {|i⟩} for H , i.e.,
X N = ci|i⟩⟨i|, (2.39) i with ci ∈ C. Conversely, any diagonalizable operator is normal. Using this diagonal representation we can define a function of an operator f(N), as
X f(N) ≡ f(ci)|i⟩⟨i|. (2.40) i
Special types of normal operators are the following:
• Unitary operator U, where U † = U −1 . (2.41) 2.2 Linear Operators 19
Based on the spectral decomposition we have
U = X eiφi |i⟩⟨i|, (2.42) i
with φi ∈ R. It is worth noting that unitary operators preserve the inner product, i.e., ⟨Uψ|Uφ⟩ = ⟨ψ|φ⟩.
• Hermitian operator H, where H† = H. (2.43)
Hermitian operators take the form
X H = λi|i⟩⟨i|, (2.44) i
with λi ∈ R, so it holds ⟨Hψ|φ⟩ = ⟨ψ|Hφ⟩. Eigenvectors corresponding to distinct eigenvalues of a Hermitian operator must be orthogonal.
• Positive operator P , where ⟨ψ|P |ψ⟩ ⩾ 0. (2.45)
Under the spectral decomposition a positive operator is written as
X P = pi|i⟩⟨i|, (2.46) i
+ P with pi ∈ R . It is worth noting that a positive operator with tr(P ) = i pi = 1 is also called a density operator.
• Projection operator Π (or projector), where
Π2 = Π = Π† , (2.47)
which can be represented as Π = X |i⟩⟨i|, (2.48) i with {|i⟩} being a basis for a subspace P ⊂ H . The action of an operator Π is to project out the component of a vector that lies within a certain subspace. Note that for rank-1 projection
operators it also holds ΠiΠj = δijΠi.
• Identity 1, where 1|ψ⟩ = |ψ⟩, (2.49)
and 1 = X |i⟩⟨i|. (2.50) i 20 Mathematical Preliminaries
Fig. 2.1 Convex and concave sets. On the left we have a convex set and on the right a concave one. As we see, for two arbitrary elements of the sets, A and B, we can always find a straight line which lies inside the set, that connects them when the shape is convex, while this is not true for a concave set.
It is often useful to decompose an operator into products of unitary and positive operators, also known as the polar decomposition. In particular, for any linear operator A ∈ H , there exist unitary U and positive operators P and K such that
A = UP = KU, (2.51) √ √ where positive operators P and K are unique, and are given as P ≡ A†A and K ≡ AA†. If A is invertible operator, then U is unique. Combining the spectral and the polar decomposition we get the singular value decomposition, i.e., for any square matrix A ∈ H , there exist unitary matrices U and V , and a diagonal matrix D with non-negative elements such that A = UDV. (2.52)
The diagonal elements of D are called singular values of A.
2.2.2 Convexity and Concavity
A Hilbert space is called convex if for any finite collection of its elements A ≡ {Ai} and any probability P distribution P ≡ {pi}, the linear combination i piAi is also an element of the same space. Extreme elements in a convex set are the ones which cannot be represented as a nontrivial combination of other points (trivial here being that only one pi is non-zero). Convexity has an interesting geometrical representation: given two elements in a convex set we can always find a line that connects them, and the line is inside the convex set, otherwise the set is concave (see Fig. 2.1). It is also worth noting that a real function f is called convex if
! X X f piAi ⩽ pif(Ai), (2.53) i i
concave and if ! X X f piAi ⩾ pif(Ai). (2.54) i i 2.2 Linear Operators 21
2.2.3 Composite operators
Let A ∈ H A and B ∈ H B be two operators. The tensor product of the operators A⊗B ∈ H A ⊗H B is defined as (A ⊗ B)(|ψ⟩ ⊗ |φ⟩) = A|ψ⟩ ⊗ B|φ⟩, (2.55) which satisfies the following properties:
• (A1 + A2) ⊗ B = A1 ⊗ B + A2 ⊗ B ,
• (λA) ⊗ B = A ⊗ (λB) ,
• (A ⊗ B)† = A† ⊗ B† ,
• tr(A ⊗ B) = tr(A)tr(B) , with λ ∈ C. An explicit matrix representation we can define is the so-called Kronecker product. Suppose A is an n × m matrix, and B is a p × q matrix, then their Kronecker product is
a11B a12B ··· a1nB a21B a22B ··· a1nB A ⊗ B ≡ , (2.56) . . .. . . . ··· am1B am2B ··· amnB where aij are elements of the matrix A. For two operators we can also define the Kronecker sum as
A ⊕ B = 1 ⊗ A + B ⊗ 1. (2.57)
A vector |ψ⟩ in the tensor product of spaces H A ⊗ H B is given by
X X |ψ⟩ = cij|i⟩ ⊗ |j⟩ = cij|ij⟩, (2.58) i,j i,j where {|i⟩} ∈ H A and {|j⟩} ∈ H B are orthonormal bases. An operator A ∈ H A ⊗ H B is given by
X jl X kl A = cik |i⟩⟨j| ⊗ |k⟩⟨l| = cij |ik⟩⟨jl|, (2.59) i,j,k,l i,j,k,l where {|i⟩},{|k⟩} are orthonormal bases in H A and {|j⟩},{|l⟩} are orthonormal bases in H B. The partial transpose of the above operator A over H A is defined as
ΓA X jl T X il A ≡ cik (|i⟩⟨j|) ⊗ |k⟩⟨l| = cjk |j⟩⟨i| ⊗ |k⟩⟨l|, (2.60) i,j,k,l i,j,k,l and over HB as ΓB X jl T X jk A ≡ cik |i⟩⟨j| ⊗ (|k⟩⟨l|) = cil |i⟩⟨j| ⊗ |l⟩⟨k|. (2.61) i,j,k,l i,j,k,l 22 Mathematical Preliminaries
The partial trace of an operator A ∈ H A ⊗ H B over H A is defined as
X trAA ≡ (⟨i| ⊗ 1)A(|i⟩ ⊗ 1) , (2.62) i and over H B as X trBA ≡ (1 ⊗ ⟨j|)A(1 ⊗ |j⟩) , (2.63) j where {|i⟩} and {|j⟩} are orthonormal bases on H A and H B, respectively.
2.2.4 Superoperators
Let us define a linear map M called superoperator, that takes as an input an operator and has an output another operator. Significant types of superoperators are the following:
• Positive, for which the output operator is a positive operator, i.e.,
M(A) ⩾ 0. (2.64)
• Completely positive, where the tensor product of the a linear map M with the identity is a positive operator, i.e., M(A) ⊗ 1 ⩾ 0. (2.65)
• Trace preserving, where the trace of the output operator is equal to the trace of the initial operator, i.e., tr[M(A)] = tr(A). (2.66) Chapter 3
Quantum Mechanics and Quantum Information
“There is no quantum world. There is only an abstract quantum physical descrip- tion. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.”
— Niels Bohr
Quantum Mechanics is the field of Science that describes the behavior of the atomic and subatomic particles that make up the universe. The counter-intuitive phenomena that emerge from those systems led the founders of the field to abandon the classical description that appeared to be completely inadequate. In Physics, the word “quantum” (plural: quanta) has been historically used to describe the notion of discrete packets of a physical entity, e.g. quanta of energy by M. Planck [32] in 1901 or quanta of light by A. Einstein [33] in 1905. The root of all the “weirdness” in this theory comes from the notion of quantum superposition, meaning that if two vectors represent possible states of a physical system, then so does their linear combination. The principle of superposition is not a new concept in Physics, e.g. in Classical Mechanics two waves can form a superposition representing another wave (different waveform, different energy etc.). However, the crucial difference in Quantum Mechanics is that the superposed state (loosely speaking) resembles with some probability each state that contributes to the superposition. Quantum superposition in practice is manifested through phenomena that can be attributed to the wave-like nature of the quantum state, e.g., the self-interference of electrons or molecules in the double slit experiment setup [34–36]. The concept of quantum superposition of a state is still considered one of the biggest mysteries in Quantum Mechanics, and a lot of different interpretations have been proposed as an attempt to understand its physical meaning, e.g., Statistical/Ensemble interpretation [37], Bayesian Interpretation (also goes by the name “QBism”) [38], Pilot-Wave Interpretation (also known as Bohmian Mechanics) 24 Quantum Mechanics and Quantum Information
[39], Many Worlds Interpretation [40], and a lot more (see Refs. [41, 42] for a comprehensive review of the subject). In this chapter, we introduce Quantum Mechanics through an axiomatic approach, deriving its results from two basic postulates: (i) definition of the quantum state and (ii) definition of its evolution (measurement included). We also present the analog to Shannon entropy for quantum states, i.e., von Neumann entropy, which constitutes a fundamental tool to study those systems. Finally, we discuss the difference in the description between states with finite and infinite dimensions.
Note on references. The development of Quantum Theory can be found in numerous textbooks, in a big range from introductory [21–23] to more advanced [24–27, 43] discussions. For a dedicated analysis on continuous-variable states see Refs. [44–47].
3.1 Quantum State
Postulate 1. A quantum state represents a statistical ensemble P ≡ {pi} of independent, identi- cally prepared copies of a quantum system, and it is described by a positive unit trace operator ρˆ, also called the density operator or density matrix, that belongs to the rigged Hilbert space N ⊂H ⊂N ×, where H is the Hilbert space, N the nuclear space, and N × the conjugate nuclear space.
Rigged Hilbert spaces (discussed in the AppendixA) are employed in the above postulate, since later in the thesis there will be states, e.g., position and momentum eigenstates, that due to normalization reasons cannot be properly defined in the usual Hilbert space.
3.1.1 Pure and Mixed States
Density operators form a convex set, so any convex combination of density operators is also a density operator, i.e., X ρˆ = piρˆi , (3.1) i describing a mixture of the corresponding statistical ensembles. An extreme point of the convex set of density operators is called a pure state and it is a projection operator, i.e.,
ρˆp = |ψ⟩⟨ψ|. (3.2)
A pure state can be fully represented by the vector
X |ψ⟩ = ci|i⟩, (3.3) i 3.1 Quantum State 25
Fig. 3.1 Pure and mixed states as a convex set [25]. With red dots we represent the pure states |ψ⟩,|φ⟩,|u⟩,|v⟩ and with the yellow dot the mixed state ρˆ. The mixed state ρˆ can be simultaneously represented as ρˆ = 1 1 √ √ √ √ x|ψ⟩⟨ψ| + y|φ⟩⟨φ| or ρˆ = 2 |u⟩⟨u| + 2 |v⟩⟨v|, where |u⟩ = x|ψ⟩ + y|φ⟩ and |v⟩ = x|ψ⟩ − y|φ⟩. with ci ∈ C and {|i⟩} denoting an orthonormal basis. Due to the spectral theorem [see Eq. (2.46)] we know that every state can be written as a convex combination of no more than d pure states, i.e.,
X ρˆm = pi|ψi⟩⟨ψi|, (3.4) i
P where d = dimH and P ≡ {pi} is a probability distribution (i.e., i pi = 1). States which are not pure are called, in short, mixed states. When the probability distribution P of a mixed state is uniform, i.e., 1 ρˆmm = d , the state is called maximally mixed. Note that Eq. (3.4) is the mathematical description of the notion of classical superposition, which involves the “classical” mixing of quantum states, which is physically different from the notion of quantum superposition that is encapsulated in Eq. (3.3). A criterion for determining whether a state is pure or mixed is the following:
tr(ˆρ2) < 1 ⇐⇒ mixed state, (3.5a) tr(ˆρ2) = 1 ⇐⇒ pure state, (3.5b) and the function tr(ˆρ2) is called the purity of a state. Two different ensembles can be described by the same density matrix, or in other words, any P P quantum state ρˆ can be written simultaneously as ρˆ = i pi|ψi⟩⟨ψi| or ρˆ = i qi|φi⟩⟨φi| as long as √ P √ pi|ψi⟩ = j uij qj|φj⟩, where uij are the elements of a unitary matrix U. In Fig. 3.1 a simple example of this non-unique decomposition is discussed through a schematic representation of the convex set of pure and mixed quantum states. The Hilbert space of a quantum system might also be described by a composite Hilbert space, i.e., N (i) i H . Having access to larger Hilbert spaces allows us to purify a mixed state in the following way. Let us assume that we have a mixed state ρˆA in Hilbert Space H A. We can always assume that state ρˆA is the reduced state of a composite pure state ρˆAB = |ψAB⟩⟨ψAB| that belongs to the Hilbert Space H AB = H A ⊗ H B, i.e., A AB AB ρˆ = trB(|ψ ⟩⟨ψ |). (3.6) 26 Quantum Mechanics and Quantum Information
Fig. 3.2 Quantum state interpretations. Each of the six distinct interpretations (see numbers in the diagram) involves the selection of one of the two possibilities from each dichotomy. Note that an ontic interpretation is necessarily objective and a subjective interpretation is necessarily epistemic.
3.1.2 Interpreting the Quantum State
Postulating the concept of the quantum state is a controversial topic, since (inevitably) one has to consider assumptions regarding fundamental physical/philosophical/mathematical concepts. Broadly speaking, there are three dichotomies (sets of two mutually exclusive alternatives) that one has to take a stand on for an exhaustive definition of the quantum state. The first one is whether a purequantum state provides a complete description of an individual system (which is the most popular approach) or alternatively it describes an ensemble of similarly prepared systems (in postulate 1 we chose to follow the ensemble perspective). The second dichotomy is related to whether a pure quantum state is ontic, i.e., state of reality, or epistemic, i.e., state of knowledge (the distinction between ontological and epistemic models in Quantum Mechanics was introduced in Ref. [48], and significant results on this issue appeared in Refs. [49, 50]). The third dichotomy is whether the probability distribution that is associated to the quantum state should be interpreted objectively via the (orthodox) frequentist approach or the closely related propensity approach [51], where the probability of an event is defined by the limit of its relative frequency in a large number of trials, or subjectively via the Bayesian approach [52–54], where the probability is associated to somebody’s knowledge or personal belief of an uncertain event. In Fig. 3.2 we summarize all the above concepts and their corresponding relations.
3.1.3 Quantum Entropy of a State
Let us have a probability distribution P ≡ {pi}. The Shannon entropy [55] (see Ref. [56] for a comprehensive discussion on classical entropy) of this distribution is given by
X H(P) ≡ − pi log2 pi . (3.7) i 3.1 Quantum State 27
Fig. 3.3 Venn diagram for the von Neumann entropies of a bipartite state ρˆAB. Note that the area of the diagram does not necessarily represent positive value of the entropies, since the conditional von Neumann entropies S(A|B) and S(B|A) can also take negative values for entangled states.
The value of H(P) is interpreted as the amount of “information” that we gain, on average, when we learn the value of an event from the distribution P. A complementary view is that H(P) quantifies our average “uncertainty” about an unobserved event from the distribution P. Quantum states ρˆ represent a statistical ensemble, so we can extend the Shannon entropy to those systems as well [57, 58].
Definition (von Neumann entropy). For a quantum state ρˆ von Neumann entropy [19] is defined as
S(ˆρ) ≡ −tr(ˆρlog2 ρˆ). (3.8)
P Writing a state ρˆ in its spectral decomposition, i.e., ρˆ = i pi|ψi⟩⟨ψi|, the von Neumann entropy reduces to the Shannon entropy, i.e., S(ˆρ) = H(P). Obviously, for a pure state, i.e., ρˆ= |ψ⟩⟨ψ|, the von
Neumann entropy vanishes, i.e., S(|ψ⟩) = 0, and for maximally mixed states, i.e., ρˆmm = 1/dimH , it takes its maximum value S(ˆρmm) = log2 dimH . So, von Neumann entropy can be interpreted as the uncertainty about the “mixedness” of a quantum state, or from a different perspective, the smallest Hilbert space to which the quantum state ρˆ can be “compressed” reliably [59]. An important property of the von Neumann entropy is that it is a concave function, i.e.,
! X X S piρˆi ⩾ piS(ˆρi). (3.9) i i
Let us have a bipartite state ρˆAB. The von Neumann entropy of the whole system is called joint entropy, defined as AB AB AB S(A,B) ≡ S(ˆρ ) = −tr(ˆρ log2 ρˆ ), (3.10) and it is in general sub-additive, i.e.,
S(A,B) ⩽ S(A) + S(B), (3.11)
A AB B AB where S(A) = S(ˆρ ) = S[trB(ˆρ )] and S(B) = S(ˆρ ) = S[trA(ˆρ )] are the entropies of the reduced states. Note, though, that the von Neumann entropy becomes additive for uncorrelated (or independent) systems, meaning that
S(ˆρA ⊗ ρˆB) = S(ˆρA) + S(ˆρB). (3.12) 28 Quantum Mechanics and Quantum Information
Analogously to the classical notion of conditional entropy we can define the quantum conditional entropy as S(A|B) ≡ S(A,B) − S(B), S(B|A) ≡ S(A,B) − S(A), (3.13) which is related to the conditional density operator ρˆA|B (or ρˆB|A) denoting the outcome state of a partial measurement on one of the subsystems. We should note that the quantum conditional entropy (contrary to its classical counterpart) can take negative values [60]. Finally, we can also define the quantum mutual entropy (also called quantum mutual information) as S(A:B) ≡ S(A) + S(B) − S(A,B), (3.14) and in Fig. 3.3 we graphically present all the above entropies in a Venn diagram.
3.2 Quantum Operation
Postulate 2. A quantum operation is a linear completely-positive map M :ρ ˆ → M(ˆρ), defined as P M ρMˆ † M(ˆρ) ≡ i i i , (3.15) P † tr i MiρMˆ i
with M = {Mi} being a set of linear operators called Kraus operators [61], satisfying
X † 1 Mi Mi ⩽ . (3.16) i
Note that we did not postulate the quantum operation as a trace preserving operation (as many authors do) because we wanted to take into account the process of measurement of a state under the same definition. Thus, Kraus operators do not necessarily satisfy the completeness relationship, i.e., P † 1 P † i Mi Mi ⩽ , and in Eq. (3.15) we have to divide by tr i MiρMˆ i so the outcome will always be normalized.
Two quantum operations M and K are equivalent if and only if their Kraus operators M = {Mi} and K = {Ki}, respectively, satisfy the following relationship
X Mi = uijKj , (3.17) j where uij are the elements of a unitary matrix U. Quantum operations provide a general mathematical formalism that allows a unified description of both the evolution and the measurement of a quantum state. Below we discuss those two types of operations. 3.2 Quantum Operation 29
3.2.1 Evolution of a Closed Quantum System
The evolution of a quantum state in a closed quantum system is described by a unitary transformation of the state ρˆ, i.e., M(ˆρ) = UρUˆ † , (3.18) where U is a unitary operator (also a Kraus operator in this context), and thus the operation is trace preserving. Note that when the initial state is pure |ψ⟩ we have
M(|ψ⟩) = U|ψ⟩, (3.19) and assuming the unitary operator U = exp[−iH/h¯], where H is a Hermitian operator corresponding to the Hamiltonian of the system (and h¯ is equal to the Planck constant divided by 2π), the Eq. (3.19) is just the solution of the well-known Schrödinger equation [17], i.e.,
d|ψ⟩ ih¯ = H|ψ⟩. (3.20) dt
3.2.2 Evolution of an Open Quantum System
The evolution of a quantum state in an open quantum system is described by the trace preserving quantum operation X † M(ˆρ) = MiρMˆ i , (3.21) i with Kraus operators M given by
Mi = ⟨ei|U|e0⟩. (3.22)
The set {|ei⟩} denotes an orthonormal basis for the environment that is described by the state |e0⟩, † and U is the unitary operator that represents the interaction between ρˆand |e0⟩, i.e., U(ˆρ⊗|e0⟩⟨e0|E)U . Note that there is no loss of generality by assuming that the environment is in a pure state |e0⟩, since we can always introduce an extra system and eventually purify it [see Eq. (3.6)]. These type of operations are also called quantum channels, and given a state ρˆ, an equivalent way to present the output of a channel M is the so-called Stinespring dilation [62], i.e.,
† M(ˆρ) = trE[U(ˆρ ⊗ |e0⟩⟨e0|E)U ], (3.23) that is also graphically depicted in Fig. 3.4. Under the Stinespring dilation, the environment has an output too, which can be represented by a complementary channel Mf, that is given analogously as
† Mf(ˆρ) = trS[U(ˆρ ⊗ |e0⟩⟨e0|E)U ], (3.24) where S denotes the system that the state ρˆ is part of. We call a channel degradable [63] if there exists another channel D such that M ◦ D ≡ Mf , (3.25) 30 Quantum Mechanics and Quantum Information
Fig. 3.4 Stinespring dilation of a quantum channel. The input state ρˆ interacts with the environmental state |e0⟩ through the unitary operator U, and that gives as the primary output the state M(ˆρ) and as the complementary output the state Mf(ˆρ). If the channel M is degradable then the relationship between the primary and the complementary channels is given by M ◦ D = Mf, if it is anti-degradable we have A ◦ Mf = M. where the symbol “◦” denotes the concatenation of the channels. We also call a channel anti-degradable when there is a channel A such that A ◦ Mf ≡ M. (3.26)
Before we introduce the notion of classical capacity for a quantum channel it is worth defining the capacity of a classical channel.
Classical Capacity of a Classical Channel
Let us have two parties, Alice and Bob. Alice is assumed to randomly draw letters {ai} associated A with a probability distribution PA ≡ {pi }. The information content of PA is quantified by the Shannon entropy [see Eq. (3.7)] and is expressed in terms of bits per letter. A given sequence of letters, i.e.,
A ≡ {a1,a2,···}, (3.27) is called a random message, which is sent to Bob through the channel N , that is assumed to be memoryless (it does not create correlations between different letters). Bob’s output message
B = N (A) ≡ {b1,b2,···}, (3.28)
B is a sequence of letters {bi} that is associated with a probability distribution PB ≡ {pi }. The number of bits per letter which are, on average, communicated to Bob is given by the classical mutual entropy (also called classical mutual information)
H(A:B) = H(B) − H(B|A), (3.29) where H(B) denotes the Shannon entropy of B, and H(B|A) the Shannon entropy of B conditioned on the knowledge of A. The classical channel capacity C(N ) is defined as the maximum mutual entropy over all of Alice’s possible inputs C(N ) ≡ maxH(A:B), (3.30) A and is expressed in bits per channel use. Note that due to the lack of restriction on the measurement accuracy in classical physics, classical capacity becomes infinite for noiseless channels, which isan 3.2 Quantum Operation 31 issue that can be resolved by taking into account the actual quantum nature of the physical systems, which always contain some level of noise.
Classical Capacity of a Quantum Channel
P Since quantum states are also associated with probability distributions, i.e., ρˆ = i piρˆi, each letter in the previous nomenclature can be substituted by a quantum state ρˆi that occurs with a probability pi. Given a quantum ensemble
Q ≡ {ρˆ1,ρˆ2,···}, (3.31) the maximum information that can be physically extracted is given by the so-called Holevo bound [5], defined as X χ(Q) = S(ˆρ) − piS(ˆρi), (3.32) i where S(·) is the von Neumann entropy [see Eq. (3.8)]. If we assume that Alice’s random message is given by the quantum ensemble Q, and that she transmits this message through a memoryless quantum channel M to Bob, then based on the Holevo bound, the maximum amount of information that he can extract from his output message is given by
X χ(Q,M) = S [M(ˆρ)] − piS[M(ˆρi)]. (3.33) i
Thus the Holevo bound χ(Q,M) can be interpreted as the optimal communication rate which is achievable over the memoryless quantum channel M for a fixed ensemble Q. Analogously to the classical capacity, we can define the (single-shot) classical capacity of a quantum channel M by maximizing over all possible ensembles Q, as
C(1)(M) ≡ maxχ(Q,M). (3.34) Q
The notion “single-shot” refers to the assumption that the messages are restricted to single-letter ensembles. However we can remove this restriction and assume the possibility of messages with multi-letter ensembles with input states that are in general entangled between n uses of the channel M⊗n. Thus, we can define the (full) classical capacity as
1 C(M) ≡ lim C(1)(M⊗n). (3.35) n→∞ n
From now on, by “classical capacity” we will refer to Eq. (3.35). Since the classical capacity is the limit of an optimization, the analytical quantification of its value is in general a difficult task.
Quantum Capacity of a Quantum Channel
Quantum channels can also be used to transmit quantum information (and hence can preserve superpos- tions), so we need an analogous expression to Eq. (3.35) that will give the number of quantum states 32 Quantum Mechanics and Quantum Information
(the type of the state depends on the protocol) per channel use that can be reliably transmitted through a memoryless channel M. In order to do so, we need to introduce the notion of coherent information. A A EA Let us assume that Alice has a quantum state ρˆ , which can be purified as ρˆ = trE(ˆρ ), where ρˆEA = |ψ⟩⟨ψ|EA. This state is sent through channel M to Bob, who receives a state ρˆB. Direct coherent information is defined as [64]
A B EB ID(M,ρˆ ) ≡ S(ˆρ ) − S(ˆρ ), (3.36) where ρˆEB ≡ [1 ⊗ M](ˆρEA). Note that an analogous quantity has also been defined, called reverse coherent information [65, 66], given by
A E EB IR(M,ρˆ ) ≡ S(ˆρ ) − S(ˆρ ). (3.37)
Analogously to the definitions of classical capacity of a quantum channel inEqs.(3.34) and (3.35), the (single-shot) quantum capacity is then given by maximizing the direct coherent information over all input states ρˆA, i.e., (1) A Q (M) ≡ maxID(M,ρˆ ), (3.38) ρˆA and respectively, the (full) quantum capacity is defined as
1 h i Q(M) ≡ lim Q(1) M⊗n,(ˆρA)⊗n . (3.39) n→∞ n
3.2.3 Measurement of a Quantum State
We call measurement of a quantum state the quantum operation
† MiρMˆ i Mi(ˆρ) = † , (3.40) tr MiρMˆ i with Kraus operators M given, due to the polar decomposition [see Eq. (2.51)], by
q Mi = Ui Ei , (3.41) where U ≡ {Ui} is a set of unitary operators and E ≡ {Ei} is a set of positive operators, i.e., Ei ⩾ 0, that satisfy the completeness relationship, i.e.,
X Ei = 1. (3.42) i
The set E is also called a quantum observable, or POVM, i.e., positive operator valued measure.
The probability distribution P ≡ {pi} of the observable E is given by
† pi = tr(ˆρEi) = tr MiρMˆ i , (3.43) 3.2 Quantum Operation 33 and for pure states ρˆ = |ψ⟩⟨ψ| the probability distribution P is given by
† pi = ⟨ψ|Ei|ψ⟩ = ⟨ψ|MiMi |ψ⟩. (3.44)
A set {Πi} is called sharp quantum observable if for every element Πi we have Ei = Πi, where 2 Πi are projector operators, i.e., Πi = Πi . Based on the spectral decomposition [see Eq. (2.44)], every Hermitian operator H can be written as
X H = λiΠi , (3.45) i so the whole set {Πi} is more conveniently represented by a single operator H, and that is why Hermitian operators are usually called just observables in the context of Quantum Mechanics. For an observable H, the probabilities of measuring an eigenvalue λi is given by
pi = tr(ˆρΠi), (3.46) which is historically called the Born rule [67], but it can be considered as a special case of the Eq. (3.43). The mean value (or 1st statistical moment) of a quantum observable, described by a Hermitian operator H, is given by H¯ ≡ ⟨H⟩ = tr(ˆρH) . (3.47)
Note that based on Eq. (3.42), the number of projector operators in the set {Πi} is equal to the dimensions of the Hilbert space of the system that is measured in, while the number of elements in a POVM observable, i.e., E, can be larger. Thus, POVM observables describe a more general type of measurement and they are usually employed in tasks related to state discrimination [68].
Uncertainty Principle of Quantum Measurements
Let us consider two Hermitian operators X and Y . We say that those two operators commute if
[X,Y ] ≡ XY − YX = 0, (3.48) where [·,·] is called the commutator of two operators, and anti-commute if
{X,Y } ≡ XY + YX = 0, (3.49)
{·,·} is called the anti-commutator. We saw before, that a sharp observable X can be represented by the set of projection operators
{Xi}. Thus, we can define the covariance matrix of an observable X as
1h i V (X) ≡ ⟨{∆Xi,∆Xj}⟩ , (3.50) 2 i,j 34 Quantum Mechanics and Quantum Information where
∆Xi ≡ Xi − ⟨Xi⟩. (3.51)
For a given observable X we can also define its variance (or 2nd statistical moment) as
Var(X) ≡ ⟨X2⟩ − ⟨X⟩2 , (3.52)
q and its standard deviation as the squared root of the variance, i.e., σX = Var(X). Let us also mention that the diagonal elements of the covariance matrix of an observable X, provide the variance of the projection operators Xi, i.e., 2 vii(X) = Var(Xi) = ⟨(∆Xi) ⟩. (3.53)
The covariance matrix is a real, symmetric and positive-definite matrix satisfying the following inequality i V (X) ± Ω(X), (3.54) ⩾ 2 where h i Ω(X) ≡ i ⟨[Xi,Xj]⟩ , (3.55) i,j is a real matrix called the commutation matrix. Given two Hermitian operators X and Y the above inequality (3.54) is equivalent to the Robertson–Schrödinger uncertainty principle [69, 70], i.e.,
1 1 Var(X)Var(Y ) ⟨{∆X,∆Y }⟩2 + |⟨[X,Y ]⟩|2 . (3.56) ⩾ 4 4
The uncertainty principle operationally implies that if we prepare an ensemble of identical states and then perform measurements on the observable X on some states and Y on others, then the variance of the results of those measurements will satisfy the inequality (3.56). For the special case of canonically conjugate operators, i.e., [X,Y ] = ih¯, we end up with the well-known Heisenberg uncertainty principle [71], i.e.,
h¯2 h¯ Var(X)Var(Y ) ⇐⇒ σ σ . (3.57) ⩾ 4 X Y ⩾ 2
3.3 Distinguishing Quantum States
Given two quantum states, i.e., ρˆ and σˆ, we are sometimes interested to know how “close” they are to each other. Thus, we can define different measures that quantify their “distance”, in the senseof operational distinguishability.
3.3.1 Trace Distance
The most important distance measure is the so-called trace distance, which for two quantum states ρˆ and σˆ is given by 1 T (ˆρ,σˆ) ≡ ∥ρˆ− σˆ∥ , (3.58) 2 1 3.3 Distinguishing Quantum States 35
where ∥ · ∥1 is the trace norm defined in Eq. (2.36). Since density matrices are positive and thus Hermitian as well, the trace distance takes the form
1 X T (ˆρ,σˆ) = |λi|, (3.59) 2 i where λi are the eigenvalues of the matrix ρˆ− σˆ, which is not necessarily a positive matrix. Trace distance is invariant under unitary transformations, i.e.,
T (UρUˆ †,UσUˆ †) = T (ˆρ,σˆ), (3.60) and it is also called a metric, since it has the nice property of satisfying the triangle inequality, i.e., for states ρˆ, σˆ, and φˆ we have T (ˆρ,σˆ) ⩽ T (ˆρ,φˆ) + T (φ, ˆ σˆ). (3.61) Based on the trace distance, we can define the so-called Helstrom bound [72], given by
1 p = [1 − T (ˆρ,σˆ)] , (3.62) e 2 which is related to the minimum error probability of distinguishing two states by applying dichotomic POVM measurements.
3.3.2 Fidelity
Another way of distinguishing two quantum states ρˆ and σˆ is fidelity, which is defined [73–75] as
rq q F(ˆρ,σˆ) ≡ tr ρˆ σˆ ρˆ , (3.63) and ranges from zero for orthogonal states to one for identical states1. A reason why fidelity defines a good notion of distinguishability is that it is equal to the minimal overlap between the probability distributions pi(ˆρ) = tr(ˆρEi) and pi(ˆσ) = tr(ˆσEi) generated by a generalized measurement or positive operator-valued measure Ei [76]. Similarly to trace distance, fidelity is also invariant under unitary transformations, i.e.,
F(UρUˆ †,UσUˆ †) = F(ˆρ,σˆ), (3.64) but fidelity is not a metric. However, we can easily construct another measure based on fidelity thatis a metric, i.e., q B(ˆρ,σˆ) ≡ 2 − 2F(ˆρ,σˆ), (3.65) called Bures distance [77].
1Note that sometimes in the literature fidelity is also defined as the squared value ofEq.(3.63). 36 Quantum Mechanics and Quantum Information
Interestingly, fidelity and trace distance are closely related to each other, as can be seenfromthe following inequality q 1 − F(ˆρ,σˆ) ⩽ T (ˆρ,σˆ) ⩽ 1 − F(ˆρ,σˆ)2 . (3.66)
3.3.3 Relative Entropy
Finally, we can define an entropic “measure of closeness” for two quantum states ρˆ and σˆ, inspired by the classical Kullback-Leibler distance [78]. This is the quantum relative entropy and is defined [79] as
S(ˆρ∥σˆ) ≡ tr(ˆρlog2 ρˆ) − tr(ˆρlog2 σˆ), (3.67) which vanishes for identical states (see Ref. [80] for an extended review on the role of relative entropy in quantum information theory). Relative entropy is not a proper measure of distance, since (contrary to trace distance and fidelity) it is not in general symmetric, i.e.,
S(ˆρ∥σˆ) ̸= S(ˆσ∥ρˆ). (3.68)
It is worth noting that the quantum relative entropy between a bipartite state ρˆAB and the product state ρˆA ⊗ ρˆB is equal to the mutual quantum entropy of those two systems, i.e.,
S(ˆρAB∥ρˆA ⊗ ρˆB) = S(A:B). (3.69)
3.4 Examples of Quantum Systems
In this section we present two quantum states that play a significant role in Quantum Mechanics and Quantum Information.
3.4.1 Finite-dimensional System
Let us have a quantum system with finite degrees of freedom. The quantum state ρˆ ∈ H d has in general the following form d−1 X ρˆ = pi|i⟩⟨i|, (3.70) i=0 where d = dimH and P ≡ {pi} is a probability distribution. The two-dimensional pure state is the the so-called qubit, i.e., |ψ⟩ = α|0⟩ + β|1⟩, (3.71) with α,β ∈ C, and |α|2 + |β|2 = 1. Its name is an abbreviation of the word “quantum-bit” and it is historically significant because it is the quantum version of the classical binary bit. Qubit systems are ubiquitous in Quantum Information, since they can model any 2-dimensional quantum system, e.g., 2-level atoms and ions, polarization of particles, quantum dots, ray paths in a two-path interferometer containing one photon, etc. States with higher (but finite) levels are usually called qudits. 3.4 Examples of Quantum Systems 37
Fig. 3.5 Qubit in a Bloch sphere. We graphically represent a two-dimensional quantum state in a 3-dimensional sphere. Based on the Eq. (3.72) the values θ and φ correspond to polar coordinates of a qubit. We depict a generic pure state ⃗rψ, which lies on the surface of the sphere, and a mixed state ⃗rρ, which has a shorter length. The maximally mixed state, i.e., ⃗rmm = 1/2 is the origin of the sphere. Also note that orthogonal states, e.g., |0⟩ ⊥ |1⟩, represented by vectors ⃗r0 and ⃗r1, respectively, are antipodal in this representation, meaning that they are on the opposite side of the sphere.
Bloch sphere
Interestingly, qubit states have a nice geometrical representation. In particular, a qubit may be also written as θ θ ! |ψ⟩ = eiζ cos |0⟩ + eiφ sin |1⟩ , (3.72) 2 2 where θ,φ,ζ ∈ R, and ignoring the global phase eiζ, since it has no observable effect, the numbers θ and φ define a point on the surface of a unit radius sphere (see Fig. 3.5), also known as the Bloch sphere. We can also represent the state |ψ⟩ as a vector
T ⃗rψ = (sinθ cosφ,sinθ sinφ,cosθ) , (3.73) that points from the origin of the sphere to a point on the surface. In the same representation, a two-dimensional mixed state ρˆ corresponds to a vector
T ⃗rρˆ = (tr[ˆρX],tr[ˆρY ],tr[ˆρZ]) , (3.74) where X,Y,Z are the well-known Pauli matrices given by
0 1 0 −i 1 0 X ≡ ,Y ≡ ,Z ≡ . (3.75) 1 0 i 0 0 −1
Note that, in general, a mixed state ρˆ is a vector ⃗rρ with length less than one, so it lies inside the 1 Bloch sphere, and the maximally mixed state ρˆmm = 2 has length equal to zero, so it reduces to a point, i.e., the origin of the sphere. 38 Quantum Mechanics and Quantum Information
3.4.2 Infinite-dimensional System
Let us now have a quantum state with infinite-degrees of freedom. The quantum state ρˆ∈ (N ⊂H ⊂N ×) has in general the form ∞ X ρˆ = pi|i⟩⟨i|, (3.76) i=0 with P ≡ {pi} being a probability distribution. A typical example of a system that can be described by such a state is the quantized bosonic mode of an electromagnetic field, e.g., a photonic mode. The Hamiltonian of those systems has the following form
1 H = X2 + P 2 , (3.77) 2 where X and P are the position and momentum operators of the quantum harmonic oscillator, also called in short quadrature operators. The quadrature operators are Hermitian operators satisfy the following commutation relation [X,P ] = ih.¯ (3.78)
Note that as a convention for the rest of the thesis we fix h¯ = 2 in order to simplify the notation, a choice that will be justified later in Sec. 4.1.1. The quadrature field operators are observables with continuous eigenspectra, with corresponding eigenstates |x⟩ and |p⟩, given by
X|x⟩ = x|x⟩,P |p⟩ = p|p⟩, (3.79) and are related to each other as follows
1 Z ∞ 1 Z ∞ |x⟩ = √ dpe−ixp/2|p⟩, |p⟩ = √ dxeixp/2|x⟩. (3.80) 2 π −∞ 2 π −∞
The eigenvectors |x⟩ and |p⟩ cannot be properly defined in the usual Hilbert space H , and that is why we defined the state ρˆ of those systems in the rigged Hilbert space N ⊂H ⊂N ×, which is discussed in the AppendixA. In matrix notation, X and P are given by
0 1 0 0 ··· 0 −1 0 0 ··· √ √ 1 0 2 0 ··· 1 0 − 2 0 ··· √ √ √ √ X ≡ 0 2 0 3 ··· ,P ≡ i 0 2 0 − 3 ··· . (3.81) √ √ 0 0 3 0 ··· 0 0 3 0 ··· . . . . . . . . . . ......
Another set of operators that can describe this system are the annihilation and creation operators defined, respectively, as follows
1 1 A = (X + iP ) ,A† = (X − iP ) , (3.82) 2 2 3.4 Examples of Quantum Systems 39 which in short are called bosonic operators (or ladder operators for reasons that will be discussed in a bit), and satisfy the following commutation relation
[A,A†] = 1. (3.83)
In matrix notation, A and A† are given by
0 1 0 0 ··· 0 0 0 0 ··· √ 0 0 2 0 ··· 1 0 0 0 ··· √ √ † A ≡ 0 0 0 3 ··· ,A ≡ 0 2 0 0 ··· . (3.84) √ 0 0 0 0 ··· 0 0 3 0 ··· . . . . . . . . . . ......
Quadrature operators can be expressed via the bosonic operators as follows
X = A† + A,P = i(A† − A). (3.85)
The bosonic operators are not Hermitian, i.e., not observables, but we can introduce the number operator, i.e., N = A†A, which is indeed Hermitian (and commutes with the Hamiltonian). The eigenstates of the number operator, i.e., {|n⟩}, drawn from the equation
N|n⟩ = n|n⟩, (3.86)
P∞ are called Fock states or number states, and form a complete orthonormal basis, i.e., n=0 |n⟩⟨n| = 1, for the associated Hilbert space H of the mode. The action of the bosonic field operators on those states yield:
† √ A |n⟩ = n + 1|n + 1⟩ for n ⩾ 0, (3.87a) √ A|n⟩ = n|n − 1⟩ for n ⩾ 1, (3.87b) A|0⟩ = 0 for n = 0, (3.87c) thus the name annihilation and creation operators, since they respectively annihilate or create one “unit” of energy. Those “packages” of energy are the so-called quanta that the whole theory took its name from. The simplest Fock state is the one which represents a bosonic mode with zero particles, i.e., |0⟩, (for example photons) and is called the vacuum state. Any Fock state can be obtained from the vacuum state by successive application of the creation operator A†, i.e.,
(A†)n |n⟩ = √ |0⟩, for n ⩾ 0. (3.88) n! 40 Quantum Mechanics and Quantum Information
Fig. 3.6 1st and 2nd quantisation. The quantum state described by the ket notation |1⟩ ⊗ |2⟩ ⊗ |3⟩ is represented for each type of quantization. The lines represent modes and the dots represent particles.
1st and 2nd Quantization
At this point it is probably worth discussing how different quantum systems can be described under the same notation, but have completely different nature (see Ref. [81] for a more in-depth discussion). In the 1st quantization (that it is usually employed in finite-dimensional systems) we specify for each particle denoted as n the mode mn that it occupies. So, the whole system is described by the Hilbert space N O H (Mi) , (3.89) i where N is the number of all particles and Mi is the number of all possible modes for each particle. On the other hand, we have the 2nd quantization (common in infinite-dimensional systems such as the photonic modes described before), where we specify for each mode m, the number of particles nm that it contains. The system now is described by the Hilbert space
M O Fi[H ], (3.90) i
L∞ (j) where Fi[H ] = j=0 H denotes the bosonic Fock space of each mode. In Fig. 3.6 we depict an example of a composite quantum state for each type of quantization to make the distinction more easily understood.
3.5 Quantum Mechanics in Phase Space
Given a classical state we can always represent it as a unique point in a two dimensional graph, called the phase space, that refers to the particle’s momentum and position. We can also assign a phase space
(joint) probability distribution function fc(x,p) that gives the number of particles per unit volume in the phase space. In order to define an analogous probability distribution fq(x,p) in quantum mechanical systems, the function fq(x,p) should satisfy the following two conditions:
1. It should be finite-valued and non-negative, i.e.,
0 ⩽ fq(x,p) < ∞. (3.91) 3.5 Quantum Mechanics in Phase Space 41
Fig. 3.7 Wigner function for Fock states. We plot the the function W(x,p) for the case of a Fock state given in Eq. (3.97) for: (a) n = 0, (b) n = 2, and (c) n = 4 photons, respectively. All quantities plotted are dimensionless.
2. Its marginal distributions should yield the position and momentum probability distributions, i.e.,
Z ∞ dpfq(x,p) = ⟨x|ρˆ|x⟩, (3.92a) −∞ Z ∞ dxfq(x,p) = ⟨p|ρˆ|p⟩. (3.92b) −∞
Unfortunately, in quantum mechanical systems we cannot find a function that satisfies both condi- tions (as proved by Wigner [82]), but we can define functions that satisfy each condition separately.
Wigner Function
Let us start by presenting the most popular quantum phase space representation of a state ρˆ, called the Wigner function, defined [83] as
1 Z ∞ W(x,p) ≡ dyeiyp/2⟨x − y |ρˆ|x + y ⟩, (3.93) 4π −∞ 2 2 where x,y,p ∈ R. Note that the above expression is a “modernized” but equivalent version to the originally defined Wigner function84 [ ]. The Wigner function is a real and normalized to 1 function, i.e., ZZ ∞ dxdpW(x,p) = 1, (3.94) −∞ but it violates in general the 1st condition defined before [see Eq. (3.91)], i.e., it can take negative or singular values, and thus cannot be associated to a classical probability distribution. That is why it is sometimes referred to as a “quasi-probability distribution”. On the other hand, its marginal distribution in one of the variables gives the probability distribution of the state in that space, i.e.,
Z ∞ Z ∞ W(x) = dpW(x,p), W(p) = dxW(x,p). (3.95) −∞ −∞
Through Wigner function we can also describe the previously defined measure of fidelity between two states ρˆ and σˆ, as ZZ ∞ 2 F (ˆρ,σˆ) = 4π dxdpWρ(x,p)Wσ(x,p). (3.96) −∞ 42 Quantum Mechanics and Quantum Information
A Fock state under the Wigner representation is written as
2 h i W(x,p) = (−1)nL (4r)exp −2 x2 + p2 , (3.97) π n
2 where Ln(x) denotes the Laguerre polynomial . In Fig. 3.7, we plot the Wigner function for several Fock states. As it is obvious, apart from the vacuum state, i.e., |0⟩, all the rest of Fock states yield a Wigner function that part of it takes negative values, so it cannot represent a probability distribution. Wigner function is not the only way to represent a quantum state in phase space. Two other useful ways to represent it are through the Q and P functions, but we will define them later in the thesis (Sec. 4.1.2), since for them we first need to discuss the concept of a quantum “coherent state”.
2Laguerre polynomials are solutions of Laguerre’s equation xy′′ + (1 + x)y′ + ny = 0, which is a second-order linear differential equation. Chapter 4
Gaussian Systems
“I have had my results for a long time, but I do not yet know how I am to arrive at them...”
— Carl Friedrich Gauss
In this chapter, we focus on a specific set of continuous-variable states, the Gaussian states. In general, “Gaussian” is called a continuous-variable state that can be fully characterized by the first two statistical moments of the bosonic/quadrature field operators. Even though this restriction might seem too narrow, it turns out that a lot of existing experimental set-ups used for Quantum Information processing are Gaussian systems. An obvious merit of those systems is that their mathematical description is relatively easier compared to non-Gaussian systems, since instead of dealing with an infinite-dimensional density matrix, Gaussian states can be fully described by their finite-dimensional covariance matrices.
Note on references. References for specific results will be given along the text, but as a whole the discussion of this chapter has been based on Refs.[26, 44–47, 85–89]
4.1 Gaussian States and Operations
Let us have a quantum system that consists of N bosonic modes [independent quantum harmonic oscillators described by a Hamiltonian given in Eq. (3.77)], corresponding to N pairs of bosonic † N field operators {Ai,Ai }i=1. Bosonic field operators can be arranged in a vectorial bosonic operator † † T R ≡ (A1,A1,...,AN ,AN ) and satisfy the commutation relation
[Ri,Rj] = ωij , (4.1) 44 Gaussian Systems
where ωij is the element of the commutation matrix [see Eq. (3.55)], i.e.,
N M 0 1 Ω ≡ , (4.2) i=1 −1 0
NN also known as the symplectic form. This system is described by the Hilbert space i Fi[H ], where L∞ (j) Fi[H ] = j=0 H denotes the infinite-dimensional Fock space for each mode. N The quadrature field operators of N modes {Xi,Pi}i=1 can also form a vectorial operator, i.e., T Q ≡ (X1,P1,...,XN ,PN ) , that satisfies the commutation relation
[Qi,Qj] = 2iωij . (4.3)
The covariance matrix for those systems is a real and symmetric matrix, defined as
1h i V ≡ ⟨{∆Qi,∆Qj}⟩ , (4.4) 2 i,j where ∆Qi ≡ Qi − ⟨Qi⟩. The uncertainty principle for those systems [see Eq. (3.54)] takes the form
V + iΩ ⩾ 0, (4.5) implying also the positive definiteness of the covariance matrix.
Definition (Gaussian state). An N-mode quantum state ρˆG is called Gaussian if it can be written as (2) e−βH ρˆG ≡ h i , (4.6) tr e−βH(2)
where β ∈ R+, and H(2) is a second-order Hamiltonian in the field operators, i.e.,
N N N (2) X (1) † X (2) † X (3) † † H ≡ gi Ai + gij Ai Aj + gij Ai Aj + h.c., (4.7) i=1 i⩾j=1 i,j=1
with g(i) ∈ C, and “h.c.” denoting the hermitian conjugate of all the preceding terms.
Note that the limiting case lim ρˆG is also included in the Eq. (4.6), representing pure Gaussian β→∞ states. From a statistical point of view, Gaussian states can be fully described by the mean value and the covariance matrix of their observables (which includes the variances of all the projection operators of a sharp observable), so we will refer to them as ρˆG(Q,V¯ ). From a geometric perspective, the Wigner function [see Eq. (3.93)] of a Gaussian state is a Gaussian distribution, thus their name.
Definition (Gaussian operation). A quantum operation is called Gaussian MG when it trans- forms a Gaussian state into another Gaussian state. Gaussian operations involve Gaussian h i unitaries, i.e., U = exp −iH(2)/2 , where H(2) is a second-order Hamiltonian in the field operators of the form of Eq. (4.7). 4.1 Gaussian States and Operations 45
T In an N-mode system, we can define the vectorial annihilation operator A⃗ ≡ (A1,A2,...,AN ) , and ⃗† T the vectorial creation operator A ≡ (A†1,A†2,...,A†N ) . A Gaussian unitary yields the following transformation (also called Bogoliubov transformation)
⃗ † ⃗ ⃗ ⃗† A → USAUS = KA + LA + ⃗α, (4.8) where ⃗α ∈ CN , and K and L are N × N complex matrices that satisfy
KLT = LKT , (4.9a) KK† = LL† + 1. (4.9b)
Regarding quadrature operators, the corresponding transformation is given by
Q → ΣQΣT + ⃗q, (4.10) where ⃗q ∈ R2N and Σ is a 2N × 2N real matrix. This type of transformation is called symplectic (see Ref. [90] for a comprehensive discussion on linear symplectic groups and their connection to Classical and Quantum Physics), and must preserve the commutation relations given in Eq. (4.3), which implies that the following relationship is satisfied
ΣΩΣT = Ω. (4.11)
Finally, the mean value of the quadrature operators and the corresponding covariance matrix are transformed as follows
Q¯ → ΣQ¯ + ⃗q, (4.12a) V → ΣV ΣT . (4.12b)
Below we define the most common set of states with the corresponding operations thatcanbe generated.
4.1.1 Vacuum States
Vacuum states (see Fig. 4.1) were defined in the previous chapter as Fock states with zero photons, i.e., |0⟩. This is the simplest pure Gaussian state, having zero mean value Q¯ = 0, and an identity as its covariance matrix, i.e., V = 1, so vacuum states saturate the uncertainty principle.
Definition (shot noise). The variance of both quadrature operators in a vacuum state |0⟩ is equal to vac Var(Qi ) = 1, (4.13) which we call the shot noise. 46 Gaussian Systems
We will keep this convention throughout this thesis. Keep in mind, though, that other authors use the convention of a shot noise variance equal to 1/2 [because they used the value of h¯ = 1 instead of h¯ = 2]. Expanded in terms of position states the vacuum state takes the form
1 Z ∞ 2 |0⟩ = √ dxe−x /2|x⟩. (4.14) 4 π −∞
The Wigner function of a vacuum state (plotted in Fig. 4.3) is given by
1 " x2 + p2 # W(x,p) = exp − , (4.15) 2π 2 with a circle, i.e., x2 + p2 = 1, as its corresponding contour (plotted in Fig. 4.1).
4.1.2 Coherent States and the Displacement Operator
A coherent state can be created by acting with the displacement operator D(α) (which is a Gaussian operator) on the vacuum state |0⟩, i.e.,
|α⟩ = D(α)|0⟩, (4.16) where the displacement operator is defined as
D(α) ≡ exp(αA† − α∗A), (4.17)
1 and α = 2 (xα + ipα) ∈ C is the complex amplitude. The displacement operator has the following property D(α) = D−1(α) = D(−α), (4.18) and applied on a bosonic operator induces the unitary transformation
D†(α)AD(α) = A + α. (4.19)
The corresponding symplectic transformation on the quadrature operators is given by
D(α)QD(α)T = Q + ⃗q, (4.20)
T with ⃗q = (xα,pα) . A coherent state is a minimum uncertainty state, since it saturates the uncertainty principle, i.e.,
Var(X)Var(P ) = 1, (4.21) 4.1 Gaussian States and Operations 47
Fig. 4.1 Contour of the Wigner function for: (a) a vacuum state in light blue, (b) a thermal zero-displaced thermal state in dark blue, (c) a coherent state in yellow, and (d) a displaced squeezed state in red. and is also an eigenstate of the annihilation operator A, i.e.,
A|α⟩ = α|α⟩. (4.22)
In the Fock basis, a coherent state is given by
∞ n 2 α |α⟩ = e−|α| /2 X √ |n⟩, (4.23) n=0 n! where the probability distribution of photons in a coherent state Pα ≡ {pn} is a Poisson distribution (see Fig. 4.2), i.e., 2 |α|2ne−|α| p = |⟨n|α⟩|2 = , (4.24) n n! where |α|2 =n ¯ is the mean number of photons on a mode. Expanded in the position basis a coherent 1 state with a complex amplitude α = 2 (xα + ipα) takes the form
∞ " 2 # 1 Z ixαpα (x − xα) |α⟩ = √ dxexp ipαx − − |x⟩. (4.25) 4 π −∞ 2 2
Coherent states have the same covariance matrix as the vacuum states, i.e., V = 1, but different mean values, or in other words a coherent state is just a displaced vacuum state. The inner product of two coherent states is given by
1 ⟨α|β⟩ = exp − |α|2 + |β|2 − 2α∗β , (4.26) 2 48 Gaussian Systems
Fig. 4.2 Probability distributions. In figure (a) we have the probability distribution of a coherent state, wherethe different colors denote different values of amplitude |α|. It worths noting that the probability distribution of the coherent state is a Poisson distribution. In figure (b) we have the probability distribution of photons of athermal state, where the different colors denote different values of mean number of photons per mode n¯. All quantities plotted are dimensionless. and the modulus squared of their overlap is equal to
|⟨β|α⟩|2 = exp −|β − α|2 , (4.27) which means that the coherent states are not orthogonal even though two states |α⟩ and |β⟩ become approximately orthogonal in the limit of |α−β| ≫ 1. The set of coherent states is in fact overcomplete, and the following completeness relationship holds
1 ZZ ∞ d2α|α⟩⟨α| = 1, (4.28) π −∞ with d2α = dRe(α)dIm(a). 1 For a coherent state |α⟩ with a complex amplitude α = 2 (xα + ipα), the Wigner function is given by 1 " (x − x )2 + (p − p )2 # W(x,p) = exp − α α , (4.29) 2π 2 2 2 with a corresponding contour (x − xα) + (p − pα) = 1, which similarly to the vacuum states is also a circle. Thus, it is just a displaced Wigner function of a vacuum state (see Fig. 4.1). The Fock states we introduced in the previous chapter had a definite number of photons but completely indefinite phase. Coherent states have an indefinite number of photons, but the productof the quadrature operators saturates the uncertainty principle. The reason why coherent states have a physical significance in Quantum Optics is that they model the field generated by a highly stabilized laser operating well above threshold.
Phase Space Representation through Coherent States
Coherent states can also lay the basis for other phase space representations of a quantum state apart from the Wigner function. As we discussed in Sec. 3.5, Wigner function satisfies only one of the two 4.1 Gaussian States and Operations 49 conditions required in order a function to be associated to a classical probability distribution, i.e., W function gives proper marginal probability distributions for marginal integrals, but it can take negative values. We can define another function, called the Husimi function or just Q function [91], as
1 Q(α) ≡ ⟨α|ρˆ|α⟩, (4.30) π which is strictly positive, but its marginal integrals are not associated with the marginal probabilities. Husimi function is upper bounded by 1 Q(α) , (4.31) ⩽ π and can be normalized to unity, i.e.,
ZZ ∞ dα2Q(α) = 1, (4.32) −∞ where d2α = dRe(α)dIm(α). Q function can be obtained by Wigner function as follows
2 ZZ ∞ 2 Q(α) = d2βW(β)e2|α−β| , (4.33) π −∞ which can be understood as a Gaussian convolution of the Wigner function, also called Gaussian smoothing. Another phase space representation can be obtained using the fact that coherent states form an overcomplete basis, as shown in Eq. (4.28). In particular, a quantum state ρˆ can be represented [92, 93] as follows ZZ ∞ ρˆ = d2αP(α)|α⟩⟨α|, (4.34) −∞ where P(α) is a quasi-probability distribution, also called the P function [92, 93]. P function is related to W function as 2 ZZ ∞ 2 W(α) = d2βP(β)e−2|α−β| , (4.35) π −∞ and to Q function as 1 ZZ ∞ 2 Q(α) = d2βP(β)e−|α−β| . (4.36) π −∞
4.1.3 Squeezed States and the Squeezing Operator
The coherent states discussed before is a special case of a more general class of minimum-uncertainty states, called squeezed states. A squeezed vacuum state can be created by applying the Gaussian operator S(r), which is called squeezing operator, on a vacuum state |0⟩, i.e.,
|r⟩ = S(r)|0⟩, (4.37) with r 2 S(r) ≡ exp A2 − A† , (4.38) 2 50 Gaussian Systems
Fig. 4.3 Wigner function representation of Gaussian states. In figure (a) we have the W function of a vacuum state. Coherent states are just displaced vacuum states so their Wigner function is just shifted respectively. In figure (b) we plot squeezed vacuum state, that is squeezed on X quadrature and anti-squeezed in P quadrature. All quantities plotted are dimensionless. where r ∈ R is the squeezing parameter. Note that the squeezing parameter is in general a complex number, representing squeezing in different directions, however, in order to simplify the discussion we defined the squeezing parameter as a real value, since different squeezing directions canbealso achieved by applying a combination of displacements and phase rotations (that will be introduced later). A squeezed vacuum state in the Fock basis is written as
q 1 ∞ (2n)! √ X n |r⟩ = n tanhr |2n⟩, (4.39) coshr n=0 2 n! and expanded in the position basis takes the form
1 Z ∞ x2e2r ! |r⟩ = √ dxexp − |x⟩. (4.40) e−r 4 π −∞ 2
The squeezing operator has the following property
S†(r) = S−1(r) = S(−r), (4.41) and applied on a bosonic operator induces the following unitary transformation
S†(r)AS(r) = Acoshr − A† sinhr , (4.42) which leads to the eigenvalue equation
Acoshr + A† sinhr |r⟩ = 0. (4.43)
In terms of the quadrature operators, the squeezing operation takes the form
ir S(r) ≡ exp (XP + PX) , (4.44) 4 4.1 Gaussian States and Operations 51 and applied on a pair of quadrature operator can be represented as the following symplectic transfor- mation e−r 0 ΣS(r) = , (4.45) 0 er
T and thus the covariance matrix is given by V = ΣS(r)ΣS(r) = ΣS(2r). The uncertainty principle for a squeezed vacuum state is given by
Var(X)Var(P ) = 1, (4.46) so when the variance of one quadrature goes below the shot noise due to squeezing the other one becomes larger in a way that the uncertainty principle remains saturated. For a squeezed vacuum state the Wigner function (seen in Fig. 4.3) is
1 x2e2r + p2e−2r ! W(x,p) = exp − , (4.47) 2π 2
x2 p2 r −r with a corresponding contour given by e−2r + e2r = 1, representing an ellipse with e and e giving the length of the major and minor axes, respectively. Applying a displacement operator on a squeezed vacuum state, i.e., D(α)S(r)|0⟩, we can create displaced squeezed states as seen in Fig. 4.1. Squeezing plays a significant role in Quantum Optics, since it can be thought of as a resource for several quantum effects such as entanglement, that will be discussed later in the thesis. Squeezed states can be created by pumping a nonlinear crystal with a bright laser.
4.1.4 Thermal States
Thermal states are defined as the Gaussian states that maximize the von Neumann entropy ofastate † ρˆG [see Eq. (3.8)] for fixed energy tr(ˆρGA A) =n ¯, where n¯ is the mean number of photons in the mode. This type of state is mixed and represents a quantum field which has no phase information. A single-mode thermal state can be written in the Fock basis as
∞ n th 1 X n¯ ρˆG (¯n) = |n⟩⟨n|, (4.48) 1 +n ¯ n=0 1 +n ¯ where the probability distribution of photons Pth ≡ {pn} for those states (depicted in Fig. 4.2) is given by n¯n p = . (4.49) n (1 +n ¯)n+1 The covariance matrix of a thermal state is given by V = (2¯n+1)1, so Eq. (4.48) can take the form
∞ n th 2 X V − 1 ρˆG (V ) = |n⟩⟨n|. (4.50) V + 1 n=0 V + 1 52 Gaussian Systems
The P function for a thermal state is given by
ZZ ∞ th 1 2 −|α|2/n¯ ρˆG (α) = d αe |α⟩⟨α|. (4.51) πn¯ −∞
4.1.5 Phase Rotation Operator
Another important Gaussian operation in the electromagnetic field is the phase rotation operator R(φ), which is defined as R(φ) ≡ exp iφA†A . (4.52)
Applied on a bosonic operator, phase rotation induces the following unitary transformation
R(φ)†AR(φ) = eiφA, (4.53) and the corresponding symplectic transformation on the quadrature operators is given by
cosφ −sinφ ΣR(φ) = . (4.54) sinφ cosφ
A phase rotation applied on a Gaussian state is responsible for rotating the contour in the graph in Fig. 4.1 by an angle φ around the origin of the plot. In continuous-variable systems, the phase is usually defined with respect to a local oscillator, and it can be implemented by increasing theoptical path length of the beam compared to the local oscillator. So far we saw single-mode operations and states, however we can also define operations that act on two or even multiple modes of the system. Below we present some of the most important ones.
4.1.6 Beam Splitter Operator
The beam splitter operator B(θ) is a Gaussian operation applied on a pair of modes, and it is defined as
h † † i B(θ) ≡ exp θ(A1A2 − A1A2) , (4.55) where A1 and A2 are the annihilation operators for each mode, and θ is a parameter related with the transmissivity of the beam splitter, i.e., τ = cos2 θ ∈ [0,1]. Applied on a bosonic operator the beam splitter operation induces the following unitary transfor- mation: √ √ A1 τ − 1 − τ A1 → √ √ , (4.56) A2 1 − τ τ A2 and the corresponding symplectic transformation on the quadrature operators is given by
√ √ τ12 − 1 − τ12 ΣB(θ) = √ √ . (4.57) 1 − τ12 τ12 4.1 Gaussian States and Operations 53
4.1.7 Two-mode Squeezing
The single-mode squeezing operation defined before can be extended to a couple of modes as follows
r S (r) ≡ exp A A − A†A† , (4.58) 2 2 1 2 1 2 where r is the squeezing parameter.
Applying a two-mode squeezer onto a couple of vacua, i.e., S2(r)|00⟩, we create the two-mode squeezed vacuum state, which in the Fock basis is given by
q ∞ |TMSV⟩ = 1 − χ2 X (−χ)n|nn⟩, (4.59) n=0 with χ = tanhr ∈ [0,1). In the limit of infinite squeezing, i.e., χ → 1, we have the well-known Einstein-Podolski-Rosen (EPR) state [94]. The unitary transformation that a two-mode squeezing operation induces in the field operators is the following † † S2(r)AiS2(r) = Ai coshr − Aj sinhr , (4.60) and the corresponding symplectic transformation on the quadrature operators is given by
coshr 0 sinhr 0 0 coshr 0 −sinhr Σ (r) = . (4.61) S2 sinhr 0 coshr 0 0 −sinhr 0 coshr
T The covariance matrix of the two-mode squeezed vacuum is given by ΣS2 (r)ΣS2 (r) = ΣS2 (2r). The way we can generate two-mode squeezing is by pumping a nonlinear crystal in the non-degenerate optical parametric amplifier regime. Then we can create a pair of photons in two different modes, known as the signal and the idler. Note that a two-mode squeezing operation is equivalent to applying a balanced beam-splitter on two single-mode and orthogonal to each other squeezed operations.
N-mode Squeezing
The squeezing operation can be extended in multiple modes [95] as follows
h i SN (r) ≡ exp r(Wˆ + − Wˆ −) , (4.62) where
N N ˆ 2 − N X †2 2 X † † W+ = Ai + Ai Aj , (4.63a) 2N i=1 N i Applying the N-mode squeezing operation on the bosonic operators yield N † 2 − N † 2 X † SN (r) AiSN (r) = coshrAi + sinhr Ai + Aj . (4.64) N N j̸=i 4.2 Symplectic Analysis of Gaussian States In this section we will discuss some important mathematical representations of multi-mode Gaussian states, and their implications. 4.2.1 Decomposition of a Gaussian State Definition (Williamson’s form). As a consequence of the Williamson’s theorem [90, 96], for every covariance matrix V [see Eq. (3.50)] of an N-mode Gaussian state, there exists a unique (up to a permutation of the symplectic spectrum) symplectic transformation Σ such that N M T V = Σ νi12 Σ , (4.65) i=1 with νi ⩾ 1 called the symplectic eigenvalues of V . Eq. (4.65) is called the Williamson’s form of a covariance matrix. The symplectic eigenvalues are given by the absolute value of the eigenvalues of the matrix iΩV , while the symplectic transformation Σ can be constructed following a method derived in Ref. [97]. Using the symplectic eigenspectrum {νi} we can write the following two invariants for the covariance matrix V , i.e., N X 2 ∆V = νi , (4.66) i=1 and N Y 2 detV = νi . (4.67) i=1 The purity of a Gaussian state is then given by 1 µ = √ , (4.68) detV and the von Neumann entropy takes the form [98] N X S(ˆρG) = h(νi), (4.69) i=1 with the auxiliary function h(·) defined as x + 1 x + 1 x − 1 x − 1 h(x) ≡ log − log . (4.70) 2 2 2 2 2 2 4.2 Symplectic Analysis of Gaussian States 55 Thermal State Decomposition Based on the Williamson’s form of a state, any Gaussian state can be written as N ¯ ¯ O th νi − 1 † ¯† ρˆG(Q,V ) = D Q U ρG U D Q , (4.71) i=1 2 where D Q¯ is the displacement operator and U is a unitary transformation, respectively, on a thermal NN th νi−1 state i=1 ρG 2 . The above decomposition is called the thermal decomposition of a state ρˆG. Bloch-Messiah Decomposition In the previous section we discussed various Gaussian operations. A Gaussian operation is called passive when the total energy or the photon number is conserved, which implies that the corresponding symplectic transformation is trace preserving, i.e., tr ΣV ΣT = tr(V ). (4.72) Those kind of operations are described by orthogonal symplectic matrices, i.e., ΣT = Σ−1. When a Gaussian operation is not passive, i.e., it does not preserve the photon-number and the corresponding symplectic is not trace preserving, is called active. The Bloch-Messiah reduction (also called Euler decomposition)[99–101], which is the singular value decomposition of a symplectic transformation [see Eq. (2.52)], decomposes a Gaussian unitary transformation into the active and passive parts of it. In particular, for a unitary U we have N O U = UK US(ri) UL , (4.73) i=1 where UK and UL are symplectic and orthogonal transformations, and US(ri) denotes the squeezing unitary operation. The corresponding symplectic transformation Σ under the Bloch-Messiah reduction takes the form N M Σ = ΣK ΣS(ri) ΣL , (4.74) i=1 where, analogously, ΣK and ΣL represent symplectic and orthogonal transformations, and ΣS(ri) is the symplectic transformation of squeezing. In Fig. 4.4, we graphically represent the Bloch-Messiah reduction for a Gaussian state. 4.2.2 Purification of a Gaussian State Given the symplectic spectrum of a Gaussian state and the corresponding thermal state decomposition we can also derive a simple expression for the purification of a mixed state. In particular, let ushavea AB ¯ Gaussian state ρˆG (Q,V ), that is mixed. We can always introduce an additional reference system of 56 Gaussian Systems Fig. 4.4 Bloch-Messiah decomposition. In this figure an arbitrary symplectic transformation Σ is graphically represented in the Bloch-Messiah (or Euler) reduction, by its main components. The operations ΣK and ΣL are symplectic and orthogonal transformations that represent the passive elements, while the array of ΣS(ri) are squeezing operations that represent the active elements of a symplectic transformation Σ. N modes and make it pure as follows AB ¯ ABE ¯′ ′ ρˆG (Q,V ) = trE[ˆρG (Q ,V )], (4.75) ABE ¯′ ′ where ρˆG (Q ,V ) is a pure Gaussian state for the composite system, with ′ VSC V = . (4.76) CT ΣT V ⊕ ⊕ LN 1 The symplectic operation Σ is given by Eq. (4.65), V = i=1 νi , and N M q 2 C = νi − 1Z, (4.77) i=1 with Z = diag(1,−1). 4.2.3 Two-mode Gaussian States A zero-mean valued N-mode Gaussian state can be fully represented by a covariance matrix V , which 1 in general has 2N × 2N elements given by vij ≡ 2 ⟨{∆Qi,∆Qj}⟩ [see Eq. (3.50)]. In this section, we focus on the special case of two-mode states, where the covariance matrix takes the full form 2 1 1 1 ⟨(∆XA) ⟩ 2 ⟨{∆XA,∆PA}⟩ 2 ⟨{∆XA,∆XB}⟩ 2 ⟨{∆XA,∆PB}⟩ 1 2 1 1 2 ⟨{∆PA,∆XA}⟩ ⟨(∆PA) ⟩ 2 ⟨{∆PA,∆XB}⟩ 2 ⟨{∆PA,∆PB}⟩ V = , (4.78) 1 1 2 1 2 ⟨{∆XB,∆XA}⟩ 2 ⟨{∆XB,∆PA}⟩ ⟨(∆XB) ⟩ 2 ⟨{∆XB,∆PB}⟩ 1 1 1 2 2 ⟨{∆PB,∆XA}⟩ 2 ⟨{∆PB,∆PA}⟩ 2 ⟨{∆PB,∆XB}⟩ ⟨(∆PB) ⟩ 4.2 Symplectic Analysis of Gaussian States 57 with ∆Xi ≡ Xi − ⟨Xi⟩. (4.79) Thus, we can write Eq. (4.78) as follows 2 1 ⟨XA⟩ 2 ⟨{XA,PA}⟩ ⟨XAXB⟩⟨XAPB⟩ 1 2 2 ⟨{PA,XA}⟩ ⟨PA⟩⟨PAXB⟩⟨PAPB⟩ V = + 2 1 ⟨XBXA⟩⟨XBPA⟩⟨XB⟩ 2 ⟨{XB,PB}⟩ 1 2 ⟨PBXA⟩⟨PBPA⟩ 2 ⟨{PB,XB}⟩ ⟨PB⟩ 2 ⟨XA⟩ ⟨XA⟩⟨PA⟩⟨XA⟩⟨XB⟩⟨XA⟩⟨PB⟩ 2 ⟨PA⟩⟨XA⟩⟨PA⟩ ⟨PA⟩⟨XB⟩⟨PA⟩⟨PB⟩ − . (4.80) 2 ⟨XB⟩⟨XA⟩⟨XB⟩⟨PA⟩⟨XB⟩ ⟨XB⟩⟨PB⟩ 2 ⟨PB⟩⟨XA⟩⟨PB⟩⟨PA⟩⟨PB⟩⟨XB⟩⟨PB⟩ In a more compact way, V can also take the form AC V = , (4.81) CT B where A = AT , B = BT , and C are 2 × 2 real matrices. The symplectic eigenvalues of this covariance matrix, based on the Williamson’s form, are given by [102] v u q 2 u∆V ± ∆ − 4detV ν = t V , (4.82) ± 2 with ∆V = detA + detB + 2detC. (4.83) The global purity of the state is given by √ µ = 1/ detV, (4.84) while local purities by √ √ µa ≡ 1/ detA, µb ≡ 1/ detB, (4.85) respectively. The uncertainty principle for a two-mode Gaussian state can take the following equivalent forms [103, 97] V > 0, detV ⩾ 1, ∆V − detV ⩽ 1. (4.86) A covariance of a two-mode Gaussian state can be transformed into a matrix with the least amount of non-zero elements under local operations. We call those matrices, the standard form of the covariance matrix, and it will be shown how much they simplify the analysis of those systems in the following chapters. 58 Gaussian Systems Definition (Standard form of a covariance matrix). Through local operations the covariance matrix V in Eq. (4.81) of a zero mean-valued two-mode Gaussian state takes the following form [104, 105] a 0 c+ 0 sf 0 a 0 c− V ≡ , (4.87) c+ 0 b 0 0 c− 0 b where [106] √ √ a = max{ detA, detB}, (4.88a) √ √ b = min{ detA, detB}, (4.88b) v u q ua2b2 − detV + (detC)2 + (a2b2 − detV + det2 C)2 − 4a2b2 det2 C c = t , (4.88c) + 2ab √ 2abdetC c = , (4.88d) − r q a2b2 − detV + det2 C + (a2b2 − detV + det2 C)2 − 4a2b2 det2 C which implies that a ⩾ b and c+ ⩾ |c−| ⩾ 0 . The local operations that lead to the standard form is just a sequence (for each mode) of a phase rotation, single-mode squeezing, and another phase rotation. The symplectic eigenvalues ν± of a covariance matrix in the standard form are given by v u 2 2 q 4 2 2 2 2 4 2 ua + b + 2c+c− − a − 2a (b − 2c+c−) + 4ab(c+ + c−) + b + 4b c+c− ν = t , (4.89a) − 2 v u 2 2 q 4 2 2 2 2 4 2 ua + b + 2c+c− + a − 2a (b − 2c+c−) + 4ab(c+ + c−) + b + 4b c+c− ν = t . (4.89b) + 2 4.3 Gaussian Measurements The general theory of measurements on a quantum state was introduced in Sec. 3.2.3. Here we focus on Gaussian measurements, which are defined as the measurements on Gaussian states with outcomes that have Gaussian statistics. In particular, let us have a Gaussian measurement that is made on N modes of an N + M Gaussian state where N,M ⩾ 1. The classical outcome from this kind of measurement forms a Gaussian distribution, and the unmeasured M modes are left in a Gaussian state. 4.3 Gaussian Measurements 59 4.3.1 Homodyne Detection Given a quantum state ρˆ, homodyne detection is a measurement process for the generalized quadrature (φ) Q = cosφX + sinφP , that yields a probability distribution Pφ ≡ {pφ} with pφ = ⟨qφ|ρˆ|qφ⟩, (4.90) where the eigenvectors |qφ⟩ are defined through the following equation (φ) Q |qφ⟩ = qφ|qφ⟩. (4.91) Let us have a mode that we want to measure that is associated with the bosonic operator 1 A = Q(0) + iQ(π/2) . (4.92) 1 2 1 1 We mix this mode with a reference mode associated with the bosonic operator 1 A = Q(0) + iQ(π/2) , (4.93) 2 2 2 2 into beam splitter B of transmissivity τ, and the outcome is √ √ A3 = τA1 − 1 − τA2 . (4.94) † Measuring the number of photons on the outcome mode using the number operator N = A3A3 we have † † † q h † † i ⟨A3A3⟩ = τ⟨A1A1⟩ + (1 − τ)⟨A2A2⟩ − τ(1 − τ) ⟨A1⟩⟨A2⟩ + ⟨A1⟩⟨A2⟩ . (4.95) q † If the reference mode is in a coherent state of large amplitude, i.e., ⟨A2A2⟩ = |β| ≫ 1, then we can neglect the first term in Eq. (4.95) and write it as † 2 q (0) ⟨A3A3⟩ ≈ (1 − τ)|β| + |β| τ(1 − τ)⟨Q1 ⟩. (4.96) As it is clear, if we do not take into account the contribution of the reference mode, the mean (0) number of photons of the outcome is proportional to the mean value of the Q1 = X1 quadrature. The variance of the photocurrent, i.e., † 2 q (0) Var A3A3 ≈ (1 − τ)|β| + |β| τ(1 − τ)Var Q1 , (4.97) (0) is also proportional to the Q1 quadrature. If we rotate the phase of the reference mode by π/2 the (π/2) mean photo-current becomes proportional to the Q1 = P1 quadrature. Schematically the homodyne scheme is depicted in Fig. 4.5 (a). Homodyne detection is a multi-purpose “tool” in optical quantum information since it can be used in various tasks related to quantum tomography [107]. As an example one can experimentally 60 Gaussian Systems Fig. 4.5 Gaussian measurements. In figure (a) we sketch the main components of a homodyne detection measurement, i.e., the input mode associated with a bosonic operator A1 that we mix in a beam splitter of transmissivity τ with the reference mode associated with a bosonic operator A2, the output mode A3 on which we perform the photo-detection, and the secondary output mode A4 that we are not concerned about. In figure (b) we depict the heterodyne measurement detection. As we see it is a dual homodyne detection scheme applied on the output modes of the initial mode A1 that is mixed on a beam splitter with an ancillary vacuum state |0⟩. reconstruct the covariance matrix of two-mode Gaussian states using a single homodyne detector [108, 106] 4.3.2 Heterodyne Detection A more general Gaussian measurement, is the so-called heterodyne detection [109]. In this type of measurement, we mix the mode that we want to measure with an ancilla vacuum state on a balanced beam splitter, and then we perform a homodyne detection on both outcomes. Heterodyne detection corresponds to a POVM projector onto coherent states, i.e., E(α) ≡ π−1/2|α⟩⟨α|. It is schematically depicted in Fig. 4.5 (b). 4.3.3 Partial Measurements We now focus on the case when we have an N-mode Gaussian state with a covariance matrix V in the form of Eq. (4.81), and we are interested in measuring through homodyne or heterodyne detection only one mode out of it. Let us call A the (N − 1)-mode block of the covariance matrix of the state, and B the single mode that we are about to measure. The covariance matrix will be transformed depending on the type of the measurement we apply on it as follows [110, 111]: • Homodyne detection. The reduced (N − 1)-mode covariance matrix of the measured state is given by V = A − C(ΠBΠ)−1CT , (4.98) where Π ≡ diag(1,0), (ΠBΠ)−1 is a pseudo-inverse matrix since (ΠBΠ) is singular, and C is the sub-matrix in Eq. (4.81) that is associated with the correlations between A and B. 4.4 Distinguishing Gaussian States 61 • Heterodyne detection. The reduced (N − 1)-mode covariance matrix of the measured state is given by V = A − C (B + 1)−1 CT . (4.99) 4.4 Distinguishing Gaussian States In Sec. 3.3, we discussed how can we measure the “closeness” of two general quantum states, using the measures of trace distance, fidelity, and relative entropy. Even though trace distance constitutes a metric there has not been derived an analytical expression to compute it in Gaussian systems yet. Below we will focus on fidelity and relative entropy, since for both of them we can provide aclosed analytical expression for the distinguishability between two N-mode Gaussian states. Every Gaussian state can be written as [112] 1 1 ρˆ(Q,V¯ ) = exp − (Q − Q¯)T G(Q − Q¯) , (4.100) Zρˆ 4 where V + iΩ!1/2 Z = det , (4.101) ρˆ 2 is the normalization factor, and G = 2iΩcoth−1(iV Ω), (4.102) is the so-called Gibbs Matrix. 4.4.1 Fidelity Let us have two quantum states ρˆ1 and ρˆ2. Their fidelity in general is given by rq q ! F(ˆρ1,ρˆ2) ≡ tr ρˆ1ρˆ2 ρˆ1 . (4.103) For two Gaussian states, i.e., ρˆ1(Q¯1,V1) and ρˆ2(Q¯2,V2), the expression of fidelity is given by [112] 1 F(ˆρ ,ρˆ ) = F (V ,V )exp − (Q¯ − Q¯ )T (V + V )−1(Q¯ − Q¯ ) , (4.104) 1 2 0 1 2 8 1 2 1 2 1 2 with 1/2 QN q 2 k=1 wk + wk − 1 F0(V1,V2) = q , (4.105) 4 det(V1+V2) 2 where wk denote the eigenvalues of the matrix W = −V iΩ. For the case of two two-mode Gaussian states ρˆG1 and ρˆG2 , the expression for the fidelity can be found in Ref. [113]. 62 Gaussian Systems 4.4.2 Relative Entropy In general the relative entropy between two states ρˆ1 and ρˆ2 is given by S(ˆρ1∥ρˆ2) ≡ −tr(ˆρ1 log2 ρˆ2) + tr(ˆρ1 log2 ρˆ1). (4.106) ¯ Assuming that the states are Gaussian with mean values Q1/2 and covariance matrices V1/2, respectively, we can express relative entropy as [114] S(ˆρGi ∥ρˆGj ) = Ξij − Ξii , (4.107) where (Q¯ −Q¯ )T G (Q¯ −Q¯ ) lndet Vi+iΩ + tr ViGi + j i i j i Ξ ≡ 2 2 2 . (4.108) ij 2ln2 It is worth mentioning that the Gibbs-matrix G becomes singular for pure states or, more generally, for mixed states with lowest symplectic eigenvalues equal to ν− = 1 [see Eq. (4.89a)]. For that cases, the functional Ξii should be computed as a limit, or in a more straightforward way just by the von Neumann entropy tr(ˆρ1 log2 ρˆ1) by substituting the symplectic eigenvalues of ρˆ1. This distance measure is the basis of a significant entanglement measure, i.e., relative entropy of entanglement, that will be introduced in the next chapter. 4.5 Gaussian Channels Gaussian channels represent a Gaussian quantum operation, i.e., they transform a Gaussian input state to another Gaussian output state. Definition (Gaussian channels). The action of an N-mode Gaussian channel G on a Gaussian state ρˆ(Q,V¯ ) is a quantum operation that transforms the state as follows [115, 116] Q¯ → X Q¯ + ⃗q, (4.109a) V → X V X T + Υ, (4.109b) where ⃗q ∈ R2N , while X and Υ = ΥT are 2N × 2N real matrices, that satisfy the complete positivity condition, i.e., T Υ + iΩ − iX ΩX ⩾ 0. (4.110) 4.5.1 One-mode Gaussian Channels One-mode Gaussian channels are ubiquitous in Quantum Information, since they model the optical communication through optical fibers and free space space. They are also characterized byEq.(4.109), 4.5 Gaussian Channels 63 where now ⃗q ∈ R2 and X and Υ satisfy T 2 Υ = Υ ⩾ 0, and detΥ ⩾ (detX − 1) . (4.111) Note that a Kraus representation for single-mode bosonic Gaussian channels can also be formulated [117]. A single-mode Gaussian channel can be fully characterized through three quantities: 1. the generalized transmissivity, i.e., τ ≡ detX , (4.112) 2. the rank of the channel, i.e., rank(G) ≡ min{rank(X ),rank(Υ)}, (4.113) 3. the thermal number n¯, i.e., √ detΥ for τ = 1 √ n¯ ≡ detΥ 1 . (4.114) 2|1−τ| − 2 for τ ̸= 1 4.5.2 Phase-Insensitive Gaussian Channels One of the most important classes of Gaussian channels are the so-called phase-insensitive (covariant), which are defined [118] as √ X = τ1, and Υ = v1, (4.115) where τ and v are scalars representing the transmissivity/gain and the added noise, respectively. More specifically we have the following special cases: • Lossy channel, L, that attenuates the incoming signal by a factor of 0 ⩽ τ < 1 adding at the same time noise given by v = (1 − τ)ε, where ε = 2¯n + 1. For thermal lossy channel we have ε > 1, and for pure lossy channel ε = 1, respectively. • Amplifier channel, A, that amplifies the state by a factor of τ > 1 adding also noise given by v = (τ − 1)ε. Analogously to the lossy channels, we have a thermal amplifier with ε > 1, and for pure amplifier channel ε = 1, respectively. • Classical additive-noise channel, N , that is the Gaussian channel that adds to the incoming signal an extra amount of noise v =n ¯. Those types of channels are an asymptotic case of either lossy or amplifier channels where τ ≈ 1. • Identity channel, 1, representing the trivial case when practically there is no channel and the input and output signal is the same state, so we have τ = 1 and v = 0. In Fig. 4.6, we graphically present this class of channels, while their corresponding Stinespring dilation [119] is plotted in Fig. 4.7. 64 Gaussian Systems Fig. 4.6 Gaussian phase-insensitive channels. The different classes of phase-insensitive Gaussian channels are presented in this graph. With blue we have the lossy channels, L, and the dark blue line indicates the specific case of pure lossy channels. With brown we have the amplifier channels, A, and, respectively, the dark brown line represents the pure amplifier channels. The central vertical grey line corresponds to the classical additive noise channels, N , and the green dot indicates the identity channel, 1. Channels above the dashed line are entanglement-breaking channels, i.e., v ⩾ 1 + |τ|, and channels below the dark blue and brown lines are non-physical. All quantities plotted are dimensionless. Quantum Capacities of one-mode Gaussian Channels In general, the calculation of the quantum capacity is a remarkably involved task but for one-mode Gaussian channels with τ ̸= 1 we have the following lower bound [115, 120] τ Q(M) max 0,log − h(ε) , (4.116) ⩾ 2 1 − τ where h(ε) can be calculated through Eq. (4.70). Interestingly, for one-mode degradable Gaussian channels, the bound in Eq. (4.116) becomes tight [121]. Examples of degradable channels with non-zero quantum capacity are the pure lossy and pure amplifier channels. Another case when the above bound becomes tight but also vanishes is the thermal lossy channels with transmissivity τ ⩾ 1/2. 4.5 Gaussian Channels 65 Fig. 4.7 Stinespring dilation for phase-insensitive Gaussian channels. On the panel (a) we have a lossy channel L that is modeled through a beamsplitter B with transmissivity 1 > τ = cosθ2. The complementary channel Le is also a lossy channel with transmissivity (1−τ). On the panel (b) we have an amplifier channel A that is modeled through a two-mode squeezer S2 with gain τ = coshr > 1. The complementary channel Ae is a contravariant channel with negative transmissivity. If we substitute the thermal states ρˆth with the vacuum |0⟩ we get a pure lossy and a pure amplifier/cotnravariant channel, respectively. The identity channel 1 is the case where τ = 1 (for either lossy or amplifier channel), so there is no interaction with the environment and no noise isinduced. Finally, the classical additive-noise channel N is a limited case with τ ≈ 1 and a highly thermal state as an input. Classical Capacities of Gaussian phase-insensitive Channels The calculation of classical capacities is a well-known problem in Quantum Information, since it is related with some famous conjectures regarding additivity. In particular, there are four conjectures that are equivalent to each other [122–124], i.e., 1. additivity of the minimum entropy output of a quantum channel, 2. additivity of the classical (also called Holevo) capacity of a quantum channel, 3. additivity of the entanglement of formation, 4. strong superadditivity of the entanglement of formation, meaning that are either all true or all false. It was recently proven that the above conjectures are in general not true [125], but interestingly, for Gaussian phase-insensitive channels the first conjecture (so the rest as well) actually holds true, as proven in Ref. [126], where the authors have also analytically derived the classical capacities of this type of channels: • Lossy channel C(L) = h[τm¯ + (1 − τ)¯n] − h[(1 − τ)¯n] . (4.117) • Amplifier channel C(A) = h[τm¯ + (τ − 1)(¯n + 1)] − h[(τ − 1)(¯n + 1)] . (4.118) • Classical additive-noise channel C(N ) = h[m ¯ + v] − h[v] , (4.119) where m¯ is mean number of photons per input mode, and h(·) the auxiliary function defined in Eq. (4.70). 66 Gaussian Systems Fig. 4.8 Thermal channel decomposition. Every thermal channel G can be decomposed as a sequence of a pure loss channel Lp followed by a pure amplifier Ap, i.e., G ≡ Ap ◦ Lp, or reversely (if it is non-entangling breaking), as a pure amplifier Ap followed by a pure loss channel Lp, i.e., G ≡ Lp ◦Ap. Decompositions of Gaussian phase-insensitive Channels It has been shown [127] that any phase-insensitive thermal Gaussian channel G with transmissivity/gain τ and noise v can be decomposed into a sequence of a pure loss channel Lp with transmissivity τℓ followed by a pure amplifier Ap with gain τa as follows: 1 + τ + v 2τ G ≡ A ◦ L ⇐⇒ τ = , and τ = . (4.120) p p a 2 ℓ 1 + τ + v For non-entangling-breaking channels [128, 129], the same thermal channel G can be also decom- posed reversely [130], i.e., a pure amplifier followed by a pure loss channel as follows: 2τ 1 + τ − v G ≡ L ◦ A ⇐⇒ τ = , and τ = . (4.121) p p a 1 + τ − v ℓ τ The concatenation of quantum channels has an interesting implication on their quantum capacities. In particular, given two quantum channels G1 and G2, we have the bottleneck inequality [115], i.e., Q(G1 ◦ G2) ⩽ min[Q(G1),Q(G2)] , (4.122) which has been used for the quantification of the quantum capacities of Gaussian channels inRef.[130]. Another interesting property of decomposable quantum channels is that the concatenation of two quantum channels G1 and G2 with zero quantum capacities can end up to a composite quantum channel G1 ◦ G2 with positive quantum capacity [131]. This property is called superactivation, and a specific example for Gaussian channels can be found in Ref. [132]. Chapter 5 Entanglement Theory “People often say the cat is both alive and dead but it can’t really be alive and dead at the same time, that doesn’t make sense, nor is it a semi-comatose zombie like my PhD students, since in these experiments we observe it either fine and healthy or completely expired” — Terry Rudolph Entanglement is a non-classical physical property, emerging from the superposition of composite quantum systems. Theoretically it can be described as the inability to separate a global quantum state of a composite system into a product of individual subsystems, and experimentally it is manifested through the correlations of the observables of different subsystems, which cannot be classically reproduced. The peculiar nature of this quantum property was initially discussed in 1935 by E. Schrödinger [133] who also is responsible for its name, followed in the same year by A. Einstein, B. Podolsky, and N. Rosen (EPR) [94] who used entanglement as a counterexample to argue against the adequacy of the Quantum Theory. More specifically, EPR proposed a thought experiment involving the wavefunction of a pure entangled state, which as they concluded does not contain the complete information about the system1 assuming (i) locality2 and (ii) physical reality3. Little progress was made on this topic for decades, until 1964 when J. Bell in his seminal paper [135] showed that Quantum Theory is not a local-realistic theory. The so-called “Bell correlations” that can arise via measurements on pure entangled states [136], and have no classical analog, revived the philosophical discussions on the nature of Quantum Mechanics, while at the same time sparked the interest of the community from an operational point of view, i.e., discussion on how entanglement can be harnessed for physical applications. 1“Every element of physical reality must have a counterpart in the physical theory”[94] 2 “The real factual situation of system S2 is independent of what is done with the system S1, which is spatially separated from the former.”[134] 3“If, without in any way disturbing the system, we can predict with certainty the value of a physical quantity, then there exists an element of reality corresponding to this physical quantity”[94] 68 Entanglement Theory Apart from the fundamental issues in physics and the deep philosophical problems that entangle- ment is connected with, in the field of Quantum Information and Quantum Computation it is usually thought of as a resource for several tasks. In this chapter, we discuss how we can detect whether a system has this property or not, and different methods through which we can assign a value to it. Quantification of entanglement is a difficult task, since various measures exist that assign avalueto different aspects of the property, and most of them lack an analytical expression. Note on references. This chapter is mainly based on Refs.[137–141]. 5.1 Separability and Entanglement Given two quantum states, i.e., ρˆA ∈ H A and ρˆB ∈ H B, we can always define a product state (or uncorrelated state) in H AB as ρˆ =ρ ˆA ⊗ ρˆB , (5.1) or we can even consider convex combinations of those, i.e., X A B ρˆ = piρˆ ⊗ ρˆ , (5.2) i which we call separable states. The product states do not form a convex set, but the separable states do, so a mixture of separable states is also a separable state. On the other hand, given a state ρˆ ∈ H AB, we cannot always decompose it through Eq. (5.2). Definition (Entangled state). A bipartite quantum state ρˆ ∈ H AB is called entangled if X A B ρˆ ̸= piρˆ ⊗ ρˆ . (5.3) i Any state that is not a product state is called correlated. It is important to note that we used the terms “correlated” and “uncorrelated” above to refer solely to quantum correlations, since states in the form of Eqs. (5.1) and (5.2) may contain classical correlations too (keep also in mind that classically correlated states are not necessarily classical themselves, e.g., discordant states). The study of classical/non-classical correlations and classical/non-classical states in a unified form is a new and interesting field in Quantum Information (see Refs. [142, 143] for a general discussion). Entanglement can also be defined in multipartite systems as well, however, then we haveto distinguish between cases where the whole system is inseparable or just parts of it. We refer the reader to Refs. [144, 145] for a recent discussion on classification and quantification of multipartite entanglement, but this is an issue beyond the scope of this thesis, so we will not worry about it. 5.2 Separability Criteria 69 5.2 Separability Criteria Given a quantum state ρˆ ∈ H AB, the question of whether it is separable or entangled is typically not so easy to answer, and thus there exist various methods to detect the existence of entanglement in a system. Pure states are significantly easier to characterize so we start by discussing those. 5.2.1 Pure States Any pure bipartite state |ψ⟩ ∈ H AB can be decomposed through the so-called Schmidt decomposition, i.e., X q |ψ⟩ = λi|uivi⟩, (5.4) i with λi known as the Schmidt coefficients, which are unique for each state, and {|ui⟩} and {|vi⟩} being A B orthonormal bases in H and H , respectively. The number of non-zero values λi is called Schmidt rank, and is equal to one for separable states, and larger than one for entangled states. Note the Schmidt decomposition can be extended into multipartite pure states as well. For an arbitrary pure state |ψ⟩ we can evaluate the Schmidt coefficients through its reduced state (tracing over any of the two subsystems). In particular, we have X ρˆA = trB|ψ⟩⟨ψ| = λi|ui⟩⟨ui|, (5.5a) i X ρˆB = trA|ψ⟩⟨ψ| = λi|vi⟩⟨vi|. (5.5b) i As we see, the Schmidt coefficients are given by the eigenvalues of the reduced density matrix ρˆA/B. Since separability requires that exactly one Schmidt coefficient is non-zero, we can relate the entanglement of a pure state |ψ⟩ to the purity of the reduced density matrices as follows A B 2 |ψ⟩ = |ψ⟩ ⊗ |ψ⟩ ⇐⇒ trρˆA/B = 1, (5.6a) A B 2 |ψ⟩ ̸= |ψ⟩ ⊗ |ψ⟩ ⇐⇒ trρˆA/B < 1, (5.6b) which is a necessary and sufficient criterion for entanglement in pure states. A state is called maximally entangled if all its Schmidt coefficients are equal to each other138 [ ]. An example of a maximally entangled state is the well-known Bell state, defined as |00⟩ ± |11⟩ |01⟩ ± |10⟩ |Φ±⟩ ≡ √ , |Ψ±⟩ ≡ √ , (5.7) 2 2 which is characterized by two equal Schmidt coefficients with value1 “ /2” each. The two-mode squeezed vacuum, defined in Eq. (4.59), is also written in the Schmidt decomposition with infinite Schmidt rank. Note that the Schmidt coefficients in Eq. (4.59) become equal to each other in the limit of χ → 1, i.e., the EPR state, so in this sense they are the extension of Bell state to infinite dimensions. 70 Entanglement Theory 5.2.2 Mixed States For mixed states, the purity of the reduced state does not constitute a criterion for detecting entangle- ment in a state, and thus we have to develop more sophisticated methods. Positive Maps AB Let us have a mixed bipartite state ρˆ∈ H and a positive map M, i.e., M(ˆρ) ⩾ 0. For any separable P A B state, i.e., ρˆ = i piρˆ ⊗ ρˆ , the extended map [M ⊗ 1] must yield another positive operator, i.e., X A B [M ⊗ 1](ˆρ) = piM ρˆ ⊗ ρˆ ⩾ 0. (5.8) i The above observation means that, if the operator [M ⊗ 1](ˆρ) has at least one negative eigenvalue, the state ρˆ is entangled. In a more compact way, for any positive map M, we have the following separability criterion X A B ρˆ = piρˆ ⊗ ρˆ =⇒ [M ⊗ 1](ˆρ) ⩾ 0, (5.9a) i X A B ρˆ ̸= piρˆ ⊗ ρˆ ⇐= [M ⊗ 1](ˆρ) < 0, (5.9b) i meaning that, the positivity of [M ⊗ 1](ˆρ) is a necessary condition for separability, while on the other hand the negativity of [M ⊗ 1](ˆρ) is a sufficient condition for entanglement. A specific example of such a positive map M is the transpose map T , i.e., T (ˆρ) ≡ ρˆT , defined in Eq. (2.30). The extended map [T ⊗ 1] is clearly the partial transposition of an operator defined in Eq. (2.60), i.e., PT (ˆρ) ≡ [T ⊗ 1](ˆρ) ≡ ρˆΓ, and the separability criterion takes the name PPT criterion [146]. Note that, in general, the positivity of the partial transpose is not a sufficient condition for separability, since there exist entangled states with positive partial transpose, called bound entangled states [147, 148], because their entanglement cannot be distilled4 into maximally entangled states. However, for low dimensional systems, e.g., 2×2 or 2×3 the criterion is both necessary and sufficient [149] (the case of Gaussian systems will be also discussed later). Entanglement Witnesses We call entanglement witness a Hermitian operator W that yields a non-negative mean value with respect to any separable state ρˆs, i.e., tr(W ρˆs) ⩾ 0, (5.10) and negative value for at least one entangled state ρˆe, i.e., tr(W ρˆe) < 0. (5.11) 4The term entanglement distillation will be properly defined later in this chapter. 5.2 Separability Criteria 71 Fig. 5.1 Entanglement witness. The line tr(W ρˆ) = 0 signifies a line (hyperplane) corresponding to the witness W . All states located to the left of the hyperplane (or belonging to it), provide a non-negative value for the witness, i.e., tr(W ρˆs) ⩾ 0, and states located to the right are entangled states detected by this witness, i.e., tr(W ρˆe) < 0. Note that the extremal points of the separable states, i.e., pure product states, are also extremal points of the set of all states (located on the border of the total set of states). Entanglement witnesses offer an easy way to detect entanglement, since they are directly mea- surable quantities. It has been proven [149] that for any entangled state ρˆe there always exists an entanglement witness detecting it, and the Hahn-Banach separation theorem gives an interesting geometrical perspective to them, as we can schematically see in Fig. 5.1. It is also worth noting that entanglement witnesses constitute a more general approach for detecting entanglement compared to the “positive maps” method discussed before, and that can be seen as follows. Let us have a positive map M, and its extension [M ⊗ 1] yields a negative operator when it is applied on a state ρˆ. That means that the operator [M ⊗ 1](ˆρ) has an eigenvector |φ⟩ with a negative eigenvalue λ < 0. Then we can construct an entanglement witness using the observable W = [M† ⊗ 1](ˆρ). For an in-depth discussion on various separability criteria and ways to construct entanglement witnesses see Ref. [141]. 5.2.3 Separability in Gaussian Systems Let us have a zero-mean valued N-mode bipartite Gaussian state ρˆG with covariance matrix V , the partial transposition over M modes of the state corresponds to Γ V ≡ (1A ⊕ TB)V (1A ⊕ TB), (5.12) where N−M M M M 1A = 12 , and TB ≡ Z, (5.13) k=1 k=1 with Z ≡ diag(1,−1) representing a transposition at the phase space level. The PPT criterion for Gaussian states takes the form [105, 150] X A B Γ ρˆG = piρˆG ⊗ ρˆG =⇒ V ⩾ 0, (5.14a) i X A B Γ ρˆG ̸= piρˆG ⊗ ρˆG ⇐= V < 0, (5.14b) i 72 Entanglement Theory meaning that if a state ρˆG is separable then the partial transpose of its covariance matrix is a positive semidefinite matrix, or in other words, if the V Γ < 0 the state is entangled. Equivalently we can check Γ Γ Γ the positivity/negativity of min{νk }, where {νk } is the symplectic eigenspectrum of the matrix V . Γ Γ In some restricted cases, the positivity min{νk } ⩾ 0 (negativity min{νk } < 0) is necessary and sufficient to separability (entanglement) of the Gaussian state. This happens for 1 × M Gaussian states [151] and for a particular class of N × M Gaussian states which are called bi-symmetric [152]. Hence, the PPT criterion is necessary and sufficient for two-mode Gaussian states. Two-mode Gaussian states For the special case of two-mode Gaussian states with a covariance matrix AC V = , (5.15) CT B the PPT criterion can be easily checked through the positivity of the lowest symplectic eigenvalue of the partially transposed covariance matrix v u q u∆Γ − (∆∆Γ )2 − 4detV min{νΓ} ≡ νΓ = t V V , (5.16) k − 2 where Γ ∆V = detA + detB − 2detC. (5.17) If the covariance matrix is in its standard form, i.e., a 0 c+ 0 sf 0 a 0 c− V = , (5.18) c+ 0 b 0 0 c− 0 b Γ with a ⩾ b and c+ ⩾ |c−| ⩾ 0, then ν− has the following expression r 2 4 2 2 q 4 2 2 2 2 a + b − 4b c+c− + b − 2c+c− − a − 2a (b + 2c+c−) + 4ab(c+ + c−) νΓ = √ . (5.19) − 2 Entanglement in two-mode Gaussian states can be experimentally manifested through the correla- tions between the observables of different subsystems. As an example, let us have two vacuum modes (0) (0) (0) (0) A and B, with quadrature field operators XA /PA and XB /PB , respectively. We can then squeeze the position quadrature of mode A and the momentum quadrature of mode B, and subsequently mix 5.3 Quantifying Entanglement 73 those two orthogonally squeezed vacuums in a (balanced) beam-splitter, getting the following modes: 1 r (0) −r (0) XA = √ e X + e X , (5.20a) 2 A B 1 −r (0) r (0) PA = √ e P + e P , (5.20b) 2 A B 1 r (0) −r (0) XB = √ e X − e X , (5.20c) 2 A B 1 −r (0) r (0) PB = √ e P − e P . (5.20d) 2 A B As discussed in Sec. 4.1.7, this state is called two-mode squeezed vacuum, and in the limit of infinite squeezing the position and momentum quadratures are perfectly correlated and anti-correlated, respectively, i.e., r → ∞ ⇒ XA − XB → 0, and PA + PB → 0. (5.21) which is the so-called EPR state. 5.3 Quantifying Entanglement Apart from witnessing entanglement in a given quantum system, it is also useful to quantify it, but assigning a value to this property is a highly non-trivial task. Given a quantum state ρˆ, our goal is to define a measure (function), i.e., E(ˆρ), that characterizes the entanglement of the system. Obviously, we must insist that this measure should vanish for separable states, while for entangled states should be positive. Also, since entanglement is a global property of the system, we expect that any local interaction should either decrease or keep this function constant. By local interaction we have to assume any kind of local manipulation of the system, which includes the case where the local operations in each subsystem are classically correlated with each other, also called Local Operations and Classical Communication (LOCC). 5.3.1 Local Operations and Classical Communication The most general local operation that is applied on one of the subsystems, for instance the subsystem A, of a bipartite state ρˆ ≡ ρˆ(0) is given by (0) (1) X 1 (0) † 1 ρˆ → ρˆ = (Ki ⊗ )ˆρ (Ki ⊗ ), (5.22) i P † with Ki being the Kraus operators that satisfy i Ki Ki = 1. Now we assume that a subsequent local operation (with corresponding Kraus operators Li) on the other subsystem, i.e., B, will be related to the outcomes of the first operation, so we have (1) (2) X 1 1 (0) † 1 1 † ρˆ → ρˆ = ( ⊗ Lij)(Ki ⊗ )ˆρ (Ki ⊗ )( ⊗ Lij). (5.23) ij 74 Entanglement Theory The next local operation (represented by Mi Kraus operators) again on the first subsystem A will give (2) (3) X 1 1 1 (0) † 1 1 † † 1 ρˆ → ρˆ = (Mijk ⊗ )( ⊗ Lij)(Ki ⊗ )ˆρ (Ki ⊗ )( ⊗ Lij)(Mijk ⊗ ), (5.24) ijk and after N such operations, we finally have a state (N) X 1 1 (0) † 1 1 † ρˆ = ( ⊗ Fijk···ℓ)···(Ki ⊗ )ˆρ (Ki ⊗ )···( ⊗ Fijk···ℓ), (5.25) ijk···ℓ where Fi are the Kraus operators of the final local operation. The sequence of those classically correlated local operations is called in short LOCC operations, which stands for local operations and classical communication, and is denoted by the symbol Λ. If an entanglement measure E is non-increasing under LOCC operations it is called monotonous under LOCC. It is also worth mentioning that separable states as defined in Eq. (5.2) can always be created by applying LOCC operations on product states. 5.3.2 Majorization Since LOCC operations cannot increase entanglement if a state ρˆ can be transformed into σˆ via LOCC operations then for an entanglement measure E we should have the following ordering E(ˆρ) ⩾ E(ˆσ). Thus, it is useful to be able to check if a state can be transformed into another through local operations. Let us have a pure bipartite state |ψ⟩ with Schmidt coefficients λ1 ⩾ λ2 ⩾ ··· ⩾ λn. We can transform [153] this state into another pure state |φ⟩ with Schmidt coefficients κ1 ⩾ κ2 ⩾ ··· ⩾ κn if and only if their Schmidt coefficients (ordered decreasingly) satisfy the following relationship λ1 ⩾ κ1 , (5.26a) 2 2 X X λi ⩾ κi , (5.26b) i=1 i=1 . . n n X X λi ⩾ κi . (5.26c) i=1 i=1 We then say that state |ψ⟩ majorizes state |φ⟩. We can also define the Schmidt vectors, i.e., ⃗ T T λ ≡ (λ1,λ2,··· ,λn) and ⃗κ ≡ (κ1,κ2,··· ,κn) , and express the above relationship in a compact way as ⃗λ ≻ ⃗κκ. (5.27) 5.3.3 Quantifying Entanglement in Pure States (Entanglement Entropy) Entanglement entropy is a measure of entanglement that for pure states satisfies all the previous requirements, and it is defined154 [ , 155] as the von Neumann entropy of either of its reduced states, 5.3 Quantifying Entanglement 75 i.e., A B X E(ˆρ = |ψ⟩⟨ψ|) ≡ S(ˆρ ) = S(ˆρ ) = − λi log2 λi , (5.28) i where λi are the Schmidt coefficients of the state ρˆ. Note that the entropy of entanglement of a pure state can also be expressed as half of the mutual information of the two subsystems, or half of the relative entropy of the state and the product state of its components, i.e., 1 1 E(ˆρ = |ψ⟩⟨ψ|) ≡ S(A:B) ≡ S(ˆρ∥ρˆA ⊗ ρˆB). (5.29) 2 2 5.3.4 Quantifying Entanglement in Mixed States Quantifying entanglement in mixed states is a considerably harder task, and the entropy of entanglement is not sufficient any more. Numerous measures have been defined in the literature, and eachofthem takes a different approach on the problem. Even though there is not a single measure for entanglement in mixed states, we can set a list of axioms that any potential entanglement measure should satisfy. An entanglement measure is a mapping E :ρ ˆ ∈ H → E(ˆρ) ∈ R+, that satisfies the following postulates: 1. E(ˆρ) vanishes if and only if the state ρˆ is separable, i.e., X ρˆ = pi|ψi⟩⟨ψi| ⇐⇒ E(ˆρ) = 0. (5.30) i 2. E(ˆρ) does not increase on average under local operations and classical communication (LOCC) Λ, also called strong monotonicity, i.e., X E(ˆρ) ⩾ piE [Λ(ˆρ)] . (5.31) i 3. E(ˆρ) reduces to the entropy of entanglement for a pure state ρˆ, i.e., tr(ˆρ2) = 1 =⇒ E(ˆρ) ≡ S(ˆρA). (5.32) If the 3rd axiom is not satisfied then the mapping E is called an entanglement monotone (this is just a convention since other authors use the terms “measure” and “monotone” interchangeably). Modifying the Axioms Note that the 2nd axiom can be weakened by restricting to a subset of LOCC operations, i.e., positive partial transpose operations (PPT) [156]. As the name suggests, for a state that has a positive partial transpose, a PPT operation will also give a state with a positive partial transpose. This modification has interesting implications on the relationship between the entanglement measures that will be defined later. 76 Entanglement Theory The entropy of entanglement, as we defined it in Eq. (6.14), is based on the von Neumann entropy [see Eq. (3.8)], which is only a specific type of a more generalized family of functions, i.e.,the quantum Rényi entropies (which is an analog to the classical Rényi entropies [157]) defined as 1 S (ˆρ) ≡ log tr(ˆρr) , (5.33) r 1 − r 2 with r ∈ (0,1) ∪ (1,∞), or the quantum Tsallis entropies [158], defined as 1 h i S (ˆρ) ≡ tr ρˆt − 1 , (5.34) t 1 − t with t ∈ (0,1) ∪ (1,∞). Note that we can further unify the Rényi and Tsallis entropies in a single function called generalized quantum entropy [159]. Von Neumann entropy is the limiting case of either r → 1 for Rényi or t → 1 for Tsallis entropy, i.e., S(ˆρ) = lim Sr(ˆρ) = lim St(ˆρ), (5.35) r→1 t→1 and thus, the 3rd axiom can be modified according to the order of Rényi/Tsallis entropy we employ for the quantification of the entropy of entanglement. An interesting example can be found160 inRef.[ ], where the Rényi entropy of order 2 is operationally linked to the Shannon entropy of a Gaussian Wigner function, and used to define entanglement measures (and general discord-like quantum correlations). Properties of Entanglement Measures Properties that entanglement measures may also satisfy are the following: • Convexity, i.e., ! X X E piρˆi ⩽ piE(ˆρi). (5.36) i i Loosely speaking, convexity captures the notion of the loss of information, but since we already imposed strong monotonicity as an axiom, convexity is just a mathematically convenient property. • Additivity, i.e., E(ˆρ⊗n) = nE(ˆρ), (5.37) where n is an integer number. Given that a measure E is additive, then we can define its regularized or asymptotic version as E(ˆρ⊗n) E∞(ˆρ) ≡ lim . (5.38) n→∞ n A much stronger condition is the strong additivity [161], i.e., E(ˆρ) = E(trAρˆ) + E(trBρˆ). (5.39) 5.3 Quantifying Entanglement 77 The notion of additivity was briefly mentioned in chapter4 when we discussed the classical capacity of the Gaussian channels. We will discuss the implications of additivity on entanglement measures later in this chapter. n n • Asymptotic Continuity. For two sequences of states, i.e., {ρˆi}i=1 and {σˆi}i=1, an entanglement measure E is called asymptotically continuous, if the following relationship holds i→0 |E(ˆρi) − E(ˆσi)| ∥ρˆi − σˆi∥1 → 0 =⇒ → 0. (5.40) log2(dimHi) It is worth noting that any entanglement monotone that is both (i) additive and (ii) asymptotically continuous, is equivalent (equal or a simple rescaling) to the entropy of entanglement on pure states. This is the reason we imposed the 3rd axiom as a concise way of combining those two useful conditions. • Monogamy. The concept of monogamy for an entanglement measure E is related with the im- possibility of unconditionally sharing entanglement among subsystems of a composite quantum system. For a tri-partite system, for instance, monogamy was conventionally associated with the following inequality [162] A|BC ABC A|B AB A|C AC E (ˆρ ) ⩾ E (ˆρ ) + E (ˆρ ), (5.41) AB ABC AC ABC with ρˆ = trC ρˆ and ρˆ = trB ρˆ . The symbol “|” denotes the corresponding bipartitions. Eq. (5.41) implies that the sum of the entanglement between systems A|B and the entanglement between systems A|C cannot exceed the entanglement between system A and the rest of the subsystems BC. Even though the validity of monogamy follows intuitively for entanglement, Eq. (5.41) is found to be falsified by many well-established entanglement measures, raising questions [163] of whether this property should be expected in quantum systems in the first place. In a recent work [164] the authors resolved the issue by introducing dimension-dependent monogamy relations. Ordering of Entanglement Measures Two entanglement measures E1 and E2 are said to generate the same order, if for all density operators ρˆ and σˆ we have that E1(ˆρ) ⩽ E1(ˆσ) ⇐⇒ E2(ˆρ) ⩽ E2(ˆσ). (5.42) It was shown [165] that entanglement measures (that satisfy the axioms that we proposed above) impose, in general, different ordering on the set of entangled states. This result implies that there is no “right” or “wrong” entanglement measure for mixed states, but only measures that quantify different aspects of entanglement. 78 Entanglement Theory Extremality of Gaussian Entanglement Even though Gaussian states is only a subset of the family of continuous-variable states, they possess some “extremal” properties that can be used to bound several functionals of non-Gaussian states. For instance, in Ref. [166] it was shown that for an arbitrary state ρˆ, any asymptotically continuous and strongly superadditive entanglement measure E, should satisfy E(ˆρ) ⩾ E(ˆρG), (5.43) where ρˆG is a Gaussian state with the same first and second moments as the state ρˆ. Below we provide a list of different entanglement measures that have been defined in the literature. We should note that this is not an exhaustive list, but only measures that are either popular or useful for the rest of this thesis. 5.4 Survey of Entanglement Measures Based on the axioms discussed in the previous section we can construct several entanglement measures that capture different aspects of the non-separable nature of a state. 5.4.1 Distillable Entanglement and Entanglement Cost Two measures of paramount importance in Quantum Information, are the distillable entanglement and the entanglement cost. Let us have n copies of an entangled state ρˆ, i.e., ρˆ⊗n, that are asymptotically transformed by means of LOCC operations Λ into m copies of a maximally entangled state, e.g., Bell state |Φ+⟩⊗m. Then, we can write ρˆ −→Λ |Φ+⟩⊗m/n . (5.44) We can also assume the reverse scenario, where m copies of a maximally entangled state, e.g., Bell state |Φ+⟩⊗m, are asymptotically transformed by means of LOCC operations into n copies of an entangled state ρˆ, i.e., ρˆ ←−Λ |Φ+⟩⊗m/n . (5.45) m The asymptotic rates limn→∞ n characterize the entanglement of the state ρˆ and can be used to define entanglement measures. + Distillable entanglement ED [154, 155] is defined as the maximal rate of Bell states |Φ ⟩ that can be distilled from a state ρˆ over all possible LOCC operations {Λi}, i.e., ( " # ) ⊗n + + rn ED(ˆρ) ≡ sup lim inf Λi(ˆρ ) − (|Φ ⟩⟨Φ |) = 0 , (5.46) r n→∞ Λi 1 where ∥ · ∥1 is the trace norm defined in Eq. (2.36). In Ref. [167] it was shown that the definition of the distillable entanglement is not restricted only to Bell states as the “reference” for maximally entangled states. Due to its asymptotic definition, distillable entanglement is additive and strongly superadditive, 5.4 Survey of Entanglement Measures 79 since restricted protocols lead to smaller rates. It was also proved to be asymptotically continuous for the set of distillable states [168], and thus it satisfies the Gaussian extremality. Apart from special cases [169, 170] distillable entanglement is quite difficult to compute analytically for mixed states. Entanglement cost [171] is analogously defined as the minimum rate of Bell states |Φ+⟩ needed to create ρˆ over all possible LOCC operations {Λi}, i.e., ( " # ) h i ⊗n + + rn EC(ˆρ) ≡ inf lim inf ρˆ − Λi (|Φ ⟩⟨Φ |) = 0 , (5.47) r n→∞ Λi 1 and similarly to distillable entanglement is additive and strongly superadditive. As we will see below, EC is relatively easier to be computed compared to the ED, due to its connection with another measure, i.e., entanglement of formation. Extremality of Entanglement Measures Even though for some trivial cases of mixed states distillable entanglement and entanglement cost may coincide [172], in general they are not equivalent [173]. Interestingly, they constitute bounds for the rest of the entanglement measures. In particular, any strongly superadditive and continuous entanglement measure satisfies the following extremality condition [174] ∞ ED(ˆρ) ⩽ E (ˆρ) ⩽ EC(ˆρ). (5.48) Thus, computing those two measures we have at least a fundamental lower and upper bound of the entanglement in the system. 5.4.2 Entanglement of Formation Entanglement of formation (EoF) quantifies the entanglement of a state in terms of the entropyof entanglement of the least entangled pure state needed to prepare it under LOCC [154, 155]. For a given state ρˆ, EoF is defined by the convex-roof extension of the reduced von Neumann entropyof |ψi⟩, i.e., ( ) X EF (ˆρ) ≡ inf piS(trB|ψi⟩⟨ψi|) , (5.49) i P over all possible ensembles of pure states {pi,|ψi⟩} that can compose the state ρˆ = i pi|ψi⟩⟨ψi|. Note that in finite-dimensional systems, theinf “ ” in the definition of Eq. (5.49) can be substituted by “min”, because then a global minimum over all possible decompositions of a mixed state is guaranteed [175–178]. Monotonicity for EoF was shown in Ref. [155], and later it was proven that any entanglement measure based on convex roof extensions is actually monotonous [179]. EoF has been proven to be asymptotically continuous [180, 181], and monogamous for any finite dimensions164 [ ]. 80 Entanglement Theory Fig. 5.2 Relative entropy of entanglement is given by the distance (measured by the quantum relative entropy) between a given entangled state ρˆe (orange dot) and the closest separable one ρˆs (yellow dot). Interestingly, entanglement cost is the regularized version of EoF [171], i.e., E (ˆρ⊗n) E (ˆρ) ≡ lim F , (5.50) C n→∞ n which implies that, if EoF is additive then it coincides with entanglement cost. Entanglement of formation is sub-additive, i.e., EF (ˆρ1 ⊗ ρˆ2) ⩽ EF (ˆρ1) + EF (ˆρ2), (5.51) so its additivity would follow from EoF’s strong superadditivity [122–124]. However, it has been proven that in general EoF is not additive [125], but there are special cases where the two measures indeed coincide, e.g., a subset of two-mode Gaussian states [182]. Entanglement of Formation in general lacks an analytical formula, apart from special cases of high symmetry [161, 183, 184] or the case of two qubits [185]. Numerical techniques for its estimation for general states have also been developed [186]. Entanglement of formation for Gaussian states will be thoroughly discussed in chapter6. 5.4.3 Relative Entropy of Entanglement Relative entropy of entanglement (REE) quantifies how much a given entangled state can be distin- guished operationally by the set of all separable states [187]. More rigorously, for an entangled state ρˆ, it is defined as follows ER(ˆρ) ≡ minS(ˆρ∥ρˆi), (5.52) ρˆi∈S where S is the set of all separable states and S(ˆρi||ρˆj) ≡ tr[ˆρi(log2 ρˆi − log2 ρˆj)] , (5.53) is a quasi-distance between states ρˆi and ρˆj, called the quantum relative entropy [see Eq. (3.67)]. Note that the relative entropy of entanglement can be easily modified by either (i) restricting the minimization over a subset of separable states, e.g., states with positive partial transpose, or non-distillable states, or (ii) employing a different distance measure, e.g., Bures distance [see Eq. (3.65)]. 5.4 Survey of Entanglement Measures 81 Relative entropy of entanglement was proven to be convex [169] and asymptotically continuous measure [188], that also reduces to the entropy of entanglement for pure states [169]. In general, ER lacks a closed formula for arbitrary states. Relative entropy of entanglement is in general non-additive, but we can define its regularized version, i.e., E (ˆρ⊗n) E∞(ˆρ) ≡ lim R , (5.54) R n→∞ n ∞ which is by construction additive. For special cases, e.g., Werner states [189], ER (ˆρ) can be analytically calculated if we restrict the minimization over states with positive partial transpose. The asymptotic relative entropy of entanglement was also proven [164] to be monogamous for any finite dimensions. 5.4.4 Squashed Entanglement Squashed entanglement [190] is an entanglement measure based on the conditional mutual information, defined as AB ABE AB E (ˆρ =ρ ˆ ) ≡ inf S(A:B|E) tr (ˆρ ) =ρ ˆ , (5.55) S E where S(A:B|E) is the conditional mutual entropy defined as S(A:B|E) ≡ S(ˆρAE) + S(ˆρBE) − S(ˆρE) − S(ˆρABE), (5.56) over all possible purifications ρˆABE of a bipartite state ρˆAB. Squashed entanglement is an asymptotically continuous measure [191] that is also convex and additive [190], but quite difficult to compute analytically. ES is also strongly superadditive, so it fulfills the Gaussian extremality condition. Squashed entanglement also constitutes a lower boundto entanglement of formation and an upper bound to distillable entanglement, i.e., ED(ˆρ) ⩽ ES(ˆρ) ⩽ EF (ˆρ). (5.57) 5.4.5 Negativity and Logarithmic Negativity Negativity is an entanglement monotone (it does not reduce to the entropy of entanglement for pure states), initially discussed in Refs. [192, 193], and properly defined in Ref. [194, 195] as ∥ρˆΓ∥ − 1 E (ˆρ) ≡ 1 . (5.58) N 2 Even though negativity is a convex monotone and quite easy to compute analytically, it is not additive or asymptotically continuous (thus it does not satisfy the 3rd axiom and is not an entanglement measure). An interesting property of negativity, also called the disentangling theorem [196], is that it can be used to detect whether a tripartite-state, with subsystems A, B, and C, can be factorized into a product state of two parts of the system, i.e., h i h i A|BC ABC A|B ABC ABC AB1 B2C EN |ψ ⟩ = EN |ψ ⟩ ⇐⇒ |ψ ⟩ = |ψ ⟩ ⊗ |ψ ⟩. (5.59) 82 Entanglement Theory One can define a closely related monotone, i.e. logarithmic negativity [195], as Γ EL(ˆρ) ≡ log2∥ρˆ ∥1 , (5.60) which contrary to negativity it is not convex [197]. Even though logarithmic negativity is additive it fails to be strongly superadditive, and thus it does not satisfy the Gaussian extremality given in Eq. (5.43). In Ref. [166] the authors give a specific example, and in Chapter6 we will also present another example where EL fails this condition. One of the merits of logarithmic negativity is that it is an upper bound to the distillable entanglement [195], i.e., ED(ˆρ) ⩽ EL(ˆρ). (5.61) As we discussed before, sometimes we can relax the 2nd axiom regarding LOCC operations, and we can ask for monotonicity under PPT operations. In Ref. [156] it was shown that for some special Γ Γ cases, i.e., Werner states, Gaussian states, and states with |ρˆ | ⩾ 0, logarithmic negativity is equal to the entanglement cost under PPT operations (instead of LOCC) for the preparation of the state ρˆ. 5.4.6 Rains Bound Finally, another interesting monotone is the so-called Rains Bound, which is defined [198] as EB(ˆρ) ≡ min{S(ˆρ∥ρˆi) − EL(ˆρi)} , (5.62) ρˆi where S(·∥·) is the relative entropy defined in Eq. (9.4), and the minimization is over all states (not just separable states S as in the case of REE), that is a non-convex set, which implies the existence of local minima, making the numerical calculation of it a hard task. Rains bound is a lower bound to the REE over PPT states and an upper bound to the distillable entanglement, i.e., PPT ED(ˆρ) ⩽ EB(ˆρ) ⩽ ER (ˆρ). (5.63) Interestingly, for Werner states, Rains Bound happens to be equal to the regularized REE under PPT [189]. Part III Results Chapter 6 Entanglement of Formation in Gaussian Systems “The most exciting phrase to hear in science, the one that heralds new discoveries, is not ‘Eureka!’ (I found it!) but ‘That’s funny...’ ” — Isaac Asimov The results in Sec. 6.4 and Sec. 6.8 have been published in: • Ref. [15]: Spyros Tserkis and Timothy C. Ralph, Quantifying entanglement in two-mode Gaussian states, Physical Review A 96, 062338 (2017). The results in Sec. 6.5 and Sec. 6.6 have been published in: • Ref. [2]: Spyros Tserkis, Sho Onoe, and Timothy C. Ralph, Quantifying entanglement of formation for two-mode Gaussian states: Analytical expressions for upper and lower bounds and numerical estimation of its exact value, Physical Review A 99, 052337 (2019). The results in Sec. 6.7 have been published in: • Ref. [1]: Spyros Tserkis, Josephine Dias, and Timothy C. Ralph, Simulation of Gaussian channels via teleportation and error correction of Gaussian states, Physical Review A 98, 052335 (2018). 6.1 Introduction Quantifying entanglement is a highly non-trivial task as we have discussed in Chapter5. Various measures have been defined associated with different operational meanings, but most of them involve an 86 Entanglement of Formation in Gaussian Systems optimization process that typically cannot be expressed as a closed formula. Among them, entanglement of formation EF (EoF) is of significant importance, due to its well-defined physical meaning; EoF quantifies the entanglement of a state in terms of the entropy of entanglement E of the least entangled pure state needed to prepare it under LOCC operations. However, an analytical expression for EoF exists only for special cases, e.g., for two qubits [185], and finding a closed formula for an arbitrary state remains an open problem to this day. In this Chapter, we derive analytical lower and upper bounds for the entanglement of formation for an arbitrary two-mode Gaussian state, and based on those bounds, we also propose a simple method for its numerical calculation. A comparison between the lower and upper bounds with the exact value of the EoF (estimated numerically) is also presented for a large set of randomly created states. We further compare the lower bound with two other (existing in the literature) lower bounds [199, 200], showing its clear advantage. Finally, we discuss the difference between entanglement of formation and the widely used entanglement monotone logarithmic negativity, showing that the latter is not a proper way of quantifying entanglement even though it is mathematically an easier task. 6.2 Two-mode Gaussian States In Chapter4, we discussed how any two-mode Gaussian state ρˆG can be fully described by its first two statistical moments. For simplicity, from now on, we will assume all states are two-mode Gaussian states with vanishing mean values, since variations in the first moment will not affect the entanglement properties of the state, and we will use the covariance matrix, denoted as ρ, to fully represent the Gaussian state ρˆG. The covariance matrix of a two-mode Gaussian state (see Sec. 4.2.3 for more details) is in general given by AC ρ ≡ , (6.1) CT B where A = AT , B = BT , and C are 2 × 2 real matrices. The global purity of the state is given by √ √ √ µ ≡ 1/ detρ, while local purities by µa ≡ 1/ detA and µb ≡ 1/ detB, respectively. In the standard form [104, 105], the covariance matrix ρsf is written as a 0 c+ 0 sf 0 a 0 c− ρ ≡ , (6.2) c+ 0 b 0 0 c− 0 b with a ⩾ b, and c+ ⩾ |c−| ⩾ 0. The symplectic eigenvalues of a covariance matrix in the standard form are given by v u 2 2 q 4 2 2 2 2 4 2 ua + b + 2c+c− ± a − 2a (b − 2c+c−) + 4ab(c+ + c−) + b + 4b c+c− ν = t , (6.3) ± 2 6.2 Two-mode Gaussian States 87 and the symplectic eigenvalues of the partially transposed states are the following v u 2 2 q 4 2 2 2 2 4 2 ua + b − 2c+c− ± a − 2a (b + 2c+c−) + 4ab(c+ + c−) + b − 4b c+c− νΓ = t . (6.4) ± 2 The elements of the covariance matrix in Eq. (6.2) can be parametrized, as discussed in Ref. [201], over the local and global purities of the state as follows: 1 z + w√ a = , c+ = µaµb , (6.5a) µa 8 1 z − w√ b = , c− = µaµb , (6.5b) µb 8 where q z = [8d2 + (β − 1)(1 + g2) − 2(β + 1)(2d2 + g)]2 − 16g2 , (6.6a) q w = [8s2 + (β − 1)(1 + g2) − 2(β + 1)(2d2 + g)]2 − 16g2 , (6.6b) with s = (a + b)/2, (6.7a) d = (a − b)/2, (6.7b) g = 1/µ, (6.7c) −1 ⩽ β ⩽ 1, (6.7d) and the parameters s, d, and g are constrained by the following inequalities: s ⩾ 1, (6.8a) |d| ⩽ s − 1, (6.8b) g ⩾ 2|d| + 1. (6.8c) Using this parametrization, every two-mode Gaussian state, instead of the parameters {a,b,c+,c−} is characterized by {µ,µa,µb,β}. In Ref. [201] it was shown that states with β = ±1 constitute two physically important types of states. In particular, states with β = 1 are called GMEMS (Gaussian maximally entangled states for fixed global and local purities), which in the conventional parametriza- tion [see Eq. (6.2)] correspond to covariance matrices with c+ = |c−|. On the other hand, states with β = −1 are called GLEMS (Gaussian least entangled states for given global and local purities), which are states that minimize the lowest symplectic eigenvalue of a state, i.e., ν− = 1. It is worth mentioning, that for GLEMS the difference between c+ and |c−|, i.e., c+ − |c−| takes its maximum possible value. 88 Entanglement of Formation in Gaussian Systems Classicality Every quantum state ρˆG can be represented in phase-space with the so-called P-function [92, 93] defined as ZZ ∞ ρˆ ≡ d2αP(α)|α⟩⟨α|, (6.9) −∞ where |α⟩ represents a coherent state and P(α) is a quasi-probability distribution. When the P function takes positive values it can be interpreted as a classical probability distribution and the corresponding state is called classical, and when the P function is negative or singular the corresponding state is called non-classical [202]. sf For two-mode Gaussian systems a state is classical if and only if σ ⩾ 14 (matrix inequality), i.e., all the eigenvalues of its covariance matrix are greater or equal to 1 [46]. 6.3 Entanglement of Formation For an arbitrary state ρˆ, the entanglement of formation is generally defined154 [ , 155] in as the convex-roof extension of the reduced von Neumann entropy of |ψi⟩, i.e., ( ) X EF (ˆρ) ≡ inf piE(|ψi⟩⟨ψi|) , (6.10) i P over all possible ensembles of pure states {pi,|ψi⟩} that can compose the state ρˆ = i pi|ψi⟩⟨ψi|, where E(|ψi⟩⟨ψi|) denotes the entropy of entanglement [154, 155], which is given by E(|ψi⟩⟨ψi|) ≡ S(trB|ψi⟩⟨ψi|). In general, the calculation of EoF is NP-hard (non-deterministic polynomial-time hard) [203], but in 2003 Giedke et al. [204] derived a closed formula for the EoF of symmetric Gaussian states ρsym, i.e., a = b, given by q q f (a − c+)(a + c−) for (a − c+)(a + c−) < 1 EF (ρsym) = q , (6.11) 0 for (a − c+)(a + c−) ⩾ 1 q where (a − c+)(a + c−) is equal to the minimum symplectic eigenvalue of the partially transposed state [see Eq. (6.4)], and f(x) is the auxiliary function (x + 1)2 (x + 1)2 (x − 1)2 (x − 1)2 f(x) ≡ log − log . (6.12) 4x 2 4x 4x 2 4x In 2004, Wolf et al. [205], defined a closely related measure, i.e., Gaussian entanglement of formation (GEoF), as Eg (ρ) ≡ inf{E[ρ (r)] | ρ = ρ + ϕ }, (6.13) F ρ pi pi i pi ρ ϕ where ρpi is the covariance matrix of a pure Gaussian state, ϕi is a positive semidefinite matrix, i.e., ϕ 0 E[ρ (r)] i ⩾ , and pi is the entropy of entanglement of the corresponding pure state with a two-mode 6.4 Lower bound for Entanglement of Formation 89 squeezing parameter r, i.e., 2 2 2 2 E[σp(r)] ≡ cosh r log2(cosh r) − sinh r log2(sinh r). (6.14) The minimization in Eq. (6.13) is a complicated task, but a method involving the solutions of a fourth-order algebraic equation was proposed in Ref. [205]. Interestingly, for two-mode Gaussian states, it has also been proven that the Gaussian entanglement of formation is actually equivalent to the entanglement of formation [206–208, 182], i.e., g EF (ρ) ≡ EF (ρ). (6.15) As it was shown in Ref. [209], apart from the symmetric states, another two classes of mixed states for which the EoF can be analytically expressed are the (previously discussed) GMEMS and GLEMS. In particular, for GMEMS we have √ √ √ √ f m − m − 1 for m − m − 1 < 1 EF (ρ) = √ √ , (6.16) 0 for m − m − 1 ⩾ 1 where f(x) is the auxiliary function defined in Eq. (6.12), and m is a parameter that for GMEMs is given by 1 for g ≥ a + b − 1 √ 2 mGMEM = g(a+b)+a+b−2 (g−ab)[(a−b)2−g+1] , (6.17) (a−b)2+4g for g < a + b − 1 and for GLEMS by √ 2 2 1 for g ≥ a + b − 1 v u q 2 2 2 2 2 2 2 2 2 2 2 2 h 2 2 i2 u 2(a +b )+(a −b ) +|a −b | (a −b ) +8(a +b ) mGLEM = a −b t , g2−1 for g ≤ 2 2 2(a +b ) 2 2 2 2 2 2 2 4 4 2a (b +g +1)+2b (g +1)−(g −1) −a −b −ε+ε− 8g2 otherwise (6.18) where q ε± = (a − b ± g − 1)(a − b ± g + 1)(a + b ± g − 1)(a + b ± g + 1). (6.19) 6.4 Lower bound for Entanglement of Formation In order to derive a lower bound for the EoF of a two-mode Gaussian state, let us use the decomposition that is based on the pure states that achieve the lowest entanglement of formation [205, 207], i.e., sf h sf i T ρ = L(r1,r2) ρp (r) + φ L (r1,r2), (6.20) with L(r1,r2) = S(r1)⊕S(r2), where S(ri) ≡ exp[riZ] is the local squeezing symplectic operation, i.e., r1 Z 0 L(r1,r2) ≡ exp , (6.21) 0 r2 Z 90 Entanglement of Formation in Gaussian Systems with Z = diag(1,−1), and φ is a positive semidefinite matrix, i.e., φ ⩾ 0. Equivalently to Eq. (6.20), we have sf 1 T T T T ρ = L(r1,r2) S2(r) 4S2 (r)+S2(r)S2(−r)φφS2 (−r)S2 (r) L (r1,r2), (6.22) | {z } | {z } sf ρp (r) θ where S2(r) is the two-mode squeezing symplectic operation given by coshr 12 sinhr Z S2(r) ≡ . (6.23) sinhr Z coshr 12 Based on Wigner’s theorem [210], since both S2(−r) and φ are positive semidefinite matrices we T know that θ = S2(−r)φφS2 (−r) is a positive semidefinite matrix, i.e., θ ⩾ 0. Setting Σ→ ≡ L(r1,r2)S2(r), (6.24) Eq. (6.22) takes the form sf 1 T T ρ = Σ→ 4Σ→ + Σ→θΣ→ , (6.25) or equivalently sf → → ρ = ρp + θ . (6.26) The operation Σ→ is schematically represented in Fig. 6.1 (a). Now, the EoF for a state in its standard form, ρsf, can also be expressed as sf sf → → EF (ρ ) ≡ inf E[ρp(ri)] ρ = ρp (ri) + θi . (6.27) ri → In other words, among all pure states ρp (ri) that satisfy the decomposition sf → → ρ = ρp (ri) + θi , (6.28) → one has the minimum amount of entropy of entanglement, i.e., the optimum pure state ρp (ro), which has a two-mode squeezing parameter ro ≡ min{ri}. Using this optimal state, we are able to calculate the EoF of the state, by substituting the parameter ro in Eq. (6.14) as follows sf EF (ρ ) ≡ E[ρp(ro)]. (6.29) Re-arranging Eq. (6.25), we get an equivalent decomposition written as sf 1 T ρ = Σ→ [ 4 + θ]Σ→ , (6.30) 1 where 4 + θ can always be interpreted as a classical state ρc, so we finally have sf T ρ = Σ→ρcΣ→ . (6.31) 6.4 Lower bound for Entanglement of Formation 91 Fig. 6.1 Decomposition of a two-mode Gaussian state. In figure (a) and (b) we present two symplectic transformations Σ→ and Σ←, given in Eq. (6.24) and Eq. (6.33), respectively. Both of them are decomposed into a sequence (direct and reverse) of a two-mode squeezing transformation S2 and two single-mode squeezing transformation S. Every state in the standard form can prepared by applying Σ→ or Σ← onto a classical state, ′ i.e., ρc and ρc sf Thus, the EoF of a state ρ is entirely dependent on the transformation Σ→, and since the local squeezing cannot affect the entanglement, the two-mode squeezing operation S2(r) is the only operation that contributes to it. So, entanglement of formation of a state ρsf quantifies the minimum amount of two-mode squeezing S2(r) that is needed in order to prepare an entangled state starting from a classical one, i.e., ρc. Having a state in its standard form ρsf, we can always apply a two-mode squeezing operation with a squeezing parameter such that the state becomes separable. Then, we can further apply local squeezing on both modes to remove any non-classicality, and thus end up with a classical state. So, for every state we have the decomposition sf ′ T ρ = Σ←ρcΣ← , (6.32) where ′ ′ ′ Σ← ≡ S2(r )L(r1,r2). (6.33) 92 Entanglement of Formation in Gaussian Systems ′ 1 ′ In Fig. 6.1 (b) we schematically depict the operation Σ←. Writing the classical state as ρc = 4 +θ , ′ with θ ⩾ 0, we have sf h1 ′i T ρ = Σ← 4 + θ Σ← , (6.34a) sf 1 T ′ T ρ = Σ← 4Σ← + Σ←θ Σ← , (6.34b) sf ← ← ρ = ρp + θ . (6.34c) So, analogously to the Eq. (6.28), every state ρsf can also be decomposed as sf ← ′ ← ρ = ρp (ri) + θi . (6.35) ′ The minimum value of two-mode squeezing in the above decomposition, i.e., r− ≡ min{ri}, can sf be found analytically by applying an operation S2(−r) onto an arbitrary state ρ , such that it becomes separable, i.e., Γ ′ sf ′ T ν−[S2(−r )ρ S2(−r ) ] = 1, (6.36) Γ with ν− being the lowest symplectic eigenvalue of the partially transposed state, given in Eq. (6.4), which is a necessary and sufficient separability condition (see Sec. 5.2.3 for more details on separability of two-mode Gaussian states). Note that the above equation has in general two solutions, but we are interested only in the smaller one of the two values, which is given by v u q uκ − κ2 − λ λ 1 t + − r− = ln , (6.37) 2 λ− where we have set κ = 2(detρsf + 1) − (a − b)2 , (6.38) and λ± = detA + detB − 2detC + 2[(ab − c+c−)±(c+ − c−)(a + b)]. (6.39) → ← ′ ′ A given pure state ρp (ri), can also be written as ρp (ri), where ri and ri are connected with each other as follows 1 −1 2r′ 2 ′ 2r′ 2 ′ r = cosh e 2i χsinh r + e 1i χcosh r , (6.40) i 2 i i with v u −2r′ −2r′ 2 u 1i 2i ′ ue + e tanh ri χ = t ′ ′ . (6.41) 2r1 2r2 2 ′ e i + e i tanh ri As it can be easily seen, ri in Eq. (6.40) has a global minimum, given by ′ ri ⩽ ri . (6.42) 6.4 Lower bound for Entanglement of Formation 93 → For the specific case of the optimum state ρp (ro), with two-mode squeezing equal to ro, there is a ← ′ corresponding ρp (ro), and according to Eq. (6.42) we have the following relationship ′ ro ⩽ ro . (6.43) ′ ′ Since r− ≡ min{ri}, ro should always be lower bounded by r−, i.e., ′ r− ⩽ ro ⩽ ro , (6.44) so we can derive a lower bound for the entanglement of formation, i.e., Lower Bound for EoF. For any entangled two-mode Gaussian state we have a lower bound for the entanglement of formation, given by Γ sf sf ν−(ρ ) < 1 =⇒ E[ρp(r−)] ⩽ E[ρp(ro)] ≡ EF (ρ ). (6.45) where r− is given in Eq. (6.37), and E[ρp(r−)] can be calculated through Eq. (6.14). This bound implies that the least amount of two-mode squeezing we need to apply to a state to make it separable is always less than or equal to the least amount of two-mode squeezing we need to create it. In general, this lower bound becomes tight when the transformation Σ→ that corresponds to sf the optimal decomposition of the state ρ is equivalent to the transformation Σ←. It is trivial to show that [S2(r),L(ri,ri)] = 0 ⇐⇒ Σ← ≡ Σ→ , (6.46) which means that when the two single-mode squeezers of either transformation Σ← or Σ→ are equal to each other, they can commute through the two-mode squeezer. That is true for both symmetric states and states with β = 1 (also called GMEMS). ′ ′ It is also worth mentioning that the single-mode squeezing parameters r1 and r2 of Eq. (6.40) can be analytically calculated for a given value of two-mode squeezing r′, i.e., v u ′ ′ ′ u(a − b)ξ+ − 2θ sinh(2r ) − (a + b)ξ− cosh(2r ) r1 = ln u q , (6.47a) t sf ω − detρ + 1 + γ(ζ1 + ζ2) v u ′ ′ ′ u(a − b)ξ+ + 2θ sinh(2r ) + (a + b)ξ− cosh(2r ) r2 = ln u q . (6.47b) t sf ω + detρ − 1 + γ(ζ1 + ζ2) 94 Entanglement of Formation in Gaussian Systems with 2 ξ± = ab − c+ ± 1, (6.48a) 2 θ = abc− − c+c− + c+ , (6.48b) ′ ′ ω = (a − b)[(a + b)cosh(2r ) + (c− − c+)sinh(2r )], (6.48c) 1 γ = [a2(b2 − 1) − ab(c2 + c2 ) − b2 + (c c − 1)2], (6.48d) 2 + − + − 2 2 2 2 2 2 2 ζ1 = a (2b − 1) − 2ab c+ + c− − 1 − b + 2c+c− + 2, (6.48e) ′ ′ 2 ζ2 = 2(a + b)(c+ − c−)sinh(4r ) − cosh(4r )[(a + b) − 4c+c−]. (6.48f) Comparison with other bounds This is not the first time a lower bound has been derived for the entanglement of formation ofGaussian states. In Fig. 6.2 we compare the performance of this bound for randomly generated states1 with the previously known lower bounds of the measure, derived in Refs. [199] and [200]. As we see, the former lower bounds deviate significantly from the real value (calculated numerically), and sometimes they even fail to detect entanglement. 6.5 Upper bound for Entanglement of Formation Since the EoF is given by the entropy of entanglement of the least entangled state of the decomposition in Eq. (6.28), the entropy of entanglement of every other pure state that satisfies this decomposition is by construction an upper bound to the entanglement of formation. As we mentioned before, every pure state can be prepared by either the transformation Σ← or Σ→ applied onto two vacua. Let us have the pure state that was prepared via the operation Σ←, with the minimum amount of two-mode squeezing, i.e., r−. Based on Eq. (6.40) this corresponds to a pure state prepared via the operation Σ→ with two-mode squeezing that we denote as r+. Since in general we have ro ≡ min{ri}, the value r+ is lower bounded as follows ro ⩽ r+ , (6.49) and analogously to the lower bound we can construct an upper bound to the EoF. 1In order to create random two-mode Gaussian states we follow the following procedure: We randomly pick a value for local purities 0 ⩽ µa/b ⩽ 1 and the parameter −1 ⩽ β ⩽ 1. Then the global purity of the state is constrained as µ µ µ 1 and we also pick a random value within this range. Finally, a check is performed to verify a b ⩽ ⩽ 1+| 1 − 1 | µa µb whether the created state is a valid state, i.e., the covariance matrix is real and positive semidefinite matrix that satisfies the Heisenberg uncertainty principle. The MATHEMATICA file with the code implementing this task can be downloaded from spyrostserkis.com 6.5 Upper bound for Entanglement of Formation 95 Fig. 6.2 Lower bounds for entanglement of formation. Entanglement of formation is in general an unbounded function, but it depends only on the optimum two-mode squeezing parameter ro. We plot with (black) dots Γ −2ro above the optimum symplectic eigenvalue νo− = e for randomly created states against the corresponding −2r value based on r−, i.e., e − . The symplectic eigenvalue is a bounded value ∈ (0,1], which shows that: (i) − sf sf EF (ρ ) ⩽ EF (ρ ), and (ii) that the bound is tight for separable and infinitely entangled states. We also depict with (blue) squares ■ [199] and (red) triangles ▲ [200] the corresponding values we get from the previously known lower bounds. The closer the dots are to the diagonal the smaller the deviation from the real value of entanglement. It is clear that our bound is, on average, tighter than the previous bounds. All the quantities plotted are dimensionless. Upper Bound for EoF. For any entangled two-mode Gaussian state we have an upper bound for the entanglement of formation, given by Γ sf sf ν−(ρ ) < 1 =⇒ E[ρp(r+)] ⩾ E[ρp(ro)] ≡ EF (ρ ). (6.50) ′ where r+ can be found via Eq. (6.40) by substituting ri → r−, and E[ρp(r+)] can be calculated through Eq. (6.14). Based on Eq. (6.40) it is obvious that the upper and lower bounds are intimately related with each other, and when the lower bound gets tight the upper bound gets tight as well, which happens for symmetric states and states with β = 1. Based on numerical calculations it seems that the upper bound ′ ′ 1 becomes tight also for the case of states with β = −1, if the condition |r1 − r2| ⩽ 2 lnν+ is satisfied, but the general validity of this argument is only conjectured. 96 Entanglement of Formation in Gaussian Systems 6.6 Numerical estimation of EOF Finding an analytical expression for the exact value of the entanglement of formation is still considered an open problem. However, if we express the EoF through the decomposition of Eq. (6.35), and take into account the upper and lower bounds derived before, we end up with a numerically straightforward optimization process, i.e., Numerical Estimation for EoF. For any entangled two-mode Gaussian state, the entanglement of formation can be computed as follows sf sf ← ← EF (ρ ) = inf E[ρp(ri)] ρ = ρp (ri) + θi . (6.51) r−⩽ri⩽r+ The problem of writing down Eq. (6.51) as a closed formula is that the function that needs to be minimized is in general non-smooth. However, for the cases of symmetric states, i.e., a = b, and states with β = 1, there is no need for an optimization of the Eq. (6.51), since we just have to set ri = r− = r+. As we mentioned before, for states where β = −1 the EoF can be calculated through Eq. (6.17). This is not the first time a method for calculating the EoF has been derived, but the onegiven in Eq. (6.51) is significantly easier for numerical calculations. A specific algorithm writtenin MATHEMATICA has also been developed2, that numerically evaluates the exact value of EoF for an arbitrary two-mode Gaussian state written in its standard form. It is also worth comparing the upper and lower bound to the actual value of EoF in order to see how close they are. In Fig. 6.3, we generate a large number of random entangled states, and for each one we calculate the percentile relative difference, given by |E − E±| δ± ≡ F F × 100%, (6.52) EF against the global purity µ of the corresponding state. As we clearly see for a random state the purity is sf ± sf inversely proportional to the relative difference between EF (ρ ) and EF (ρ ). It is also apparent that the upper bound is on average closer to the exact value than the lower bound. Thus, besides the cases mentioned above, the upper and lower bounds can also be faithfully used for analytical calculations of the EoF for states with high purities, e.g., 0.8 ≲ µ ⩽ 1. 6.7 EoF of the Choi-state Gaussian channels describe the decoherence introduced by the environment on a quantum state, and for many quantum communication protocols they represent the basic models of communication lines such as optical fibers or free space [44]. Let us assume that a two-mode squeezed vacuum state is sent through a phase-insensitive single-mode Gaussian channel G with parameters τ > 0 and v = |1 − τ|ε defined in Sec. 4.5.2. Then, the optimum squeezing parameter r, i.e., ro, that needs to be substituted in 2The MATHEMATICA file with the code can be downloaded from spyrostserkis.com 6.8 Comparison with Logarithmic Negativity 97 Fig. 6.3 Lower and upper bound for randomly created states. In this figure we plot the percentile relative difference between both the upper δ+ (blue dots) and lower δ− (red crosses) bound and the actual value of the entanglement of formation, given in Eq. (6.52), against the purity of randomly created entangled states. It is sf ± sf apparent that the less the purity the larger the difference between EF (ρ ) and EF (ρ ). We also observe that on average the upper bound is closer to the real value than the lower bound. All quantities plotted are dimensionless. Eq. (6.29) to calculate the entanglement of formation is given by p 1 3 + 2v2 − (1 − τ)2cosh(4r)τ(3τ + 2) + 4v(1 + τ)cosh(2r) − 4 v2 − (1 − τ)2 sinh(2r)[v cosh(2r) + 1 + τ] ro = ln √ . 4 2[v − 2 τ sinh(2r) + (1 + τ)cosh(2r)]2 (6.53) A special type of state that will be found to be of significant interest in the next Chapter isthe so-called Choi-state [211, 212], which is defined as the maximally entangled state Φ passing through a channel, i.e., [1 ⊗ G](Φ). In continuous-variable systems Φ can be considered the EPR state, i.e., a two-mode squeezed vacuum with infinite amount of squeezing, passing through a single-mode channel. For the special case of the phase-insensitive Gaussian channels G discussed before the parameter ro is given by h q i 2v v − v2 − (1 − τ)2 − (1 − τ)2 Choi 1 τ ̸= 1 =⇒ ro = ln √ , (6.54a) 4 (1 − τ)4 1 4 τ = 1 =⇒ rChoi = ln . (6.54b) o 4 v2 Note that for v ⩾ 1 + |τ|, we have an entanglement-breaking channel [128, 129], i.e., the entangle- ment of any state vanishes through the channel, and thus ro = 0 by definition. 6.8 Comparison with Logarithmic Negativity In Sec. 5.4.5, we defined an entanglement monotone called logarithmic negativity, which foran arbitrary state ρˆ is given [195] by Γ EL(ˆρ) ≡ log2 ∥ρˆ ∥1 , (6.55) 98 Entanglement of Formation in Gaussian Systems Fig. 6.4 Comparison between entanglement of formation (solid blue line) and logarithmic negativity (dashed red line). Assuming a pure state ρp(r) with squeezing parameter r = 1 is sent through a pure lossy channel with transmissivity 0 ⩽ τ ⩽ 1, we compare the two measures. The deterministic upper bounds (upper lines), i.e., the amount of entanglement assuming an infinitely squeezed state is sent through the same channel, are also depicted, since they provide further insight regarding the qualitative differences between entanglement of formation and logarithmic negativity. The deterministic bound for logarithmic negativity can be found in ref. [213]. Specifically, for logarithmic negativity, the deterministic bound of a state with transmissivity value τa, can also be reached by sending the squeezed state (r = 1) through a channel of transmissivity τb, with τb > τa. However, in contrast, entanglement of formation predicts that we cannot reach the deterministic bound with a squeezed state (r = 1) regardless of how much we raise the transmissivity. This is a critical difference, since the two quantifiers disagree on whether a physical upper bound has been reached or not. All the quantities plotted are dimensionless. Γ where ρˆ denotes the partially transposed density matrix ρˆ, and ∥ · ∥1 is the trace norm of an operator defined in Eq. (2.36). Logarithmic negativity has been extensively used in the literature for the quantification of entanglement, since it is an easily computable entanglement monotone. In Gaussian systems, for a state with a covariance matrix ρ, it can also be expressed [201] as h Γ i EL(ρ) = max 0,−log2 ν−(ρ) . (6.56) As it is clear from the above definition, the only parameter that the logarithmic negativity depends Γ on is ν−(ρ), which is also the only parameter that the PPT criterion depends on, as we discussed in Sec. 5.2.3. Thus, from an operational point of view, for a given state ρ, the logarithmic negativity quantifies the maximum possible violation of the PPT separability criterion. Even though as an entanglement monotone logarithmic negativity has its merits (discussed in Sec. 5.4.5) from a resource perspective it is not in general connected to the squeezing of the entangled state. In Fig. 6.4, we present an example where it fails to satisfy the extremality of entanglement [see Eq. (5.48)], which is expected since logarithmic negativity is not asymptotically continuous. That results in an inconsistent behavior of EL, which, for finite squeezing, can either be an upper or lower bound of EF , depending on the channel that the state is sent through. A specific example of how EL 6.9 Conclusion 99 can lead to a qualitatively different evaluation of the entanglement sent through a physically relevant channel compared to EF is also shown in Fig. 6.4. The previous theoretical example is complemented by the results of an experiment discussed in Ref. [214]. In that work, a distillation protocol, based on the noiseless linear amplification (NLA), that is discussed in AppendixB, was applied on two-mode Gaussian states in order to increase their entanglement. Several measures and monotones, i.e., entanglement of formation, squashed entanglement, relative entropy of entanglement, logarithmic negativity, were employed to quantify the entanglement of the states, plotted against the success probability of the distillation protocol. Logarithmic negativity surpasses the deterministic bound of maximum entanglement at a relatively large probability of success (indicating entanglement distillation), while all the other entanglement measures are unable to surpass it (indicating no entanglement distillation). Thus, we conclude that logarithmic negativity alone is insufficient to unambiguously verify the distillation of entanglement of a state, at least with these types of distillation protocols. 6.9 Conclusion In this Chapter, we derived an analytical lower and upper bound to the entanglement of formation of an arbitrary two-mode Gaussian state. Based on those bounds we were able to construct a simple numerical procedure for the estimation of the measure. We also introduced an alternative interpretation for the measure in Gaussian systems, showing that entanglement of formation of a state is equal to the minimum amount of two-mode squeezing we need to apply to a classical state in order to prepare it through LOCC operations. Based on this approach, the quantification of entanglement of other families of states, e.g., multipartite Gaussian [by extending the decomposition of Eq. (6.31) into the special case of an N-mode state created by an N-mode squeezer (followed by single-mode squeezers) applied on an N-mode classical state] or non-Gaussian states (for the cases where we can show that the non-Gaussianity is introduced after the application of the two-mode squeezer), may also become feasible. Finally, we provided evidence that another entanglement monotone, i.e., logarithmic negativity, even though typically preferred by researchers due to its simple formula, may inconsistently estimate the entanglement of a state. Chapter 7 Continuous-Variable Teleportation and Error Correction “I argue that quantum mechanics is fundamentally a theory about the represen- tation and manipulation of information, not a theory about the mechanics of nonclassical waves or particles." — Jeffrey Bub The results of this chapter have been published in: • Ref. [1]: Spyros Tserkis, Josephine Dias, and Timothy C. Ralph, Simulation of Gaussian channels via teleportation and error correction of Gaussian states, Physical Review A 98, 052335 (2018). 7.1 Introduction Quantum decoherence is an inevitable feature of any realistic quantum system, leading to errors in the information encoded on it. This decoherence process is mathematically modeled through quantum channels, which induce a corresponding transformation on the states passing through them. Understanding and correcting these errors is the main theoretical and technological barrier to quantum computation and communication overtaking their classical counterparts. Gaussian states (discussed in Chapter4) constitute the optimal states for several quantum protocols, and are widely used in the photonics community due to the well-established techniques for realizing them. The most typical kind of decoherence induced on those states along their propagation through optical fibers or free space is also Gaussian, and thus being able to error correct this typeofnoiseis vital for quantum communication purposes. 102 Continuous-Variable Teleportation and Error Correction A key tool in quantum information theory is quantum teleportation (see Ref. [215] for recent advances). With entanglement as a resource, teleportation allows a quantum state to be moved from one place to another using classical communication. Realistically, though, teleportation will not lead to a perfect reconstruction, and thus the whole process can be thought of as a quantum channel that induces noise on the initial state [216, 217]. For Gaussian channels, it has been shown that the inverse is also true. Every Gaussian channel can be simulated by a teleportation protocol if the resource state is either a maximally entangled state passing through the channel that we want to simulate, i.e., the Choi-state, or a state that is at least equally entangled to the Choi-state [211, 218, 114, 219]. If, by distillation techniques, a resource state can be established across the channel with more entanglement than the Choi-state, then the simulated teleportation channel may be less decohering than the physical channel, i.e., the state passing through the channel may be error corrected. Quantitatively identifying the resources required for error correction is a key open problem in Continuous-Variable (CV) quantum information. In CV quantum information the Choi-state is not a physical one, since its creation would require infinite energy. Recently it was asked if physical states (states with finite mean energy), withthe same entanglement as the Choi-state, can be found that are able to perform channel simulation via teleportation. Logarithmic negativity (see Sec. 5.4.5) was used as the entanglement quantifier in that analysis [220], and the result was that the least resourceful (in the context of entanglement) finite- energy states are in general mixed and their entanglement is not equal but higher than the corresponding Choi-state. In this chapter, we show that this result was an artifact of the use of logarithmic negativity as the quantifier of entanglement. Employing instead entanglement of formation as the quantifier (see Sec. 5.4.2), we prove that all phase-insensitive Gaussian channels can be simulated via teleportation with a physical resource state equally entangled to the Choi-state, and we find the ones with the minimum mean energy. The only exception is the identity, which is expected, since the identity channel represents an ideal teleportation, which by definition requires maximal entanglement, and thus infinite energy. We also propose an experimentally accessible way to construct such finite mean energy resource states that can either simulate the initial channel (for pure lossy/amplifier channels) or simulate another channel that decoheres the entanglement of formation of an incoming state by the same amount (for thermal lossy/amplifier channels). Further, we show that resource states with entanglement morethan the Choi-state are able to simulate channels (thermal lossy channels as an example) that decohere an incoming state less than the initial channel, and thus error correct the quantum states. This error correction protocol generalizes a previous one which was restricted to pure lossy channels [221, 222]. 7.2 CV Teleportation Quantum teleportation was initially introduced for discrete variables [13, 223], and then extended to CV systems [224–226] (see also [227] for a universal approach on teleportation). There are also protools for hybrid situations [228, 229]. 7.2 CV Teleportation 103 Let us assume that we want to teleport a (zero mean-valued) single-mode Gaussian state with covariance matrix σin from one place (laboratory 1) to another (laboratory 2). Regardless of the exact teleportation protocol, a necessary quantum resource in order to achieve that is a two-mode entangled state with covariance matrix ρ (for the purposes of this work also Gaussian with zero mean value), shared between the two laboratories, i.e., a state with a covariance matrix given (in the standard form) by a 0 c 0 0 a 0 −c ρ = . (7.1) c 0 b 0 0 −c 0 b The way we manipulate quantum correlations in order to achieve a successful teleportation depends on the protocol. In the standard CV teleportation protocol [225], on which we will focus in this chapter, one mode of the resource state is mixed with the input state through a balanced beam-splitter in laboratory 1, followed by a Bell-type measurement, i.e., dual homodyne detection, HD, (measuring the X quadrature on one mode and the P on the other), and the results are sent to laboratory 2 through a classical channel, CC. Finally, in laboratory 2, a displacement operation proportional to the results of these measurements, D, is applied to the other mode of the resource state in order to reconstruct the input state, i.e., teleport it. Graphically this protocol is depicted in Fig. 7.1 (a). The quality of the teleportation protocol is limited by the amount of entanglement that is pre- shared between the two parties, and perfect teleportation is achieved only in the limit of maximum entanglement resources. In CV systems, though, maximum entanglement is unphysical, since that would require infinite energy. Thus, realistically, instead of achieving a perfect state teleportation, we always end up with a slightly noisy copy of the target state. This process can be modeled as the decoherence that a quantum channel (completely positive trace-preserving map) induces to a transmitted state [216, 217]. The covariance matrix transformation when a phase-insensitive single-mode Gaussian channel G acts on a single-mode Gaussian state σin is given by [116] T σout = G(σin) = X σinX + Υ, (7.2) √ where X = τ1 and Υ = v1. Significant phase-insensitive Gaussian channels (also depicted in Fig. 4.6) are the following: • the lossy channel L with transmissivity 0 < τ < 1 and noise v = (1 − τ)ϵ (pure lossy Lp for ϵ = 1, thermal lossy for ϵ > 1), • the amplifier channel A with gain τ > 1 and noise v = (τ − 1)ϵ (pure amplifier Ap for ϵ = 1, thermal amplifier for ϵ > 1), • the classical additive noise channel N with τ = 1 and noise v > 0, and • the identity channel 1 with τ = 1 and v = 0, representing the ideal nondecohering channel. 104 Continuous-Variable Teleportation and Error Correction Fig. 7.1 Teleportation and channel simulation. The teleportation protocol [225] is represented in figure (a) via its basic components: i) the dual homodyne detection, HD, between the resource state, ρ, and the initial state σin, ii) the classical channel, CC, iii) the displacement, D, and iv) the output state, σout. In figure (b) we have the corresponding channel that the teleportation protocol simulates. 7.3 Channel Simulation In general, any Gaussian channel can be simulated via a quantum teleportation protocol using an appropriate resource state [211, 218, 114]. In particular, with a Gaussian resource state of the form of Eq. (7.1), the standard CV teleportation protocol corresponds to a Gaussian phase-insensitive channel with transmissivity τtel and noise vtel given by: τtel = λ, (7.3a) √ vtel = aλ − 2c λ + b, (7.3b) where λ ⩾ 0 is the experimentally accessible gain. Note that we assumed an ideal dual homodyne measurement detection. The decoherence that a Gaussian channel induces onto a Gaussian signal passing through it can always be associated to the decoherence that a teleportation protocol would induce to the same signal. In order for those two processes to be effectively equivalent for the output signal1, the resource state ρ used for the teleportation should have entanglement (computed under any proper entanglement measure) equal or greater to the corresponding Choi-state [211, 218, 114, 219], i.e., Choi E(ρ) ⩾ E(G ). (7.4) Since the Choi-state is an unphysical one, our goal is to find a physical resource state ρ able to simulate a Gaussian channel G through teleportation. In order to do so, we assume a resource state of the form given in Eq. (7.1) that satisfies Eq. (7.3). It is also convenient to express the symplectic 1By “effectively equivalent” we mean that one cannot tell if the decoherence of the output signal stems from a given channel or the teleportation protocol that substituted the channel. 7.3 Channel Simulation 105 eigenvalues through the covariance matrix elements [see Eq. (6.3)]. Then, assuming τ ⩾ 0 and τ ̸= 1 we solve this system for a,b and c and we end up with Resource States for Channel Simulation. A given Gaussian phase-insensitive channel G with parameters τ and v can be simulated through CV teleportation with two sets of resource states of the form of Eq. (7.1), i.e., plus and minus, with covariance matrix elements given by |1 − τ|(ν − ν ) + (1 + τ)v ± 2γ a = + − , (7.5a) ± (1 − τ)2 τ|1 − τ|(ν − ν ) + (1 + τ)v ± 2γ b = + − , (7.5b) ± (1 − τ)2 τ|1 − τ|(ν − ν ) + 2τv ± (1 + τ)γ c = + √ − , (7.5c) ± τ(1 − τ)2 with q γ ≡ τ(v − |1 − τ|ν−)(v + |1 − τ|ν+). (7.6) Note that for 0 < τ < 1, we have states with a > b, while for τ > 1 we have a < b. These elements are expressed in terms of the channel parameters τ and v and may vary over the symplectic spectrum with the constraints 1 ⩽ ν− ⩽ 2¯n + 1, (7.7a) ν− ⩽ ν+ , (7.7b) where n¯ is the mean number of photons per mode of the Gaussian channel. Resource states with reversed symmetry for each case, i.e., a < b for τ < 1 and a > b for τ > 1, can be retrieved by interchanging the values of ν− and ν+, but then the corresponding symplectic spectrum is given by 1 ⩽ ν± ⩽ 2¯n + 1. For an additive-noise Gaussian channel with τ = 1 and variance v > 0 we use the limit τ → 1 for the class in Eqs. (7.5a)-(7.5c) and get the following parametrization ν2 + 2ν (ν − v) + (ν + v)2 a = − − + + , (7.8a) 4v ν2 + 2ν (ν + v) + (ν − v)2 b = − − + + , (7.8b) 4v (ν + ν − v)(ν + ν + v) c = − + − + . (7.8c) 4v For finite values of ν± we get every physical state that can simulate a given channel G with parameters τ and v. Entanglement of formation (discussed in Sec. 5.4.2 and Ch.6) quantifies the minimum entanglement resources needed to prepare a state, and thus we consider it as the appropriate measure for calculating the minimum required resources for a channel simulation. Based on that measure, the optimal resource state ρopt (least resourceful state) for any channel simulation is the pure state with ν−(ρ−) = ν+(ρ−) = 1, since the entanglement (calculated by entanglement of formation) Choi EF (ρ±) is minimized and becomes equal to the corresponding Choi-state, EF (ρopt) = EF (G ), and 106 Continuous-Variable Teleportation and Error Correction thus saturates Eq. (7.4). The mean energy per mode, i.e., tr(ρ ) − 4 ⟨aˆ†aˆ⟩ = ± , (7.9) ρ± 8 also takes its minimum possible value for this resource state, and the two-mode squeezing parameter χ ≡ tanhr ∈ [0,1) is given by √ q 2 τ − (v + 1 − τ)(v − 1 + τ) χ = . (7.10) opt τ + v + 1 We should note that a similar analysis was performed in Ref. [220], where the minimization of the entanglement resources was based on logarithmic negativity. The conclusion in that work was that the optimal states are mixed states with higher values of mean energy (see Eq. 9 in Ref. [220] that corresponds to ν− = 1 and ν+ > 1), which do not in general saturate Eq. (7.4). One other restriction was that pure lossy and pure amplifier channels could not be simulated with those optimum states. In our analysis, on the other hand, using entanglement of formation as the quantifier, we find that the optimal resource states [given by Eq. (7.10)] are equally entangled to the corresponding Choi-state, they have the minimum possible energy for this level of entanglement (since they are pure states), and they can simulate all phase-insensitive channels (including pure lossy and pure amplifier). Finding the optimal resource state is a theoretical result that gives us physical insight, but has limited practical application, since, in a realistic scheme, physical limitations exist in both the creation and the transmission of this state. In particular, let us assume that the two laboratories are separated by a channel G. Instead of sending directly a state through this channel, the two laboratories can apply a teleportation protocol. However, the resource state that laboratory 1, for instance, has created needs necessarily to pass through the same decohering channel that we started with, G, since this is the environmental decoherence that is beyond our control. This channel, though, will decrease the entanglement of the resource state leading to a simulated channel, in general different from the initial one. For that reason, laboratory 2 needs to apply an entanglement distillation protocol in order that the entanglement of the resource state can be enough for the desired simulation. Interestingly, laboratory 2 can distill the resource state even more and thus make the teleportation protocol simulate a less decohering channel, Gc, that practically error corrects the state that we wanted to send through channel G. The whole process is schematically illustrated, step by step, in Fig. 7.2. In the next section we discuss this realistic construction of the resource state and the error correction protocol in detail. 7.4 Error Correction of Gaussian States During any quantum protocol, decoherence have to be taken into account since the added noise may severely impact the whole process. Developing ways of correcting those errors induced during a protocol is a process that is called error correction. Broadly speaking, error correction refers to the concept where an added redundancy to the initial state may help in the detection and correction of potential errors. In particular, the redundancy can be added via two main ways: (i) embedding the 7.4 Error Correction of Gaussian States 107 Fig. 7.2 Channel simulation through entanglement distillation. Figure (a) represents the channel, G, that we want to error-correct. Figure (b) shows the resource state ρ, which is sent through the same channel, G, and then through an entanglement distillation process before it is used in a teleportation protocol (see also Fig. 7.1). In figure (c) we have the effective transformation for a successful distillation process, i.e., ρe, and finally in figure ′ (d) the simulated channel Gc, which leads to a state σout. state into a higher dimensional space (which is the standard way of performing error correction based on measurements called “syndromes” that reveal possible errors), or (ii) by teleporting the state over the noise environment. The equivalence of those two procedures (at least for DV systems) has been shown in [155] and several protocols have been developed for both DV [155, 230, 231] and CV states [232–234, 221, 235–237]. Ideally, an error correction protocol would entirely reconstruct the state by removing all induced errors, so the fidelity [73, 113, 112] (discussed in Sec. 3.3.2 in general and in Sec. 4.4.1 for Gaussian systems) between the input and output state of the protocol would be equal to 1. Reaching this value for fidelity is in principle possible in DV codes (asymptotically), since for example a Bell state mightbe employed during a teleportation-based protocol and thus effectively the initial state is passing through the identity channel. However, in CV codes fidelity equal to 1 cannot be reached, since that would require a maximally entangled state, i.e., the EPR state, which is unphysical. Comparing non-unity fidelities cannot in general provide a consistent analysis, since the overlap between two probability distributions fails to take into account important features of the state, e.g., classicality, separability, Gaussianity etc. In Ref. [238] the authors discussed this issue in detail (see also Ref. [216] for the specific case of teleportation), and in Fig. 7.3 we also provide a characteristic 108 Continuous-Variable Teleportation and Error Correction Fig. 7.3 Problems with fidelity. Let us assume that a pure state with squeezing parameter equalto ζ = 0.8 is going through both a thermal amplifier channel A(τ2,ε2 = 2.5) and a thermal lossy channel L(τ1,ε1 = 1.01). For different values of τ1 and τ2 we calculate the fidelity of the input/output state and we get two sets of states: (i) the set with F1 < F2, colored with blue (on the left of the solid line, sections I and IV) and (ii) the ones with no entanglement left, colored with brown (on the top of the dashed line, sections I and II). As we can see there is an overlap between those two sets, i.e., section I, where we have both an entanglement-breaking situation and F1 < F2. All quantities plotted are dimensionless. example of why this comparison might be problematic. In particular, let us assume that we send a state via an entanglement-preserving thermal lossy channel L(τ1,ε1) and we calculate the fidelity between the input and output state, F1. Now, before the thermal lossy channel, we introduce an amplifier channel, A(τ2,ε2), in way that the whole channel A(τ2,ε2)◦L(τ1,ε1) is entanglement-breaking, and we calculate again the fidelity, F2. We might expect that always F1 > F2, since for the second case all the quantum correlations are gone and the state passing through an entanglement-breaking channel is useless for quantum communication, but we can easily find certain channels L and A for which F1 < F2. Since full reconstruction of the initial state is impossible in CV systems, we will focus below on a teleportation-based error correction scheme, that aims to reduce the noise as much as possible. The scheme depends on the entanglement resources used during the teleportation, and in general the success of the protocol is related with the amount of entanglement that can be ultimately transmitted through the quantum channel G (in the limit of infinite entanglement the protocol becomes asymptotically optimal and the state is effectively passing through an error-free channel, i.e., identity channel). The reason why we choose distribution of entanglement as the figure of merit for the error-correction stems from the fact that this task is of significant interest in quantum repeaters239 [ –241], which are the building block for a large scale quantum network. 7.5 Error correction protocol 109 7.5 Error correction protocol Let us assume that in laboratory 1 we have a Gaussian two-mode entangled state σin, and we want to send one mode of it to laboratory 2. We model the decoherence that is induced due to the environment to the state through a Gaussian channel G. After the transmission through this channel, the laboratory 2 gets the output state σout. This process leads to a noisy version of the initial state σin, so we aim to develop a protocol to reduce this noise as much as possible. Laboratory 1 prepares a resource state, that we assume is initially a pure entangled two-mode Gaussian state ρ with a covariance matrix given by 1+χ2 2χ 2 0 2 0 1−χ 1−χ 1+χ2 2χ 0 0 − 1−χ2 1−χ2 ρ = 2 , (7.11) 2χ 0 1+χ 0 1−χ2 1−χ2 2χ 1+χ2 0 − 1−χ2 0 1−χ2 with squeezing parameter χmin < χ < 1, where χmin is given in Eq. (7.10). One mode of the above state is sent through the same channel G with transmissivity τ and additive noise v = |1 − τ|ε to laboratory 2, and thus it is transformed into √ 1+χ2 2 τχ 2 0 2 0 1−χ 1−χ √ 1+χ2 2 τχ 0 2 0 − 2 G(ρ) = √ 1−χ 1−χ . 2 τχ 1+χ2 (7.12) 2 0 2 τ + |1 − τ|ε 0 1−χ √ 1−χ 2 τχ 1+χ2 0 − 1−χ2 0 1−χ2 τ + |1 − τ|ε At laboratory 2, an entanglement distillation protocol is performed on the received state G(ρ). Distillation protocols are in general probabilistic, so assuming a successful distillation protocol the two laboratories establish a new resource state, that we call effective resource state, given by 2 √ 1+χe 2 τeχe 2 0 2 0 1−χe 1−χe 2 √ 1+χe 2 τeχe 0 2 0 − 2 √ 1−χe 1−χe ρe = 2 . (7.13) 2 τeχe 1+χe 2 0 2 τe + |1 − τe|εe 0 1−χe 1−χe √ 2 2 τeχe 1+χe 0 − 2 0 2 τe + |1 − τe|εe 1−χe 1−χe Then, one of the two modes of the initial state σin is teleported under the effective resource state ρe, so it is effectively passing through a channel that we call error-corrected channel Gc with corresponding ′ τc and εc. Thus, the output state σout is less decohered (less noisy) than the output state that we would get without the protocol, σout. 7.5.1 Pure Lossy and Amplifier Channels Using this protocol for pure lossy or amplifier channels an equally decohering channel would necessar- ily be identical with the initial one, since they are only transmissivity/gain dependent, (ε = 1). This 110 Continuous-Variable Teleportation and Error Correction can be achieved by solving Eq. (7.3), for v = ±(1 − λ) (plus sign for loss and minus for amplification). Assuming a successful distillation we get an effective state of Eq. (7.13), and using this effective state as a resource state, shared between the two laboratories, we apply a teleportation protocol in order to get a simulated channel, Gc. The simulated channel is 2 Lc ≡ L with τc = λ = τeχe , (7.14a) τe Ac ≡ A with τc = λ = 2 . (7.14b) χe A trivial way to achieve that is when the resource state is the Choi-state, i.e., χe → 1 and τe = τ. 2 2 However, any resource state with parameters such that τeχe = τ for lossy channels and τe/χe = τ for amplifier, would be equally entangled to the Choi-state, leading to the same amount of distributed ′ entanglement. In order to induce less decoherence and start the error correction, i.e., EF (σout) > EF (σout), we would need λ = τc > τ, which requires resource states with entanglement greater that the corresponding Choi-state, i.e., Choi EF (ρe) > EF (G ). (7.15) 7.5.2 Thermal Lossy and Amplifier Channels Assuming now that we have a thermal channel, we have to solve again Eq. (7.3), for v = ±(1 − λ)θ (plus sign for thermal loss and minus for thermal amplification), with θ ⩾ 1. The teleportation part of the protocol with the effective resource state simulates the following quantum channel: 0 < λ ⩽ 1 ⇒ Lc with {τc = λ,εc = θ}, (7.16a) 1 ⩽ λ ⩽ λmax ⇒ Ac with {τc = λ,εc = −θ}, (7.16b) where 2 2 √ εe(τe − 1) χe − 1 + χe(τe + λ) − 4χe τeλ + τe + λ θ = 2 , (7.17) (λ − 1)(χe − 1) and q 2 (εe + 1)(1 − τe) 3τe − εe(1 − τe) − 1 − 2 2τe(εe + 1)(1 − τe)(χe − 1) λmax = + 2 . (7.18) 2 2χe Simulating an identical thermal channel can be achieved only in the unphysical limit of the Choi- state. However, if we focus only on the decoherence, we can always create thermal channels that ′ induce less decoherence than the initial one, i.e., EF (σout) > EF (σout), as long as Choi EF (ρe) > EF (G ). (7.19) 7.6 Error Correction with NLA 111 7.6 Error Correction with NLA The most challenging part of the protocol from an experimental point of view is the distillation of entanglement, that induces the effective parameters {χe,τe,εe} needed for the above simulations. A distillation technique useful for CV systems is the noiseless linear amplification (NLA) [242, 243], that is discussed in the AppendixB (see Refs. [ 110, 111, 211] for other distillation protocols on Gaussian states). The critical quantity of NLA is its amplification parameter g > 1. For a thermal lossy channel, the effective parameters have been computed in Refs. [244, 245], and are given by v u 2 u2 + (g − 1)[ε(τ − 1) + τ + 1] χ = χt , (7.20a) e 2 + (ε − 1)(τ − 1)(g2 − 1) 4g2τ τ = , (7.20b) e {ε + 1 + (ε − 1)[(τ − 1)g2 − τ]}{(ε + 1)(1 − τ) + [τ + 1 + ε(τ − 1)]g2} τ + 1 + ε[2 + ε(1 − τ)] + (ε − 1)[τ + 1 + ε(τ − 1)]g4 εe = . (7.20c) [ε + 1 − (ε − 1)g2]2 − τ (ε2 − 1)(g2 − 1)2 It has already been shown [221, 222] that for pure lossy channels, i.e., ε = 1, in order to simulate ′ equally decohering channels under NLA, i.e., τc = τ ⇒ EF (σout) = EF (σout) we need g = 1/χ and 2 ′ λ = τeχe, while for g ⩾ 1/χ we get τc > τ and thus EF (σout) > EF (σout), so we start the error correction. For thermal lossy channels, in order to reach that bound, we need the same NLA gain, i.e., ′ g = 1/χ, but we also have to optimize over the teleportation gain, i.e., maxλ{EF (σout)} = EF (σout). As expected, g = 1/χ is also the point when the resource state has entanglement equal to the Choi- Choi Choi state, i.e., EF (ρe) = EF (L ). For NLA gain greater than 1/χ, we get EF (ρe) > EF (L ) and ′ thus maxλ{EF (σout)} > EF (σout), which implies error correction. A specific example for the error correction of a thermal lossy channel induced to a Gaussian state is presented in Fig. 7.4. The NLA gain is not an arbitrary parameter. Its minimum value is by construction equal to 1, but it is also upper bounded by a finite value gmax, beyond which the output state becomes unphysical. Two conditions that need to be satisfied in order for the output distilled state to be physical arethe following: v u 2 u τ(1 − ε) + (ε + 1)[1 + (τ − 1)χ ] 0 χ < 1 =⇒ 1 g g = t , (7.21) ⩽ e ⩽ ⩽ χ τ − 1 + ε(τ − 1)(χ2 − 1) + (τ + 1)χ2 and v u q u 2 2 u(1 − ε )(1 − τ) + 2 (ε − 1)τ ε 1 =⇒ 1 g g = t , (7.22) e ⩾ ⩽ ⩽ ε (ε − 1)[τ + 1 + ε(τ − 1)] So, the maximum attainable NLA gain is given by gmax = min{gχ,gε}, (7.23) 112 Continuous-Variable Teleportation and Error Correction Fig. 7.4 Error correction with ideal NLA. An initial thermal lossy channel with τ = 0.5 and ε = 1.05 induces noise into one mode of a pure two-mode squeezed state σin with squeezing parameter ζ = 0.5. We apply the protocol using a resource state ρ with squeezing parameter χ = 0.5. In figure (a) we present both the (optimized ′ over teleportation gain) entanglement of the output state, maxλ{EF (σout)}, (red solid line), and the entanglement of the distilled resource state, EF (ρe), (red dashed line), against the NLA gain, g. With solid blue and dashed blue lines we have the entanglement of the output state without the protocol, EF (σout), and the deterministic upper Choi bound of entanglement for this channel, EF (L ), respectively. We observe that for g = 1/χ the entanglement of the Choi-state is reached and from then on and until we reach gmax we are into the error correction area (light ′ blue shaded), i.e., maxλ{EF (σout)} ⩾ EF (σout). In figure (b) the contour lines indicate equally decohering channels (with parameters τ and v), i.e., channels that decohere the entanglement by the same amount. The yellow triangle represents the initial channel. Applying the protocol without distilling the resource state, i.e., g = 1, we get the channel shown with the red dot. Increasing the NLA gain, and specifically for g = 1/χ, we simulate a channel (yellow dot in the graph), that decoheres the state by the same amount as the initial channel (both channels lie on the same dashed contour line). The best channel we can simulate is achieved for gmax, and is represented by the blue dot. With red/yellow/blue diamonds we indicate the corresponding simulated channels of an error correcting protocol based on a less entangled resource state, i.e., χ′ = 0.45. Thus, we can visually interpret error correction as the process of simulating a channel “closer” to the identity (represented by the green dot) than the initial one. All quantities plotted are dimensionless. and thus the overall condition for error correction is given by gmax > 1/χ. (7.24) In Fig. 7.5 we present the set of channels that can be error corrected with this protocol for different resource states, taking into account both the maximum attainable NLA gain, gmax > 1/χ, and the entanglement-breaking condition, v ⩾ 1 + τ. As expected, the success of the error correction protocol is intimately related with the entanglement power of the resource state. We should note at this point that the above observation, i.e., that the point when the error-correction coincides with the point when the effective resource state has entanglement equal to the Choi-state, is intimately related with the entanglement measure used in this analysis, i.e., entanglement of formation. If, instead, we had picked logarithmic negativity as the entanglement quantifier, then there is no apparent connection between the entanglement of the resource state and the error correction protocol. In other words, using logarithmic negativity we can easily find situations where the deterministic upper 7.6 Error Correction with NLA 113 Fig. 7.5 Error correction range. We plot the range of all the possible channels with parameters 0 ⩽ τ ⩽ 1 and 1 ⩽ v ⩽ 2 that can be error corrected with the protocol, based on both the NLA condition, gmax > 1/χ, and the entanglement-breaking condition, v ⩾ 1 + τ. It is apparent that for increasing values of squeezing parameter χ, the set of channels is increased as well. All quantities plotted are dimensionless. bound of entanglement has been surpassed while the overall output has not been error corrected. That implies that from this point of view logarithmic negativity in general “overestimates” the entanglement which might lead to wrong conclusions when it is used for example in distillation protocols giving a potentially false-positive result. Reaching that physical bound with entanglement of formation is a significantly harder task (see Ref. [214] for a similar result), but as soon as we have reached it then the distilled entangled state is objectively more entangled since it is useful to perform tasks such as error correction discussed above. Error Correction with Realistic NLA One way the NLA can be experimentally implemented with linear optics, consists of N modified quantum scissors devices (see AppendixB). So far, we assumed the ideal case for the NLA procedure, i.e., N → ∞. However, the error correction protocol discussed before is not limited by this assumption, so in Fig. 7.6 we present a more realistic case when the NLA consists of a single quantum scissor, and we compare it with the ideal case. As it is expected the real NLA needs a higher gain compared to the ideal one. 114 Continuous-Variable Teleportation and Error Correction Fig. 7.6 Error correction with a single quantum scissor. The thermal lossy channel that we want to error correct has transmissivity τ = 0.01 and noise ε = 1.0002. Both the resource and the initial state have squeezing ′ parameter equal to χ = ζ = 0.5. The red solid line depicts maxλ{EF (σout)} for the ideal NLA, and the brown QS ′ dashed one depicts the corresponding maxλ{EF (σout)} for the realistic NLA with one quantum scissor. As we see the NLA gain needed to cross the value EF (σout) is greater for the realistic NLA compared with the ideal one. All quantities plotted are dimensionless. 7.7 Conclusion In this chapter, we showed that every phase-insensitive Gaussian channel can be simulated through teleportation with a physical resource state [Eqs. (7.5a)–(7.5c) and (7.8a)–(7.8c)] and we derived analytical expressions for all the possible states able to perform this task. We also identified the optimal resource states [Eq. (7.10)], i.e., the states with the minimum requirements in energy and entanglement, using entanglement of formation as the entanglement measure. This result clarifies a previously published work [220], where, using logarithmic negativity as the entanglement quantifier, pure channels were not able to be simulated with the optimal resource states. This discrepancy is not surprising, since logarithmic negativity is not a proper entanglement measure (even though it is an entanglement monotone) and it is not the first time that its problematic (inconsistent) behavior has been observed [166, 15, 214]. We also showed that resource states with entanglement equal to the Choi-state can simulate channels that decohere an incoming state in the same way as the initial one and we used that fact to generalize an error-correction protocol for noise induced by thermal lossy channels. The next step is to extend this error-correction protocol to other useful Gaussian channels. We should also note that recently this finite-energy analysis of resource states has found practical ap- plications in private communication, where relative entropy of entanglement of the resource states constitutes an upper bound of the capacity of the simulated channels [246–248]. Finally, the consistent behavior of entanglement of formation identified here implies that it can also be used as a faithful quantifier in concatenated error-correction protocols, such as quantum repeaters [239–241], where the incoming state of the first part becomes a resource state of the next part and soon. Chapter 8 All-optical Teleportation and Quantum Key Distribution “So, what is quantum mechanics? Even though it was discovered by physicists, it’s not a physical theory in the same sense as electromagnetism or general relativity. In the usual ‘hierarchy of sciences’ – with biology at the top, then chemistry, then physics, then math – quantum mechanics sits at a level between math and physics that I don’t know a good name for. Basically, quantum mechanics is the operating system that other physical theories run on as application software (with the exception of general relativity, which hasn’t yet been successfully ported to this particular OS). There’s even a word for taking a physical theory and porting it to this OS: ‘to quantize’.” — Scott Aaronson The results of this Chapter have been presented in the following preprint, which is under review for publication: • Ref. [3]: Spyros Tserkis, Neda Hosseinidehaj, Nathan Walk, and Timothy C. Ralph, Teleportation- based collective attacks in Gaussian quantum key distribution, Physical Review Research 2, 013208 (2020). 8.1 Introduction In Gaussian quantum key distribution eavesdropping attacks are conventionally modeled through the universal entangling cloner scheme [249, 250, 119], which is based on the premise that the whole environment that two distant parties (Alice and Bob) share is under control of the adversary (Eve), i.e., the eavesdropper purifies the system. This assumption implies that the eavesdropper haseither 116 All-optical Teleportation and Quantum Key Distribution access to an identity (noiseless) channel or infinite amount of entanglement in order to simulate such an identity channel. In this chapter, we discuss how this assumption can be challenged by proposing a teleportation- based eavesdropping attack, where the eavesdropper is not assumed to have access to the shared channel, that represents the unavoidable noise due to the environment. We show how, under this scheme, Eve’s information depends on the amount of entanglement she can prepare, distribute and distill in order to successfully perform the all-optical teleportation protocol [226]. Employing collective measurements, and using a resource state with the least required amount of entanglement, Eve’s information reaches the bound for optimal individual attacks [251, 252]. Using the same setup and taking the limit to infinite entanglement for the resource state, Eve’s information approaches the ultimate bound foran eavesdropping attack, also known as the Holevo bound [5]. We identify the distributed entanglement used as a resource for teleportation as the operationally critical quantity capturing the limitations of a realistic Eve (e.g. for a given separation of Alice and Bob one can estimate a minimum amount of loss, and Eve’s quantum information processing resources that bound the amount of pure entanglement that could be distilled across such a lossy channel). Under this limitation we evaluate the secret key rate that Alice and Bob achieve conditioned on their belief on how powerful (in entanglement resources) Eve is. Note that other types of physical limitations on Eve’s capabilities for CV-QKD systems have also been studied in the context of Eve’s restricted quantum memory [253], and Eve’s restriction to receive all the photons lost in transmission [254]. 8.2 Cryptography Throughout history, private communication has been regarded as an important task for reasons spanning from mere personal information exchange to military secrets and secure financial transactions. In any cryptographic scheme (classical or quantum), we assume two distant parties, Alice and Bob, who intend to transmit a message, avoiding a potential eavesdropper, Eve. Alice encodes the initial message into a cyphertext by applying a secret key to it, in order to make it incomprehensible to Eve, and sends it to Bob through a public classical channel. Bob receives the cyphertext and decodes it using the same (already established) secret key. One-time pad protocol [255] (graphically depicted in Fig. 8.1) is a famous example of a cryp- tographic scheme, which has been proven by C. Shannon to be unconditionally secure [256]. It is based on two main assumptions, i.e., that (i) Alice and Bob already share the same secret key, and (ii) that the secret key is random. The randomness part can be moderated using “sufficiently random” secret keys, i.e., pseudo-random keys, but the main challenge is how can we securely distribute a key between the two parties, given that an adversary, might intervene during this process attempting to copy it (a pre-established key is an impractical assumption). Classically, copying a signal is totally feasible, and that is why an entirely secure classical public-key distribution is impossible. On the other hand, copying a quantum signal is forbidden by the no-cloning theorem [257–259], implying the possibility of an always detectable eavesdropping attack. Current classical key distribution schemes, e.g., RSA, are based on the inability of classical computers to efficiently (in a polynomial time) solve a 8.3 Quantum Key Distribution 117 Fig. 8.1 One-time pad scheme. In this figure we graphically represent the one-time pad scheme that consists of two parties, Alice and Bob that want to secretly transmit a message though a public channel. The public channel is assumed to be under an eavesdropper control, called Eve. In this example, we assume that the message that Alice has to transmit is the bit sequence “101101010”, on which she applies (by modular addition) a pre-established (random) secret key, i.e., “001101011”. She sends the encrypted-text, i.e., “100000001”, through the public channel to Bob, who applies the same secret key (by modular addition) to the encrypted-text in order to retrieve the initial message. difficult mathematical problem, i.e., factorizing large numbers. However, a quantum computer could efficiently tackle this problem using the Shor’s algorithm14 [ ], and thus new ways of cryptography must be developed to overcome this threat. 8.3 Quantum Key Distribution Quantum Key Distribution (QKD) [260–262] is one of the most prominent quantum communication protocols, which enables two parties to establish a shared (random) secret key for cryptographic purposes. It was originally developed for discrete-variable (DV) quantum systems [12, 263], but has also been extended to the continuous-variable (CV) regime [264–266]. A clear advantage of the CV-QKD schemes (see Refs. [267–271] for recent advances) over their DV counterparts is the low-cost telecom optical components needed, which are already available for classical communication. We generally consider the following assumptions for Eve in a QKD scheme [260–262]: 1. Eve has full access to the quantum channel between Alice and Bob. 2. Eve has unlimited computational power. 3. Eve can monitor the public classical channel, but she cannot modify the messages (authenticated channel). 4. Eve has no access to Alice’s and Bob’s laboratories. The most powerful attack Eve can asymptotically perform is the so-called collective attack [272], where she prepares and interacts a set of individual and identical quantum systems with the quantum signals sent from Alice to Bob. She then stores the output ensemble into her quantum memory for a future collective measurement (an individual attack would rely on individual measurements respectively). Unconditional security of a QKD protocol can be achieved by upper bounding the information Eve can extract, also known as the Holevo bound [5]. 118 All-optical Teleportation and Quantum Key Distribution Prepare-and-measure Scheme A generic QKD protocol in “prepare-and-measure” (PM) scheme consists of: • quantum communication, where Alice encodes classical information into conjugate quantum basis states, which are sent through an insecure quantum channel to Bob, who measures the received quantum states in a randomly chosen basis, resulting in two sets of correlated data between Alice and Bob, and • classical communication (also called classical post-processing) over a public but authenticated classical channel, where Alice and Bob extract a secret key from the correlated data they collected during the previous step. 8.4 Gaussian CV-QKD In a fully Gaussian CV-QKD protocol [59, 44] (in the PM scheme) Alice encodes a classical random variable “a” (drawn from a Gaussian distribution) onto Gaussian quantum states, squeezed states [273] or coherent states [274], and sends them through an insecure quantum channel to Bob, who measures the received quantum states using homodyne or heterodyne detection to obtain a classical random variable “b”. Gaussian (collective or individual) attacks are asymptotically optimal [275–278], and the asymp- totic secret key rate against optimal collective attacks is given by [279, 280] K = βH(a:b) − S(x:E), (8.1) where H(a:b) is the classical mutual information between Alice and Bob, and S(x:E) is the maximum quantum mutual information between Alice (x≡a) and Eve (in the direct reconciliation1 where Alice is the reference of the reconciliation in the classical post-processing), or between Bob (x≡b) and Eve (in the reverse reconciliation where Bob is the reference of the reconciliation). The coefficient 0 ⩽ β ⩽ 1 is the reconciliation efficiency [267–271]. Note that the maximum amount of information Eve can possibly extract from the collective attack is upper bounded by the Holevo bound χ(x:E) [5], i.e., S(x:E) ⩽ χ(x:E). (8.2) Gaussian States and Channels Similarly to the previous chapter, here we also assume that the Gaussian states are fully described by their covariance matrices (states with vanishing mean value) in the standard form. A two-mode 1The term reconciliation refers to the classical information that is sent from one user to the other, which is a form of error correction in order the two parties to ensure that both keys are identical. 8.5 Entangling Cloner Attack 119 squeezed vacuum that will represent Alice’s state is given by 1+ζ2 2ζ 2 0 2 0 1−ζ 1−ζ 1+ζ2 2ζ 0 0 − 1−ζ2 1−ζ2 σin = 2 , (8.3) 2ζ 0 1+ζ 0 1−ζ2 1−ζ2 2ζ 1+ζ2 0 − 1−ζ2 0 1−ζ2 with 0 ⩽ ζ < 1 being the squeezing parameter. The covariance matrix transformation for a two-mode Gaussian state σin passing through a single- mode phase-insensitive Gaussian channel G is given by [116] T σout = G(σin) = (1 ⊕ X )σin(1 ⊕ X ) + (0 ⊕ Υ), (8.4) √ where X = τ1, with τ > 0 representing the transmissivity or the gain, and Υ = v1, with v = |1−τ|ε > 0 representing the additive noise (ε = 1 for pure and ε > 1 for thermal channels, respectively). Entanglement-based Scheme Each Gaussian PM scheme can be represented using an equivalent entanglement-based scheme [59], where Alice prepares a pure Gaussian entangled state, i.e., a two-mode squeezed vacuum state σin with squeezing parameter 0 < ζ < 1, keeping one mode while sending the second mode through the quantum channel. If Alice applies a homodyne (heterodyne) detection, the second mode of the entangled states is projected onto a squeezed (coherent) state. While in the experimental demonstration of CV-QKD, PM scheme is preferred, the entanglement-based scheme is favored for the security analysis. Before we propose our scheme, let us briefly review the current way we model individual/collective attacks. 8.5 Entangling Cloner Attack In the entangling cloner setup [249, 250, 119], the whole channel G is associated with a potential eavesdropper that has full control of the environment. Eve uses a two-mode squeezed vacuum state and mixes one arm of it with Alice’s signal in a beam-splitter with transmissivity equal to the transmissivity of the quantum channel. For the collective attack, one of the outputs is sent directly to Bob while the rest are stored in her quantum memory. Finally, she collectively measures the stored ensemble to gain the maximum information about the distributed key. This scheme is schematically represented in Fig. 8.2. An assumption that has been taken in this setup is that Eve is able to noiselessly transmit the output signal of the beam-splitter to Bob, i.e., the output signal passes through an identity channel 1. Obviously, this is a really strong assumption, since Eve has to deal with some unavoidable decoherence due to environmental reasons that go beyond her control, but even theoretically that is impossible because simulating identity channels through teleportation in CV systems (discussed in Sec. 7.2) requires an infinite amount of entanglement. 120 All-optical Teleportation and Quantum Key Distribution Fig. 8.2 Entangling cloner attack. In this figure we present the entangling cloner attack, where Eve hasfull control of the environment and simulates the channel G through a beam-splitter Bτ (with the same transmissivity as the channel). One input of the beam-splitter is Alice’s signal and the other is one arm of Eve’s state φ′. With ′ M1−2 we represent the measurements that Eve performs later on her quantum memory on her state µ , and 1 the identity channel. For the case of optimal individual attacks it has already been shown [251, 252] that they can be realistically modeled through the standard CV-teleportation protocol [225], however this protocol is dependent on individual Bell-type measurements, and thus it cannot be directly used for collective attacks. In order to realistically model an optimal collective attack, we propose below an eavesdropping scheme, based on the all-optical teleportation protocol [226] that is measurement-free. 8.6 All-optical Teleportation Protocol The all-optical teleportation protocol was introduced in Ref. [226] and it is graphically presented in Fig. 8.3 (see Sec. 7.2 for more details and references on other quantum teleportation protocols). We assume access to a resource state, which is a (zero mean-valued) Gaussian two-mode state fully represented by its covariance matrix ρ, i.e., a 0 c 0 0 a 0 −c ρ = . (8.5) c 0 b 0 0 −c 0 b In this protocol one arm of the resource state is fed into a parametric amplifier along with the input state in laboratory 1. The output amplified signal is directly sent to laboratory 2, which means thatit has to go through some decoherence that can be modeled as a Gaussian (for our purposes) channel G, that is initially shared between the two laboratories. Finally, in laboratory 2, we mix the signal with the second arm of the resource state on a beam-splitter, where the induced transmissivity is inversely proportional to the amplification applied in laboratory 1. Let us assume that we want to teleport a single-mode Gaussian state σin. In order to do so, we use a two-mode Gaussian resource state with a covariance matrix ρ of the form of Eq. (8.5). The initial state can be represented by a combined three-mode covariance matrix σin ⊕ ρ. The amplification is 8.6 All-optical Teleportation Protocol 121 Fig. 8.3 All-optical teleportation protocol. In this protocol, the basic components are: (i) a two-mode squeezer Sg, (ii) a beam-splitter Bt, (iii) the decoherence that the signal has to go through modeled with a quantum channel G, and (iv) the entangled resource state ρ. achieved by a two-mode squeezer [281], where the two inputs are the initial state and one arm of the resource entangled state. The corresponding symplectic transformation Sg that the amplifier induces is given by √ √ g 0 g−1 0 √ √ 0 g 0 − g−1 √ · √ g−1 0 g 0 S = √ , (8.6) g √ 0 − g−1 0 g 1 0 · 0 1 2 where g = cosh r ⩾ 1, with r ∈ R being the two-mode squeezing parameter. Note that the symplectic transformation Sg is applied to the initial three-mode state σin ⊕ ρ, where the identity sub-matrix of Sg indicates that the second arm of the resource state remains unaffected at this stage. Applying enough amplification to surpass the quantum limit we end up with a signal thatcan directly be sent to the other laboratory, but it still needs to go through some decoherence due to the environment. Let us assume that this decoherence is a thermal channel G, with transmissivity/gain τ and noise v. The subsequent attenuation can be modeled with a beam-splitter, where in one port we feed in the previously amplified state (that is decohered through the environment) and in the other the secondarm of the resource entangled state. The symplectic transformation of the beam-splitter Bt is given by √ √ t 0 − 1−t 0 √ · √ 0 t 0 − 1−t 1 0 B = · · , (8.7) t 0 1 √ √ 1−t 0 t 0 √ · √ 0 1−t 0 t with a transmission ratio equal to t = λ/(gτ). Applying on the initial state σin ⊕ ρ the two-mode squeezer Sg we get the amplified state T Sg(σin ⊕ ρ)Sg . (8.8) 122 All-optical Teleportation and Quantum Key Distribution This amplified state transforms according to Eq.(8.4) due to the decoherence into T G[Sg(σin ⊕ ρ)Sg ]. (8.9) Finally, this decohered state goes through the final beam-splitter Bt and evolves into T T Bt{G[Sg(σin ⊕ ρ)Sg ]}Bt . (8.10) Tracing out mode 2 (the second output of the amplification) and mode 3 (the second output ofthe attenuation) from this state (see Fig. 8.3) we get the output state T T σout = tr23Bt{G[Sg(σin ⊕ ρ)Sg ]}Bt . (8.11) This teleportation protocol corresponds to a Gaussian phase-insensitive channel with transmissivity τtel and noise vtel given by τtel = λ, (8.12a) s λ(g − 1)(gτ − λ) λ(aτ + b − v) v = aλ − 2c − + b. (8.12b) tel τg2 τg In the limit of infinite amplification, i.e., g → ∞, Eq. (8.12) reduces to Eq. (7.3) and the output signal of the all-optical teleportation protocol becomes equivalent to that of the standard CV teleportation protocol. For finite amplification and the same amount of entanglement resources, the all-optical teleportation protocol will always correspond to an equally or a slightly more noisy effective channel than the standard protocol, but its big advantage over the standard protocol is that there is no need for individual Bell-type measurements during the teleportation process. The significance of this advantage is crucial for the eavesdropping attack that is discussed below. 8.7 All-optical Teleportation Attack In this type of attack, that is schematically represented in Fig. 8.4, we start by assuming that there exists a physical quantum channel, G, between Alice and Bob, through which all the participants (including Eve) must send their signals. Given this limitation, Eve (who is allowed to establish stations arbitrarily close to Alice’s and Bob’s laboratories) performs an all-optical teleportation protocol over this channel G. In general, any Gaussian channel can be simulated via a quantum teleportation protocol using an appropriate resource state [211, 218, 114] (see also Sec. 7.3). Given a Gaussian phase-insensitive channel G the set of all resource states that can simulate it have been derived in Ref. [1] (see also Ref. [220] for special cases). 8.7 All-optical Teleportation Attack 123 For our purposes, we assume that Eve can prepare, distribute and distill a pure two-mode squeezed vacuum state given by 1+γ2 2γ 2 0 2 0 1−γ 1−γ 1+γ2 2γ 0 0 − 1−γ2 1−γ2 ρ = 2 , (8.13) 2γ 0 1+γ 0 1−γ2 1−γ2 2γ 1+γ2 0 − 1−γ2 0 1−γ2 with squeezing parameter 0 ⩽ γ < 1. One arm of it is used for the initial amplification (performed on Eve’s station close to Alice’s side) through the two-mode squeezer Sg with gain g > 1 [281]. Employing a beam-splitter Bη with transmissivity 0 ⩽ η ⩽ 1, we mix the second arm of ρ with one arm of another two-mode squeezed vacuum state 1+κ2 2κ 1−κ2 0 1−κ2 0 2 1+κ 2κ 0 1−κ2 0 − 1−κ2 φ = 2 , (8.14) 2κ 1+κ 1−κ2 0 1−κ2 0 2κ 1+κ2 0 − 1−κ2 0 1−κ2 with squeezing parameter 0 ⩽ κ < 1 (that is prepared and used on Eve’s station close to Bob’s side). One output of Bη is headed to another beam-splitter Bt with transmissivity t = 1/g for the final attenuation of the signal before it is forwarded to Bob2. The final step for Eve is to perform a collective measurement on the modes that she hasstoredin her quantum memory, that we denote as a quantum state µ. Theoretically, the maximum information that she can extract from those measurements is given by [59] S(x:E) = S(µ) − S(µ|x), (8.15) where S(·) denotes the von Neumann entropy, that can be calculated through the symplectic eigenvalues νi of an N-mode state, via N X νi + 1 νi + 1 νi − 1 νi − 1 S(σ) ≡ log2 − log2 , (8.16) i=1 2 2 2 2 and S(µ|x) is the von Neumann entropy for Eve’s quantum system conditioned on Alice or Bob’s measurement. The challenging part for Eve is to prepare, distribute and distill pure entangled states ρ over the corresponding distance between Alice and Bob. Thus, we assess her performance through the amount of entanglement of her state ρ. The least amount of entanglement needed for the simulation of the (non-entanglement-breaking) channel G (discussed in Sec. 7.3), and consecutively for any teleportation-based attack is given by E(ρ) ⩾ E(γmin), (8.17) 2The reason why the transmissivity takes the value t = 1/g is because we imposed on the all-optical teleportation protocol to simulate the exact same channel as the one that represents the environment, G. 124 All-optical Teleportation and Quantum Key Distribution Fig. 8.4 All-optical teleportation attack. In this attack, Eve has no access to the quantum channel G and she performs an all-optical teleportation over the signal sent from Alice. With ρ we denote Eve’s distilled resource state. The all-optical teleportation consists of a two-mode squeezer Sg with gain g, which is in Eve’s first station close to Alice’s laboratory, and a beam-splitter Bt with transmissivity t = 1/g, which is in Eve’s second station close to Bob’s laboratory. One mode of Eve’s resource state ρ is sent to the first station as an input of Sg. The other mode of ρ is sent to the second station and is mixed on a beam-splitter Bη with another state φ, before it becomes an input to Bt. Finally Eve performs the measurements M1−3 on her state µ that was stored in the quantum memory. For a pure channel G, both the entangling cloner and the all-optical teleportation attack need to substitute the state φ with a single-mode vacuum state |0⟩. where √ q 2 τ − (v + 1 − τ)(v − 1 + τ) γ = , (8.18) min τ + v + 1 with E being the entropy of entanglement [154, 155], given by3 2 2 2 2γ log2 γ + 1 − γ log2(1 − γ ) E(γ) ≡ . (8.19) (γ2 − 1)ln2 The reason we employ the entropy of entanglement to quantify the entanglement of the resource state ρ is because it is assumed to be pure. However, any proper entanglement measure, e.g., entangle- ment of formation [154, 155], relative entropy of entanglement [187], squashed entanglement [190] etc, reduces to Eq. (8.19) for pure states (see discussion in Sec. 5.3.4). We should note that for a pure lossy channel G the two-mode squeezed vacuum state φ is reduced to a single-mode vacuum |0⟩ for both the entangling cloner and the all-optical teleportation attack. Analysis of the Protocol For our numerical calculations we assume that the quantum channel that Alice and Bob initially share is a thermal lossy channel G with τ = 0.25, corresponding to approximately 30km of optical fiber, and ε = 1.01. Alice’s state is a two-mode squeezed vacuum with squeezing parameter ζ = 0.7, and the reconciliation efficiency is set equal to β = 0.95. Alice and Bob measure their modes with heterodyne 3Note here that we express the entropy of entanglement through the squeezing parameter γ = tanhr ∈ (0,1]. 8.7 All-optical Teleportation Attack 125 Fig. 8.5 Eve’s information and key rate. In figure (a) with the solid blue line we plot the amount of information S(b:E) Eve can extract in the reverse reconciliation scenario against the entanglement of her pure resource state E(ρ), parametrized over the squeezing parameter γ [see Eq. (8.19)]. The minimum value of entanglement E(γmin) corresponds to about 4.2dB of squeezing. The horizontal red dashed line represents the Holevo bound χ(b:E), that is the maximum amount of information an eavesdropper can physically extract. In the limit of infinite entanglement, i.e., E(γ → 1) → ∞, we see that this bound is reached. With the green dot-dashed line we indicate the maximum amount of information Eve can extract through an optimal individual attack [251, 252]. In figure (b) we have the corresponding key rate that Alice and Bob measure given Eve’s collective attack. Again, the red dashed line is the minimum possible key rate that they can extract, and the green dot-dashed represents the key rate based on the optimal individual attack. In both plots we indicate the non-physical areas with grey color. All quantities plotted are dimensionless. detection, and Eve performs an all-optical teleportation attack in the limit of g → ∞ (the protocol works for finite amount of gain g as well but with less success for Eve). Fig. 8.5 (a) shows Eve’s information for this protocol in the reverse reconciliation scenario, calculated via Eq. (8.15) and maximized over the parameters {η,κ} that can simulate the channel G, as a function of the entanglement of state ρ. Let us mention here that in case of a pure lossy channel G there is no need for an optimization, since the transmissivity can be just set equal to η = τ/γ2. The value for the Holevo bound that Eve has to reach is given by [59] χ = S(σout) − S(σout|b), (8.20) where S(σout|b) is the von Neumann entropy of the entangled state shared between Alice and Bob conditioned on Bob’s measurement. Fig. 8.5 (b) shows the achievable secret key rate calculated by Alice and Bob, based on Eq. (8.1) and the mutual information, which for this protocol is given by aτ + v + 1 H(a:b) = log . (8.21) 2 τ + v + 1 We also plot the corresponding values for Eve’s information and secret key rate under an optimal individual attack [251, 252]. We observe that when Eve uses the minimum required amount of entanglement resources, i.e., E(γmin), the attack reduces to the optimal individual attack. At this point, the beam-splitter Bη is not interacting with the signal, i.e., η = 1, and the squeezing parameter κ becomes irrelevant. 126 All-optical Teleportation and Quantum Key Distribution Eve’s information monotonically increases with the amount of entanglement, and it approaches the Holevo bound in the limit of infinite entanglement, i.e., E(γ→1) → ∞. In this extreme point, the beam-splitter Bη has transmissivity equal to the channel’s transmissivity, i.e., η = τ, and the state φ q has a squeezing parameter κ = (ε − 1)/(ε + 1). It is worth noting that the teleportation part of the protocol at this stage is operating under the Choi-state [212] (maximally entangled state sent through the channel). The notion of optimality in the extreme case of infinite entanglement E(γ→1) → ∞ is justified by the fact that a physical bound is reached i.e., the Holevo bound, that we cannot surpass. A meaningful question to ask at this point may be the following: for a given finite amount of entanglement what is the optimal collective attack? To this day, a physical limit that upper bounds the amount of classical information that can be extracted from a quantum channel under the use of finite entanglement has not been established. Thus, even though numerical searches indicate that the scheme proposed in this chapter seems to operate optimally for any value of entanglement, we will forgo making such a strong claim. Note that the key rate calculated in Fig. 8.5 is for the asymptotic regime. However, if we include the finite-size effects282 [ , 283], there would be some circumstances that while positive finite key rates cannot be generated from optimal collective attacks [under the unrealistic assumption of infinite entanglement with E(γ→1)], by considering optimal individual attacks [under the assumption that Eve can distill a pure entangled state with E(γmin)] we are able to move from insecure regime to secure regime, and generate non-trivial positive finite key rates. So, having access to a maximally entangled state is the trade-off in order to reach optimality in the teleportation-based attack without purifying the system, which is the assumption taken for the entangler cloner scheme. Both assumptions are non-realistic, however, the advantage of this type of attack compared to the entangling cloner is that it can be operated in the finite entanglement regime, in which it outperforms the optimal individual attack. Thus, the all-optical teleportation attack we introduced in this chapter can be thought of as a universal eavesdropping scheme for Gaussian QKD, that can be reduced to either optimal individual or optimal collective attack depending on the available entanglement resources, without assuming Eve has access to the entire environment. Finally, the fact that Eve needs an extremely large amount of entanglement in order to approach the optimal collective attack showcases the robustness of CV QKD protocols. Interestingly, even the minimum amount of entanglement given in Eq. (8.18) is arguably beyond current technological capabilities [284]. In DV protocols there is no need for a similar (finite entanglement) analysis, since maximally entangled states, i.e., Bell states, are not unphysical, and thus the entangling cloner scheme can operate without any unrealistic assumptions. Future Directions Apart from the proposed all-optical teleportation attack, another possibility for Eve is to use the hybrid type of teleportation introduced for qubits in Ref. [228] (and then generalized for qudits in Ref. [229]). In this scheme (see Fig. 8.6) a continuous variable state splits up to N two-dimensional states (qubits), that can probabilistically be teleported through a DV teleportation protocol (the DV teleportation 8.8 Conclusion 127 Fig. 8.6 Hybrid teleportation protocol. In this protocol, the basic components are: (i) the beamsplitters that split and later recombine the signal, and (ii) the conventional DV teleportations, with their corresponding DV resource states, Bell measurements BM, and final corrections through unitary operations U. should also be adjusted in a measurement-free scheme similar to Ref. [223]). In this scenario the need for infinite amount of entanglement of a single state is compensated with the need of infinite copiesof Bell states. With any finite amount of splitting though, i.e., finite amount of Bell states, this protocol would simulate a non-Gaussian channel due to the inevitable truncation of the initial state, so it would be interesting to investigate if this type of limitation gives any benefit to the eavesdropper. 8.8 Conclusion In this chapter, we showed that optimal collective attacks in continuous-variable QKD are always based on an extremely strong assumption that takes different forms depending on the way we model the eavesdropper, i.e., full-system purification, simulation of an identity channel, access to a resource state with infinite amount of entanglement, access to infinite copies of Bell states. However,the requirement of having access to a resource state with infinite amount of entanglement can be “tamed”, and a teleportation-based scheme can be modeled that operates in the regime between the optimal individual and optimal collective attacks depending on the available entanglement resources. Chapter 9 Secret Key Capacity of Gaussian Channels “In the beginning was the bit” — Seth Lloyd The results of this chapter have been published in: • Ref. [4]: Riccardo Laurenza, Spyros Tserkis, Leonardo Banchi, Samuel L. Braunstein, Timothy C. Ralph, and Stefano Pirandola, Tight bounds for private communication over bosonic Gaussian channels based on teleportation simulation with optimal finite resources, Physical Review A 100, 042301 (2019). Note on contributions. The paper in Ref. [4] is a follow-up work to Ref. [285]. The advances of Ref. [4] compared to Ref. [285] is that it employs results that were derived in Ref. [1]. The manuscript of Ref. [4] has been written and edited by the following authors: Spyros Tserkis, Stefano Pirandola, Leonardo Banchi and Riccardo Laurenza. The authors Samuel L. Braunstein and Timothy C. Ralph have also contributed by supervising the project (along with Stefano Pirandola). 9.1 Introduction The performance of a given channel can be assessed by its capacity for a given task, such as the transmission of classical information, quantum information, entanglement etc. Another useful type of capacity is the secret-key capacity of a quantum channel, that represents the maximum number of secret bits that two authenticated remote users may extract at the ends of the channel under LOCC operations. This capacity is particularly important because it upper-bounds the secret-key rate of any point-to-point protocol of quantum key distribution (QKD). In the continuous-variable context, those key rates are achievable by QKD protocols and are implemented with bosonic modes of the electromagnetic field, which are conveniently prepared in 130 Secret Key Capacity of Gaussian Channels Gaussian states (introduced in Chapter4). These quantum states are transmitted through optical fibers or free-space links, which are typically modeled as one-mode Gaussian channels, to be considered as the direct effect of collective Gaussian attacks. Exploring the ultimate achievable rates of CV-QKD and explicitly quantifying the secret-key capacity of a channel has been a very active research area [249, 286–289, 215, 290–292]. A lower bound to the secret-key capacity of the thermal lossy channel was given in Ref. [120] in terms of the reverse coherent information [65, 66]. For a pure lossy channel of transmissivity τ, this work showed that the rate of an optimal point-to-point QKD protocol can achieve a linear scaling of 1.44τ bits per channel use. An upper bound to the secret-key capacity was derived in Ref. [293], by resorting to the squashed entanglement, confirming the τ scaling in a pure-lossy channel. More recently, a tighter upper bound has been established in Ref. [114], by employing teleportation simulation over asymptotic resource states, namely, the asymptotic Choi-states of these channels. For a pure lossy channel, the lower and upper bounds of Refs. [120, 114] coincide, so that the secret-key capacity of this channel is fully established. In this chapter, we show that teleporting over a class of finite-energy resource states [1] allows us to closely approximate the ultimate bounds for increasing energy, so as to provide increasingly tight upper bounds to the secret-key capacity of one-mode phase-insensitive Gaussian channels. We then show that an optimization over the same class of resource states can be used to bound the maximum secret-key rates that are achievable in a finite number of channel uses. 9.2 Secret-key capacity and Bounds A general secret key generation protocol is based on adaptive LOCCs Λi, that is schematically depicted in Fig. 9.1. Each transmission through the quantum channel G is interleaved between two of such LOCCs. In particular, let us assume that two distant parties, Alice and Bob, have in their laboratories two local registers of quantum systems (modes), a and b, which are in a quantum state ρˆa ⊗ ρˆb. Before the first transmission, both parties apply an adaptive LOCC Λ0. In the first use of the channel, Alice picks a mode a1 from her register a and sends it to Bob through the channel G, who gets the output mode a2 in his register b. The parties apply another adaptive LOCC Λ1, and then there is a second transmission and so on. After n uses of the channel G, and a sequence of LOCCs {Λ0,Λ1,...,Λn} n ϵ the output state is given by ρˆab, which is ϵ-close to a target private state [294] with nRn bits. This ϵ procedure characterizes an (n,ϵ,Rn)-protocol P. The secret key capacity of the channel G is defined by ϵ K(G) ≡ suplimRn , (9.1) P n,ϵ where we have taken the limit of large n, small ϵ (weak converse) and optimized over P. 9.2.1 Upper Bound of Secret-ket Capacity Given a phase-insensitive Gaussian channel G with transmissivity τ and additive noise v, we can always (as discussed in Sec. 7.3 and can be schematically seen in Fig. 9.2) simulate it via a quantum 9.2 Secret-key capacity and Bounds 131 Fig. 9.1 Adaptive QKD protocol. In the first step, Alice and Bob prepare the initial separable state ρˆab of their local registers a and b by applying an adaptive LOCC Λ0. After the preparation of these registers, there is the first transmission through the quantum channel G. Alice picks a quantum system from her local register a1 ∈ a, which is therefore depleted as a → aa1; then, system a1 is sent through the channel G, with Bob getting the output b1. After transmission, Bob includes the output system b1 in his local register, which is augmented as b1b → b. This is followed by Alice and Bob applying another adaptive LOCC Λ1 to their registers a and b. In the second transmission, Alice picks and sends another system a2 ∈ a through the quantum channel G with output b2 received by Bob. The remote parties apply another adaptive LOCC Λ2 to their registers and so on. n This procedure is repeated n times, with the output state ρˆab being finally generated for Alice’s and Bob’s local registers. teleportation protocol using a resource state ρˆ with a covariance matrix ρ of the form of Eqs. (7.5a- 7.5c). Replacing each transmission through the channel by the corresponding teleportation protocol n ⊗n and “stretching” the the adaptive protocol into a block form we may write ρˆab = Λ(ˆρ ) for a trace- preserving LOCC Λ. Then, the secret key capacity can be upper-bounded by the relative entropy of n entanglement (REE) of the resource state ρˆab, but since REE is monotonic under Λ (data processing) and sub-additive over tensor-products, we have K(G) ⩽ ER(ˆρ). (9.2) 9.2.2 Gaussian Relative Entropy of Entanglement Relative entropy of entanglement ER, as discussed in Sec. 5.4.3, quantifies how much a given entangled state can be distinguished operationally by the set of all separable states. More rigorously, for a state ρˆ, it is defined [187] as follows ER(ˆρ) ≡ minS(ˆρ∥ρˆi), (9.3) ρˆi∈S where S is the set of all separable states and S(ˆρi∥ρˆj) ≡ tr[ˆρi(log2 ρˆi − log2 ρˆj)] , (9.4) is a quasi-distance between states ρˆi and ρˆj, called the quantum relative entropy (see Sec. 3.3.3 for more details). In general, ER lacks a closed formula for arbitrary states, and there are only special 132 Secret Key Capacity of Gaussian Channels Fig. 9.2 Finite-resource simulation of bosonic Gaussian channels. In panel (a), we depict a phase-insensitive Gaussian channel G transforming the input state σˆin into the output state σˆout. In panel (b), we show its simulation by means of a teleportation LOCC Λ. Its basic components are: (i) a dual Homodyne Detection (HD) between the input state σˆin and the resource state ρˆ =ρ ˆτ,v as in Eqs. (7.5a)-(7.5c); (ii) the classical communication (CC) of the Homodyne Detection outcomes; and (iii) a conditional phase-space displacement D with suitable gain [225] which provides the output teleported state σˆout. cases where an analytical expression is known. Even numerically this optimization is quite challenging, since the quantum relative entropy is also not easy to be expressed for arbitrary states. Given a Gaussian entangled state ρˆG we can define the Gaussian relative entropy of entanglement (GREE) as ERG (ˆρG) ≡ min S(ˆρG∥ρˆGi ), (9.5) ρˆi∈SG where the optimization now runs over only the Gaussian separable states SG, so it is by construction an upper bound of the actual measure of relative entropy of entanglement, i.e., ER(ˆρ) ⩽ ERG (ˆρ). (9.6) As we have seen in Sec. 4.4.2, quantum relative entropy in Gaussian states has a closed formula, which makes the quantification of the measure an easier task. Let us have a two-mode Gaussian state ρˆG with vanishing first moment, that can be fully described by the covariance matrix ρ in the standard form, i.e., a 0 c+ 0 0 a 0 c− ρ = . (9.7) c+ 0 b 0 0 c− 0 b The calculation of GREE through Eq. (9.5) involves in general the optimization over 3 parameters [295, 296], that can only be done numerically. However, we can define an upper bound to the GREE by fixing a candidate closest separable state114 [ , Supp. Note 4]. Specifically, for a Gaussian state ρˆG ∗ with covariance matrix ρ of the form of Eq. (9.7), we pick a separable state ρˆsep that has a covariance ∗ matrix ρsep, with the same diagonal blocks as ρ, but where the off-diagonal terms are replaced as 9.3 Upper Bounds for Bosonic Gaussian Channels 133 follows q a 0 (a − 1)(b − 1) 0 q ∗ 0 a 0 − (a − 1)(b − 1) ρ ≡ q . (9.8) sep (a − 1)(b − 1) 0 b 0 q 0 − (a − 1)(b − 1) 0 b The physical justification for this particular choice as the candidate separable state isthefact that this state: (i) contains the maximum correlations among the separable states as quantified by its (unrestricted, generally non-Gaussian) quantum discord [297], and (ii) it lies on the border of separability. ∗ Using the separable state ρˆsep we can then write the upper bound ∗ ∗ ER(ˆρ) ⩽ ER(ˆρ) ≡ D(ˆρ∥ρˆsep). (9.9) A specific algorithm written in MATHEMATICA has also been developed1 for the calculation of this upper bound of the Gaussian relative entropy of entanglement. 9.3 Upper Bounds for Bosonic Gaussian Channels Since we are able to analytically calculate an upper bound for the Gaussian relative entropy of entanglement of a state, we can further upper bound the secret-key capacity in Eq. (9.2) as follows ∗ K(G) ⩽ ER(ˆρ) ⩽ ER(ˆρ). (9.10) The tightest bound (that has been found so far) is reached by minimizing the GREE over the whole 2 class of resource states R(ν−,ν+) that can simulate a phase-insensitive Gaussian channel, derived in Eqs. (7.5a-7.5c) and Eqs. (7.8a-7.8c). Then, for any symplectic eigenvalue ν− and ν+, we have the following bound ∗ K(G) ⩽ Bν−,ν+ ≡ min E(ˆρ∥ρˆsep). (9.11) ρˆ∈R(ν−,ν+) The above bound takes its minimum value when the resource state is the asymptotic Choi-state of the channel [114]. For thermal lossy and thermal amplifier channels this state corresponds to ν− = 2¯n + 1, (9.12a) ν+ → ∞, (9.12b) while for additive noise channels Choi-state corresponds to ν± → ∞. (9.13) 1The MATHEMATICA file with the code can be downloaded from spyrostserkis.com 2 The parameters ν− and ν+ denote the symplectic eigenvalues of the covariance matrices of the two-mode Gaussian states. 134 Secret Key Capacity of Gaussian Channels Thus, based on Eq. (9.11), monotonically decreasing bounds can be created in the following way: For thermal lossy and thermal amplifier channels, we fix the lowest symplectic eigenvalue to be equal to ν− = 2¯n + 1, so for increasing ν+ (increasing energy and decreasing purity) we monotonically approach the minimum value obtained for ν+ → ∞, i.e., B2¯n+1,∞ ≡ lim B2¯n+1,ν+ B2¯n+1,ν+ . (9.14) ν+→∞ ≲ For additive noise channels, we set ν− = ν+ = ν, and for increasing ν (increasing energy) we monotonically approach the minimum value for ν → ∞, i.e., B ≡ lim B B . (9.15) ∞,∞ ν→∞ ν,ν ≲ ν,ν In the next section, we demonstrate the behavior of these bounds for the various phase-insensitive Gaussian channels. 9.3.1 Thermal Lossy Channel A thermal lossy channel L can be modeled as a beam-splitter operation with transmissivity 0 ⩽ τ ⩽ 1, that mixes the input state with a thermal state with variance 2¯n + 1. The channel is called pure lossy Lp if the secondary state is the vacuum, i.e., n¯ = 0. The secret-key capacity of the thermal lossy channel L is upper bounded by n¯ τ −log2[(1 − τ)τ ] − h(¯n) for n¯ < 1−τ K(L) ⩽ B2¯n+1,∞(L) = , (9.16) 0 otherwise where we set h(x) ≡ (x + 1)log2(x + 1) − xlog2 x. (9.17) For the pure lossy channel Lp (n¯ = 0) we have the exact formula K(Lp) = B1,∞(Lp) = −log2(1 − τ). (9.18) In Fig. 9.3 we compute the bound Bν−,ν+ for a thermal lossy channel by fixing ν− = 2¯n + 1 under n¯ = 1 and increasing ν+. As we can see, the finite-resource bound B2¯n+1,ν+ approaches B2¯n+1,∞(L) for increasing values of ν+ and this approximation can be made as close as needed according to Eq. (9.14). In Fig. 9.3, we also show the corresponding performance for a pure lossy channel Lp as well. 9.3.2 Thermal Amplifier Channel A thermal amplifier channel A can be modeled by a parametric amplifier [281], also called two-mode squeezing operation, with gain τ ⩾ 1, which is applied to the input state together with a thermal state 9.3 Upper Bounds for Bosonic Gaussian Channels 135 Fig. 9.3 Upper bounds to the secret-key rate capacity of lossy and amplifier channels (secret bits per channel use versus transmissivity 0 ⩽ τ ⩽ 1 or gain τ ⩾ 1). In panels (a) and (c) we show the results for pure lossy and pure amplifier channels, while panels (b) and (d) show the corresponding results for thermal lossy andthermal amplifier channels with n¯ = 1. In the panels the lower blue line indicates the infinite-energy bound B2¯n+1,∞ of Ref. [114] while the green dashed line is the approximate finite-energy bound B˜ of Ref. [285], which is computed over the class of states of Ref. [220]. The black dashed line corresponds to our finite-energy bound B1,1 computed with a pure resource state (ν± = 1). Note that, for pure lossy channels, this bound B1,1 coincides with the previous finite-energy bound given in285 [ ]. Then, the red dashed line is our finite-energy bound Bν−,ν+ , computed with ν− = 1, ν+ = 100 for pure lossy and pure amplifier channels, and with ν− = 2¯n + 1, ν+ = 500 for thermal lossy and thermal amplifier channels. As we see for increasing values of ν+, and thus increasing simulation energy, we can approximate B2¯n+1,∞ as closely as we want, while keeping the energy of the resource state finite (although large). All quantities plotted are dimensionless. with variance 2¯n + 1. The secret key capacity of the thermal amplifier A is upper bounded by τ−1 1 −log2 τ n¯+1 − h(¯n) for n¯ < τ−1 K(A) ⩽ B2¯n+1,∞(A) = . (9.19) 0 otherwise For the pure amplifier Ap (n¯ = 0) we have the exact formula 1 K(A ) = B (A ) = −log 1 − . (9.20) p 1,∞ p 2 τ In Fig. 9.3 we compute the bound Bν−,ν+ for a thermal thermal channel by fixing ν− = 2¯n + 1 under n¯ = 1 and increasing ν+. We see that we can closely approximate B2¯n+1,∞(A) as much as we want. In the same figure, we also plot the bound B1,∞(Ap) for a pure amplifier channel. 136 Secret Key Capacity of Gaussian Channels Fig. 9.4 Upper bounds to the secret-key capacity of the additive-noise Gaussian channel (secret bits per channel use versus added noise v). The lower blue line indicates the infinite-energy bound B∞,∞ of Ref. [114]. Then, we show our improved finite-energy bound Bν,ν which is plotted for pure resource state, i.e., ν = 1 (black dashed line) and for ν = 10 (red dashed line). Note that the previous bound given in [285] coincides with our finite-bound B1,1. For increasing values of ν we can approximate B∞,∞ as closely as we want, while keeping the energy of the resource state finite (despite being large). All quantities plotted are dimensionless. 9.3.3 Additive Noise Channel The additive noise Gaussian channel N can be modeled as the asymptotic case of either lossy or thermal channel, where τ ≈ 1 and a highly thermal state, i.e., classical, at the secondary input. Its secret-key capacity is upper-bounded by v−2 v 2ln2 − log2 2 for v < 2 K(N ) ⩽ B∞,∞(N ) = . (9.21) 0 otherwise In Fig. 9.3 we present how the bound Bν,ν behaves for increasing values of ν, and how closely it can approximate the infinite-energy bound B∞,∞(N ). 9.3.4 Finite uses of the channel Apart from the asymptotic secret key capacity, the parametrization of resource states R(ν−,ν+) [see Eqs. (7.5a)-(7.5c) and Eqs. (7.8a)-(7.8c)] can also be used to bound the maximum finite-size key rate ϵ that is achievable by an (n,ϵ,Rn)-protocol P, i.e., a QKD protocol which is implemented for a finite number n of times with security ϵ. In particular, using the methods of channel simulation discussed before for a phase-insensitive Gaussian channel G, we can derive the following bound [114, 298] 1 ⊗n K (G) Dϵ ρˆ⊗n∥ ρˆ∗ , (9.22) n,ϵ ⩽ n h sep ϵ where 0 < ϵ < 1 and Dh is the hypothesis testing relative entropy [299]. Then, based on Ref. [299] we have ⊗n r ϵ ⊗n ∗ ∗ ∗ Dh ρˆ ∥ ρˆsep = nS ρˆ∥ρˆsep + nV ρˆ∥ρˆsep F (ϵ) + O (logn) , (9.23) 9.3 Upper Bounds for Bosonic Gaussian Channels 137 Fig. 9.5 Secret-key bits versus number n of uses of a thermal lossy channel L with transmissivity τ = 0.01 corresponding to 100km of standard optical fiber and thermal numbern¯ = 0.0011 corresponding to δ ≃ 0.1 excess −2 noise. We assume a security parameter ϵ = 10 . We plot the optimized finite-size bound Φn,ϵ(L) computed from Eq. (9.26b) (solid red line) which approaches the asymptotic value B2¯n+1,∞(L) of Eq. (9.16) for large n (red dashed line). The optimal resource state ρˆ may have low energy at finite n. For instance, at n = 2 × 103, this state has spectrum ν− ≃ 0.5005 and ν+ ≃ 1.66832. For comparison, we also plot the bound (solid blue 7 line) that we would obtain with a resource state of high energy, namely ν− ≃ 0.5011 and ν+ ≃ 1.99561 × 10 . where F is the inverse of the cumulative Gaussian distribution, given by F (ϵ) = sup{a ∈ R |f(a) ⩽ ϵ} , (9.24a) Z a f(a) = (2π)−1/2 dxexp(−x2/2) . (9.24b) −∞ Through Eqs. (9.22) and (9.23), we have v u ∗ ! uV ρˆ∥ρˆsep logn K (G) S ρˆ∥ρˆ∗ + t F (ϵ) + O , (9.25) n,ϵ ⩽ sep n n as also discussed in Ref. [246]. We should note that the bound in Eq. (9.25) is valid as long as the third moment is finite, a condition that is certainly satisfied by energy-constrained zero-mean Gaussian states. It is important to remark that the actual value of the third moment has to be carefully considered in order to apply Eq. (9.25) at small number of uses n. In other words, the scaling O n−1 logn may actually be affected by a large pre-factor, so that it becomes effective only for very large n. Thus, Eq. (9.25) has to be interpreted as an approximate bound when applied to relatively small n. Consider a phase-insensitive Gaussian channel G and optimize the finite-size bound in Eq. (9.25) over the entire class of resource states R(ν−,ν+). Then we have logn! K (G) Φ (G) + O , (9.26a) n,ϵ ⩽ n,ϵ n v u uV ρˆ∥ρˆ∗ ∗ t sep Φn,ϵ(G) ≡ min S ρˆ∥ρˆsep + F (ϵ) . (9.26b) ρˆ∈R(ν−,ν+) n 138 Secret Key Capacity of Gaussian Channels In Fig. 9.5 we plot this approximate bound for the case of a thermal lossy channel L (similar calculations hold for the rest of the phase-insensitive channels as well). For our calculations we compute the finite-size optimized bound Φn,ϵ(L) for its n-use ϵ-secure secret-key capacity Kn,ϵ(L), assuming the numerical value ϵ = 10−2 (smaller values can be considered but with further approximations, unless n is of the order of ϵ−2). It is apparent that the bound converges for increasing n. In particular, the plot refers to a distance of 100km in standard optical-fiber at the loss rate of 0.2dB/km and assumes an excess noise of δ ≡ (1 − τ)τ −1n¯ ≃ 0.1. 9.4 Conclusion In this chapter, we improved the finite-energy upper bounds to the secret-key capacities of one-mode phase-insensitive Gaussian channels3, using an optimization over all possible resource states derived in Ref. [1] (and discussed in Chapter7) that can simulate a channel. We have also shown that the finite-energy bounds can approximate the asymptotic bound[114] as close as wanted. The advantage of a finite-energy analysis is that it removes the need for using an asymptotic simulation at the level of the resource state, which is numerically a hard task. The same approach can also be extended to repeater chains and quantum networks [300, 301]. Another interesting extension might be a finite-energy analysis in an adaptive quantum metrology regime and quantum channel discrimination [302, 303], for applications such as quantum sensing [304]. 3phase-sensitive Gaussian channels have not been considered because it is not yet known how to simulate them through teleportation protocols. Part IV Conclusion Chapter 10 Concluding Remarks and Future Outlook “Most of the propositions and questions of philosophers arise from our failure to understand the logic of our language. And it is not surprising that the deepest problems are in fact not problems at all." — Ludwig Wittgenstein This thesis investigated the topic of Gaussian quantum information by (i) deriving new mathematical tools for the quantification of entanglement and the secret-key capacity of Gaussian channels, and by (ii) proposing a new models for error-correction of Gaussian states and eavesdropping Gaussian quantum key distribution attacks. The major results of this thesis and their corresponding future outlook are given below. 10.1 Quantification of Entanglement in Gaussian States In chapter6 we included results from three different publications [ 15, 1, 2], regarding the quantification of entanglement in two-mode Gaussian states. In particular, we focused on a specific entanglement measure, i.e., entanglement of formation, which apart from special cases it lacks an analytic closed formula. We derived analytical lower and upper bounds of the measure that lie relatively close to its actual value (estimated numerically), especially for high purities of mixed states. We also discussed how entanglement of formation for those systems can be easily calculated numerically based on the aforementioned bounds. Finally, we compared this measure with another entanglement quantifier, i.e., logarithmic negativity, showing that the latter has in general an inconsistent behavior even though it is relatively easier to compute. 142 Concluding Remarks and Future Outlook Future Perspectives The method used to derive the lower and upper bounds can be briefly described as follows: The entanglement of formation is linked to the minimum squeezing resources needed for the creation of a quantum state starting from a classical one. Instead of estimating this amount of resources, which turns out to be a really hard task, we calculate the minimum amount of squeezing resources that would make this state separable. This method is not limited to two-mode Gaussian states, and thus it can potentially extended in other set of states, e.g. multi-partite Gaussian states or non-Gaussian states. 10.2 Error Correction of Gaussian States under Gaussian Noise In chapter7 we included the results of Ref. [ 1], dealing with error-correction of Gaussian states. More specifically, when a Gaussian state passes through a Gaussian channel it gets decohered duetothe noise induced from the environment, e.g., optical fiber, free-space etc. If this quantum state was used to encode quantum information for a protocol then this noise is manifested as errors in the final result. Using standard quantum teleportation in conjunction with entanglement distillation we proposed an error-correction scheme that aims to moderate the decoherence induced from the channel. A common figure of merit for those type of protocols is fidelity, we show however, its potential problematic behavior and we instead use another method to evaluate the performance of the error-correction. In particular, the noise that is induced to a quantum state can always be associated with a quantum channel. We show how the protocol (when successful) corresponds to another quantum channel and we evaluate how this new (less decohering) channel can degrade the entanglement of state that is sent through it. Future Perspectives The error-correction protocol discussed in this chapter is able to create a less decohering channel between two distant parties. However, a possible extension of the protocol would be its concatenation in order to create a more robust link among several parties. Thus, this type of protocol might serve as a building block of a continuous-variable quantum repeater for broader applications such as a quantum network. 10.3 Teleportation-Based Eavesdropping Attacks In chapter8 we presented the results of Ref. [ 3], where we discussed the ways an eavesdropping attack can be modeled in a quantum key distribution scenario. The current way of modeling the adversary is by assuming full access to the shared quantum channel between the parties that they want to establish the secret key. We challenged this assumption and showed that the eavesdropper can optimally attack the system even without having full control of the system if an all-optical teleportation is performed over the channel (that is associated with the environmental noise that is beyond anyone’s control). We related the amount of entanglement resources used during the all-optical teleportation with the amount of information that can be extracted from the system, and we calculated the minimum 10.4 Bounds for Secret-Key Capacities of Gaussian Channels 143 necessary resources for any teleportation-based attack. We finally argued in favor of the robustness of the continuous-variable quantum key distribution since the amount of entanglement that is needed to achieve an optimal attack is infinite. Future Perspectives It is tempting to assume that the all-optical teleportation attack upper bounds the amount of information Eve can extract from the system under any given entanglement resources, however, this has to be proven rigorously, which is an interesting future path. It would also be interesting to be investigated if this type of attack can be used in other type of cryptographic protocols within the continuous-variable regime or even extended in discrete-variable scenarios. 10.4 Bounds for Secret-Key Capacities of Gaussian Channels In chapter9 we discuss the results of a publication [ 4], regarding the quantification of the secret key capacity of phase-insensitive Gaussian channels. The secret key capacity of a channel cannot in general be measured but can be lower and upper bounded. The tighter known upper bound for this type of channels involves a method called “simulation of channel through teleportation” that assumes an asymptotic state, i.e., the Choi state. We show that this bound can be closely approximated with the same method under finite-energy resource states. 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[23, 25], we extend this discussion into infinite-dimensional Hilbert spaces. Let us have a continuous complex function ψ(x) with a domain 0 ⩽ x ⩽ L. In order to represent ψ(x) as a vector (see Fig. A.1), we can discretize the interval L into n equal parts of length ∆x = L/n, such that for each value of the function ψ(x) there corresponds an element ψi in the vector, i.e., ψ0 = ψ(0), (A.1a) ψ1 = ψ(∆x), (A.1b) . . ψn = ψ(L = n∆x). (A.1c) Since the interval L is continuous, we can consider the limit of an infinite amount of n discrete parts, i.e., n → ∞ (and thus making ∆x infinitesimally small, i.e., ∆x → 0), so we have the following correspondence ∞ X ψ(x) ←→|ψ⟩ = ψi|i⟩, (A.2) i=0 that belongs into the infinite-dimensional Hilbert space H ∞. For finite n the inner product between two vectors (that represent continuous complex functions with a domain 0 ⩽ x ⩽ L) would be n X ∗ ⟨φ|ψ⟩ = φi ψi∆x, (A.3) i 164 Rigged Hilbert Space Fig. A.1 Discretization of a real continuous function. A continuous function ψ(x) corresponds to an infinitely Pn dimensional vector |ψ⟩ = limn→∞ i=0 ψi|i⟩, if we discretize the interval L into equal n parts of length ∆x = L/n and then associate the value of the function at this point with an element ψi. and the basis vectors |x⟩ in this context would take the the form T |xi⟩ = (0,0,··· ,1,··· ,0,0) , (A.4) where the number one is in the “i-th” place. As we take the limit n → ∞ we end up with Z L ⟨φ|ψ⟩ = φ∗(x)ψ(x)dx, (A.5) 0 where the completeness relationship for the eigenbasis |x⟩ requires that Z L |x⟩⟨x|dx = 1. (A.6) 0 Applying an arbitrary vector |ψ⟩ form the right, and the basis vector ⟨x′| from the left in the above equation we get Z L ⟨x′|x⟩⟨x|ψ⟩dx = ⟨x′|ψ⟩ ≡ ψ(x′), (A.7) 0 representing a projection of the vector |ψ⟩ along the basis |x′⟩. Let us define the inner product of the eigenbasis vectors as a function δ(x′,x) = ⟨x′|x⟩, that we call Dirac delta function, which according to the orthonormality condition should vanish if x ̸= x′. Restricting the integral to an infinitesimal region ϵ near x′ we get Z x′+ϵ δ(x′,x)ψ(x)dx = ψ(x′), (A.8) x′−ϵ 165 and as x′ approaches x, the value of function ψ(x′) should also approach the value of ψ(x), and thus Z x′+ϵ x′ → x ⇒ δ(x′,x)dx = 1, (A.9) x′−ϵ which implies that δ(x′,x) should approach infinity in this limit. As a result, the norm ofthose eigenvectors |x⟩ diverges to infinity, i.e., q ∥x∥2 = ⟨x|x⟩ → ∞. (A.10) This normalization issue (among other similar issues) for Hilbert spaces containing continuous functions as elements demands a modified mathematical framework, that is presented below. Rigged Hilbert Space Let us start by the Hilbert space H , which as we discussed in chapter2, consists of all the vectors X |ψ⟩ = λj|ej⟩, (A.11) j where λj ∈ C, {|ej⟩} is a basis in H , and the vectors |ψ⟩ admit finite-valued inner products X 2 ⟨ψ|ψ⟩ = |λj| < ∞. (A.12) i We can define the nuclear space N , that has elements given by X |φ⟩ = µj|ej⟩, (A.13) j with coefficients µj ∈ C, restricted by the following set of conditions X 2 m |µj| j < ∞ for m = 0,1,2,... (A.14) j It is clear that the space N is a subspace of H , i.e., N ⊂H , since its elements are restricted under Eq. (A.14), which as a condition is stronger than the corresponding for the Hilbert space given in Eq. (A.12). We can also define the conjugate space of N , i.e., N ×, which consists of the elements X |χ⟩ = νj|ej⟩, (A.15) j P ∗ 1 such that the inner product ⟨χ|φ⟩ = j νj µj is a continuous functional and convergent for all |φ⟩ ∈ N . Such a condition, though, makes the space N × larger than the Hilbert space H , i.e., H ⊂N ×. In total, the triplet of spaces N ⊂H ⊂N × is called a rigged Hilbert space (also called a Gelfand triplet) 1The notion of dual space, N ′, as defined in Sec. 2.1, can be considered as the complex conjugate of N ×. 166 Rigged Hilbert Space [20], and interestingly the eigenvectors of any Hermitian operator [such as the one defined in Eq. (3.79)] exist in the extended space N ×. Appendix B Noiseless Linear Amplification “Quantum phenomena do not occur in a Hilbert space. They occur in a labora- tory.” — Asher Peres Noiseless Linear Amplification (NLA) is a non-deterministic measurement procedure [242, 305, 213], described (in an idealized fashion) by the operator gnˆ, that implements the number state transfor- mation, gnˆ|n⟩ = gn|n⟩, (B.1) where g ≥ 1 is an experimentally accessible gain. Interestingly, NLA can be used to distill entanglement, since when, for instance, it is applied to a single mode of a two-mode squeezed vacuum state, it probabilistically gives q ∞ q ∞ gnˆ 1 − χ2 X χn|nn⟩ → 1 − χ2 X gnχn|nn⟩. (B.2) n=0 n=0 Thus, when the NLA succeeds, the squeezing parameter is increased, i.e., χ → gχ, which implies entanglement distillation. When the NLA is experimentally implemented with linear optics, it consists of N modified quantum scissors devices [306], as seen in Fig. B.1. The input state is split on an array of beam-splitters with each mode then being passed through an individual quantum scissor. The modes are then coherently recombined to form the output state, with the correctly amplified state being achieved only when each quantum scissor heralds successful operation. In the ideal case, the NLA operation is given by gnˆ. However, using N quantum scissors the corresponding operation is given by [242, 240] ˆ nˆ TˆN := ΠN g , (B.3) 168 Noiseless Linear Amplification Fig. B.1 Noiseless linear amplification. Each quantum scissor operation (QS) consists of two beam splitters. The input signal is mixed on a balanced beam splitter with an ancilla signal and both outputs are measured using photon detectors D1 and D2. The ancilla signal is one of the two outputs of a single photon passing through a tunable beam splitter with ratio ξ, while the other signal is the overall output of the quantum scissor. Successful quantum scissor operation is heralded when a single photon is detected at D1 and none at D2 or vice versa. Using N quantum scissors and two N splitters (one to divide and another to recombine the signal), we can approximate the ideal NLA in the limit of N → ∞. ˆ where ΠN is the truncation operation defined as !N/2 N ˆ 1 X N! ΠN := 2 n |n⟩⟨n|. (B.4) 1 + g n=0 (N − n)!N 169 This operation leads to a state truncation in the photon number basis to order N, with q g = (1 − ξ)/ξ , (B.5) being a gain controlled by a tunable beam-splitter ratio, ξ. The physical construction of the NLA reduces to the ideal one only in the unphysical asymptotic limit of N → ∞. Successful operation of the NLA decreases exponentially with N. If we consider the simplest setup, i.e., the case where the NLA consists of a single quantum scissor, then we have the following transformation ˆ α|0⟩ + gβ|1⟩ T1(α|0⟩ + β|1⟩ + γ|2⟩ + ···) = q , (B.6) 1 + g2 with all higher order terms being truncated. The effect of this truncation is to introduce a small amount of excess noise to the output state. Due to the nature of this operation, the effect of this truncation on large amplitude input states is severe and will result in a large amount of excess noise. Therefore, when the NLA is implemented with a single quantum scissor, the error correction protocol performs best in the high-loss regime.