Qubit Carriers with Internal Degrees of Freedom in a Non-Factorable State

Qubit carriers with internal degrees of freedom in a non-factorable state

Martina Mikov´a,Helena Fikerov´a,Ivo Straka, Michal Miˇcuda, Miroslav Jeˇzek,Miloslav Duˇsek,and Radim Filip Department of Optics, Faculty of Science, Palacky University, 17. listopadu 12, 771 46 Olomouc, Czech Republic

A directly measurable parameter quantifying effective indistinguishability of particles as a resource for quantum information transfer and processing is proposed. In contrast to commonly used overlap of quantum states of particles, defined only for a factorable states, this measure can be generally applied to any joint state of the particles. The relevance of this generalized measure for photons produced in parametric down-conversion has been experimentally verified. The simplest linear- optical quantum-state-transfer protocol, for which this measure directly determines fidelity of the transferred state, was experimentally tested. It has been found that even if some degrees of freedom of two particles are entangled, the particles can still serve as good carriers of qubits.

PACS numbers: 03.67.Mn, 42.50.Ex, 03.67.Lx

I. INTRODUCTION uniquely on indistinguishability of particles. We consider transfer of a state of a source single-photon qubit (S) to a single-photon target qubit (T ). The transfer is performed In last few decades quantum physics has offered novel by a partial exchange of photons, optimal measurement applications in information and communication technol- on S, and conditional feed-forward correction on T . We ogy. Their performance crucially depends on the quality show that fidelity of the transferred state depends di- of elements of quantum information – qubits [1]. Nec- rectly on D even if the internal degrees of freedom of the essary conditions for high-fidelity qubits are their coher- particles are entangled. In more complex quantum proto- ence [2, 3], which appears when no information is leak- cols, the quality of information processing may depend on ing into an environment, and indistinguishability [4–6]. a nontrivial combination of the effective indistinguisha- By effective indistinguishability of two (spatially sepa- bility and the properties of other resources. rated) particles we mean that all their internal degrees of freedom not used to carry information are identical. For factorable state ρA ⊗ ρB of two particles A and B II. OPERATIONAL MEASURE OF it means that the action of a flip (exchange) operator, INDISTINGUISHABILITY F (ρA ⊗ ρB) F = ρB ⊗ ρA, does not change the state. The mean value of the flip operator for a factorable state Let us have two particles, S and T , carrying the same equals to the overlap of states of individual subsystems, qubit states, let ρE,ST denote the state, not necessarily Tr [F (ρA ⊗ ρB)] = Tr [ρAρB] [7]. This is a hint for an in- separable, of all of the other (inaccessible) degrees of free- distinguishability measure. But these considerations are dom. The internal environmental degrees of freedom can still valid only for factorable states. Direct measurement even be entangled with an external environment. Clearly, of the overlap was already suggested for qubits [8] and they are responsible for distinguishability of the particles. harmonic oscillators [9]. For the simplest case of two pho- Let us define a measure |D| describing an effective indis- tons, it was measured by Hong-Ou-Mandel (HOM) type tinguishability, where interferometry [10]. There are many other related two- photon experiments [11–17]. Indistinguishability of par- D = Tr [FA ρE,ST ] . (1) ticles is crucial in a number of quantum protocols which were intensively studied in recent years and are still of a P Operator FA = m,n |ψniShψm| ⊗ |ψmiT hψn| is a flip

arXiv:1209.0908v2 [quant-ph] 11 Apr 2013 great interest [18–28]. operator acting on the joint environment of both par- In this paper, we propose a directly measurable param- ticles, which exchange basis states corresponding to a eter, D, quantifying effective indistinguishability of par- given observable A, where A|ψni = an|ψni. Properties ticles which can be used for an arbitrary state. Effective of Tr [FA ρE,ST ] follows from the features of operator FA. † indistinguishability can be defined by means of the flip Operator FA is both hermitian FA = FA and unitary † † operator exchanging relevant degrees of freedom of the FAFA = FAFA = 1 [7]. Since FA is commuting with particles. Full flip of particles corresponds to complete any local unitary transformation UES ⊗ UET , it is invari- exchange of their quantum states. On the other hand, ant to a choice of operator A and therefore, we can con- transfer by quantum teleportation relies both on the par- sider F instead FA as a basis-independent operation. F ticle indistinguishability and entanglement. Therefore, can be also expressed as a difference, F = Πsym − Πanti, to show how distinguishability of particles used as infor- of orthogonal projectors Πsym and Πanti onto the sym- mation carriers affects quantum information processing metric and anti-symmetric subspace of the total space, without the influence of other imperfections of resources, respectively. Therefore it is directly linked to indistin- we designed a quantum-state-transfer protocol depending guishability of the environmental states. Clearly, F is a 2 dichotomic observable (with eigenvalues ±1) and we get Thus parameter |D| quantifies effective indistinguishabil- −1 ≤ Tr [F ρ] ≤ 1. Parameter D is invariant under sym- ity of resources for quantum information processing. metric local unitary transformations UES ⊗ UET (indistinguishability cannot change if the particles go through the same unitary channels). In contrast, entanglement of III. PHOTONIC QUBIT TRANSFER the environments have to be invariant under more general unitary transformations UES ⊗VET , where UES may To demonstrate relevance of effective indistinguishabil- differ from VET . The conceptual difference is that a lo- ity |D|, we have proposed and experimentally tested the cal unitary applied on the environment of one particle simplest example of a quantum information transfer, in can make its state distinguishable from the state of the which |D| alone directly determines quantum fidelity of other particle, although it does not change the amount the transferred states. It manifests a clear operational of correlations and entanglement between the environ- meaning of the above defined effective indistinguishabil- ments. Any symmetric state of two particles satisfying ity. ρE,ST = F ρE,ST = ρE,ST F has D = 1, irrespective of We consider only the equatorial states of qubit S repre- its entanglement. States related by the permutation op- sented by a dual-rail superposition of single photon states 0 eration, ρE,ST = F ρE,ST F , have the same values of D. P 1 For separable state ρE,ST = n pn ρn,ES ⊗ ρn,ET of two |ΨiS = √ [|0, 1iS + exp(iθ)|1, 0iS], (2) particles, D is always positive semi-definite. If a twirling 2 transformation is applied to any input state ρE,ST (i.e., if identical random unitaries are applied to both qubits) where phase θ may be unknown during the transfer. This parameter D does not change and the resulting state is state should be transferred to target qubit T represented the Werner state which is fully parameterized by D. For by another single photon, which is in state factorable states D = Tr [ρ ρ ] and it reduces to the ES ET 1 overlap of S and T particles. Notice however, that in |ΦiT = √ (|0, 1iT + |1, 0iT ) (3) general D is not equal to the overlap. The problem of 2 the overlap lies in the assumption that quantum states of at the beginning. All other degrees of free- all, even unaccessible, degrees of freedom of two bosons dom are described by a density matrix ρ = are factorized. Such assumption cannot be operationally E,ST P c |ψ i hψ | ⊗ |ψ i hψ |, where i, j, k, l are certified, except one achieves an unrealistic complete to- i,j,k,l ij,kl i ES j k ET l multi-indices over many different sub-degrees of freedom mography of a joint quantum state corresponding to all of the joint “environment” of photons and |ψ i denote degrees of freedom of both particles. x basis states of each photon. We consider that all physi- Alternatively, the flip operator can be rewritten as cal differences between the particles are contained in this TA F = |Ψ˜ +iST hΨ˜ +| representing an entanglement wit- environmental state. So the overall initial state of the ˜ P qubits reads ρini = |ΨiShΨ| ⊗ |ΦiT hΦ| ⊗ ρE,ST . ness, where |Ψ+i = k |ψkiS|ψkiT is an unnormalized In general, an imperfect interaction between qubits symmetric state. Thus D < 0 is a witness of entangle- can also limit quality of the transfer. We there- ment in ρE,ST (due to the presence of an anti-symmetric fore consider implementation without any direct in- component) [7, 9, 29]. However, we do not focus on en- teraction. To transfer a quantum state, we swap tanglement. Instead, indistinguishability is in our atten- two rails between S and T (see Fig. 1). It means, tion. basis states |0, 1iS|1, 0iT |ψiiES|ψkiET are changed Measure |D| characterizes resources, i.e. quantum sys- to |0, 1iS|1, 0iT |ψkiES|ψiiET (here the environmen- tems, information is encoded in. Among other condi- tal basis states are swapped). While basis states tions, theory of quantum information processing requires |1, 0iS|0, 1iT |ψiiES|ψkiET remain unchanged. All other all the resources to be in the same states which are de- possibilities are excluded by post-selection considering coupled from each other (their total state must be fac- only single photon in S and single photon in T . This torable). In our notation ρE,ST = ρE,S ⊗ ρE,T with conditional “partial exchange” process entangles qubits ρE,S = ρE,T . Then they can be used to represent the together with the basis states of the environment, re- ideal qubits (or qudits). But this strict condition is not sulting in the following total state of two photons: always fulfilled in practice. In case of linear optical quan- P i,j,k,l cij,kl|Ψi,kihΨj,l|, where tum information processing, |D| = 1 says that the resources behave in the same way as if they fulfilled the 1 |Ψi,ki = √ [|1, 0iS|0, 1iT |ψiiES|ψkiET upper condition even if they actually do not. It means, 2 they can be used for encoding and processing qubits (qu- + exp(iθ) |0, 1iS|1, 0iT |ψkiES|ψiiET ] . (4) dits) even if some of their degrees of freedom are, e.g., entangled. Interestingly, this is exactly the case of tra- After tracing out the environmental states ditional photon pairs generated by SPDC. Information we gain a partially entangled state of qubits 1 is usually encoded into polarization or spatial degrees of ρST = 2 {|1, 0iSh1, 0| ⊗ |0, 1iT h0, 1| + |0, 1iSh0, 1| ⊗ freedom but frequency degrees of freedom are entangled. |1, 0iT h1, 0|+[D exp(iθ)|0, 1iSh1, 0|⊗|1, 0iT h0, 1|+h.c.]}, 3

P where D = i,j cij,ji = Tr [F ρE,ST ] is a phase damp- IV. DIRECT MEASUREMENT OF ing (decoherence) parameter which is equivalent to INDISTINGUISHABILITY Eq.(1). To complete the transfer, we measure qubit S by the projective measurement, ΠS,± = |±iSh±|, To experimentally determine |D| one can advanta- where |±i = √1 (|0, 1i ± |1, 0i ). Then the S 2 S S geously use the standard HOM-type experiment [5], target qubit is transferred to two possible condi- similarly as it has been used for the measurement ± 1 of an overlap [9, 10]. Particles enter a balanced tional states ρT = 2 {|0, 1iT h0, 1| + |1, 0iT h1, 0| ± [D exp(iθ)|0, 1iT h1, 0| + h.c.]}, respectively to the mea- unitary mixer which symmetrically transmits and re- surement result. By conditional application of π-flip, flects them between modes S and T . If basis state |1, 0iT → −|1, 0iT whereas |0, 1iT remaining unchanged, |1iS|1iT |ψiiES|ψkiET is in the input then the corre- we reach state sponding post-selected output state (one particle in S and the other one in T mode) is proportional to 1 2 |1iS|1iT (|ψiiES|ψkiET + |ψkiES|ψiiET ). For input state |1iSh1| ⊗ |1iT h1| ⊗ ρE,ST the probability of coin- + 1 + D 1 − D ⊥ ⊥ ρT ≡ ρ = |ΨiShΨ| + |Ψ iShΨ |, (5) 1 P 1 T 2 2 cidence detection, PC = 2 1 − i,j cij,ji = 2 (1 − D), is directly proportional to parameter D. Experimental quantum information processing and transfer often uses photonic qubits [30, 31] encoded where |Ψ⊥i = √1 [|0, 1i − exp(iθ)|1, 0i ] is the orthog- S 2 S S into photons generated by spontaneous parametric down- onal complement to |ΨiS. State (5) corresponds to the conversion (SPDC). These photons represent a typical original qubit state, |ΨiS, disturbed by decoherence, with example of qubit carriers with internal degrees of free- its off-diagonal elements (in the computation basis) re- dom which may exhibit complex behavior [32, 33]. State duced by factor D. The sign of D does not play a prin- of two photons created by SPDC, which propagate in well cipal role. If it is priori known it can be simply com- defined spatial and polarization modes, can be described pensated by the same feed-forward mechanism. Thus the by the following formula: quality of this basic quantum transfer can be measured by Z |Tr [F ρE,ST ]|. However, perfect transfer with |D| = 1 can −i ω ∆t correspond to three very different environmental states |ψi = φ(ω)φ(ω0 − ω) e |ωia|ω0 − ωib, (6) ω ρE,ST : (i) Product of pure perfectly overlapping single- particle states (D = 1), (ii) Symmetric maximally entan- where |ωix represents a single photon at spatial mode x gled state (D = 1), or (iii) Anti-symmetric maximally en- and frequency ω, ∆t denotes time delay in mode a, and tangled state (D = −1). Although |D| = 1 for all these φ(ω) is a spectral amplitude function. This function is cases, it varies differently when these states undergo a mainly determined by the employed spectral filter. In local operation UES ⊗ VET . our case the filter is the same both for mode a and b. Let us suppose this state enters a balanced (50:50) beam From the presented point of view, particles S and T in splitter (BS) with input modes a, b and output modes state |1iSh1|⊗|1iT h1|⊗ρE,ST can be called effectively in- c, d. The input state is transformed to the output one distinguishable if any qubit carried by particle S (encoded by a unitary transformation, |ϕi = UBS|ψi, and the cre- into the degrees of freedom which are supposed to be un- ation operators of input modes can be expressed by the der experimentalist control like spatial or polarization creation operators of output modes as follows modes) can be perfectly transferred to a qubit carried by particle T and vice versa. The above described operation 1 a†(ω) = √ i c†(ω) + d†(ω) , transfers a quantum state between two particles similarly 2 as quantum teleportation does. However, entanglement † 1 † † is not used as a resource here, so it is not limited by the b (ω) = √ c (ω) + i d (ω) . 2 amount of entanglement (opposite to Ref. [29]). What is important is indistinguishability of particles S and T Let us further suppose that behind the BS and it can be well characterized by parameter |D|. we make a coincident measurement. Coin- cidence rate can be calculated as R(∆t) ∝ Parameter D plays the same role also in the comple- R R (−) (−) (+) (+) hϕ| Ec (t1)E (t2)E (t2)Ec (t1) |ϕi, where mentary task to quantum state transfer – quantum era- t1 t2 d d (+) R −i ω t sure. Due to the symmetry of state ρST , to concentrate Ex (t) ∝ ω x(ω) e is a positive-frequency part phase information back to qubit S we can apply the same of an electric field operator with x being an annihila- (−) (+) † type of measurement (but now on qubit T ) and feed- tion operator and Ex (t) = [Ex (t)] [2]. Here we forward strategy (on qubit S). Since the reconstructed assume the coincidence window to be infinitely large. state of qubit S has the same structure as above with D In practice it is about 1 ns but the two photons arrive given by Eq. (1), |D| represents the same upper limit on in the interval of about 100 fs which is much shorter. the quality of quantum erasure. In this notation the flip operator can be expressed as 4

is retarded by ∆t in a delay line (DL) with adjustable PM 0 0 length. By means of polarization controllers the both DL FC FC + Da0 qubit I photons are set to have the same polarizations. Qubit Da1 - states are encoded into spatial modes of individual pho- 1 1 Photon U tons. Each basis state corresponds to a single photon pair PM /2 in one, |0, 1i, or in the other, |1, 0i, of two optical fibers. 1 1 Dd0 Initial equatorial states of both qubits are prepared using PM qubit II PM fiber couplers (FC and VRC) with splitting ratio 50:50 Dd1 VRC 0 0 VRC and integrated electro-optical phase modulators (PM). State preparation Qubit transfer Output state analysis In the experiment with the qubit-state transfer, the source qubit, in “unknown” equatorial state (2), was represented by qubit I and the target qubit, in state (3), was FIG. 1. (Color online) Scheme of the experiment. FC – fiber represented by qubit II. In the quantum erasure experi- couplers, VRC – variable ratio couplers, PM – phase mod- ment, the source was qubit II, while the target was qubit ulators, DL – delay line, D – detectors. The couplers and I. phase modulators in the State preparation stage enable us to The key part of our device is the swap of two rails be- prepare required qubit states (each qubit is represented by tween qubits I and II followed by measurement on qubit I. a single photon which may propagate in two optical fibers). This measurement is performed in basis √1 (|0, 1i±|1, 0i) They do not affect the environmental degrees of freedom. The 2 middle PM applies conditional phase shift depending on the using a fiber coupler with fixed splitting ratio 50:50 and result of the auxiliary measurement. It is a part of the pro- two single photon detectors (silicon avalanche photodi- tocol. The rightmost PM and VRC serve for output state odes). When detector Da1 clicks, phase correction π is tomography. applied on qubit II by means of electronic feed forward [34]. Feed forward uses a direct signal from detector Da1 (5 V pulse). The signal is modified by a passive voltage R R F = |ω1ic|ω2id hω2|chω1|d. It can be shown by a divider to circa 1.5 V and then it is lead to a lithium- ω1 ω2 straightforward calculation that niobate phase modulator (1.5 V corresponds to the phase shift of π). States of output qubit II are characterized R(∆t) ∝ 1 − Tr [F |ϕihϕ|] = 1 − D (7) by quantum tomography. Different measurement bases R R are set by a phase modulator and variable ratio coupler with Tr(X) = hω1|chω2|d X |ω1ic|ω2id. For entan- ω1 ω2 (VRC). Photons are counted by detectors Dd0 and Dd1. gled input state (6) we obtain Small differences in detector efficiencies are corrected nu- merically in the data sets. Z 2 2 ω0 + ω ω0 − ω i ω ∆t The whole experimental setup consist of two intercon- Tr [F |ϕihϕ|] ∝ φ φ e . ω 2 2 nected Mach-Zehnder interferometers. Lengths of their arms are balanced by motorized air gaps (not shown in If φ(ω) is a rectangular function of width v and central the figure). To reduce a phase drift caused by envi- frequency ω0/2 then D = Tr [F |ϕihϕ|] = sinc(∆t v). ronmental influences (temperature fluctuations etc.) the Clearly, in such an experimental situation the role of whole setup is covered and also actively stabilized. Af- the “environment” is played by the frequency degrees of ter each 3 s period of measurement the phase drifts are freedom (in our experiment qubits are encoded into spa- determined and compensated by adding a proper correct- tial modes). Parameter D can really be measured only ing voltage on phase modulators. The HOM dip, which by means of a beam splitter and coincidence detection we use to characterize the properties of input photons, is and it can be varied by changing delay ∆t between the measured at the last VRC [5]. two photons. Its negative values correspond to partially entangled states containing vectors from anti-symmetric subspace. VI. RESULTS

V. EXPERIMENTAL SETUP We have tested both the qubit-state transfer and quantum erasing. But here we will discuss only the qubit- Our setup is depicted in Fig. 1. Photon pairs are cre- state transfer because the results of quantum erasing ated by collinear frequency-degenerate type-II SPDC in are quite similar (as can be expected from the sym- a beta barium borate crystal pumped at 405 nm. Both metry of the tasks). The target, qubit II, was pre- photons pass through the same band-pass interference pared in state (3) and the source, qubit I, was prepared filter of approximately rectangular shape with central in state (2) with phase θ = 0, 30, 60, 90, 120, 150, 180 frequency 810 nm and spectral width (FWHM) 2.7 nm. degrees, in sequence. At the output we have made Then they are separated by a polarizing beam splitter measurement on the target qubit (qubit II) in three different bases: {|0, 1i, |1, 0i}, { √1 (|0, 1i ± |1, 0i)}, and and coupled into single-mode fibers. One of the photons 2 5

1.0 0.55 vironments” of our photons are entangled. The values of hΨ|SρT |ΨiS lower than 0.5 mean that the roles of states ) ⊥ T |ΨiS and |Ψ iS were swapped (see Eq. 5).

ρ 0.9

( 0.50 1+D

g According to the theory, overlap hΨ| ρ |Ψi = i S T S 2 e 1+D 1−D x 0.8 and eigenvalues of ρ are and , see Eq. (5). a T 2 2 0.45 m HOM dip ∆t [ps] Fig. 2 shows the overlap and the maximal eigenvalue

0.10 0.05 0.00 0.05

, -1 0 1

S 0.7 as functions of parameter D. Each point represents an ® 1.0

Ψ average over all 7 phases. Vertical error bars visualize | T l

ρ 0.6 standard deviations obtained from the ensembles of mea- e r S | R 0.5 surements with different phases. Due to various experi- Ψ 0.5 mental imperfections (phase fluctuations, drift of split- 0.0 ting ratios, etc.) they are greater than standard de- 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 viations calculated purely from Poissonian photo-count D =Tr[Fρ ] =1 R distribution. But on the graph they are mostly smaller E,ST − rel than the size of the symbols. Horizontal error bars re- flect (Poissonian) statistical fluctuations of coincidence FIG. 2. (Color online) Dependence of the quality of qubit- rates R(∆t) and R(2 ps). Average output state fidelity, p√ rec√ 2 state transfer on parameter D. Blue circles denote the overlap Tr ρT ρT ρT , (averaged over all phases and all of output and input states, hΨ|S ρT |ΨiS . Green squares de- delays) was 99.2 ± 0.8%. The measured HOM dip is note maximal eigenvalues of output states ρT . Straight lines shown in the lower right inset of Fig. 2. Relative measure- are theoretical predictions. The upper left inset magnifies the ment error was less than 6 % in its minimum and less than area where D is close to zero. The lower right inset shows the 2 % for maximal values. Dip visibility was 96.4 ± 0.4%. measured Hong-Ou-Mandel dip, Rrel denotes relative (normalized) coincidence rate.

VII. CONCLUSIONS { √1 (|0, 1i ± i|1, 0i)}. Each measurement consisted of 15 2 three-second measurement intervals interlaced by active We can conclude that effective quantum indistin- stabilization. The results were used to reconstruct output guishability as a key resource for quantum information rec processing can be quantified by a directly measurable density matrices, ρT , by means of maximum-likelihood quantum tomography [35–37]. Reconstruction of density parameter |Tr [F ρE,ST ]| for any state ρE,ST of two par- matrices has enabled us to calculate various quantities ticles. We have demonstrated that this parameter rep- including purity, Uhlmann fidelity (with respect to theo- resents a bound on the quality of real-world implemen- retical output states), eigenvalues, and overlap with cor- tation of quantum transfer protocols. If other resources, responding input states. like quantum entanglement, are required then the im- Each such a measurement set was repeated 16 times pact of their imperfections is combined with the effect of with different delays ∆t between the input photons (cor- (in)distinguishability in a nontrivial way. Their coexis- responding to different positions in HOM dip). For each tence is a subject of a current investigation. ∆t we have also evaluated parameter D = 1 − Rrel. It was obtained from the coincidence rate between detec- ACKNOWLEDGMENTS tors Dd0 and Dd1, R(∆t), normalized with respect to the coincidence rate measured separately in the position far from the dip, R(2 ps), i.e., Rrel = R(∆t)/R(2 ps). Nega- This work was supported by PalackýUniversity (PrF- tive values of D correspond to the positions in the raised 2012-019). R. F. acknowledges support from the Czech “shoulders” of the HOM dip. They reveal that the “en- Science Foundation (P205/12/0577).

[1] M. A. Nielsen, I. L. Chuang, Quantum Computation and [6] Ch. Santori, D. Fattal, J. Vuˇckovi´c,G. S. Solomon, and Quantum Information (Cambridge Univ. Press, 2000). Y. Yamamoto, Nature 419, 594 (2002). [2] L. Mandel, E. Wolf, Optical Coherence and Quantum Op- [7] R. F. Werner, Phys. Rev. A 40, 4277 (1989). tics (Cambridge Univ. Press, 1995). [8] A. Ekert, C. M. Alves, D. K. L. Oi, M. Horodecki, P. [3] W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003). Horodecki, and L. C. Kwek, Phys. Rev. Lett. 88, 217901 [4] A. Peres, Quantum Theory: Concepts and Methods (2002). (Kluwer Academic, 1993). [9] R. Filip, Phys. Rev. A 65, 062320 (2002). [5] C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. [10] M. Hendrych, M. Duˇsek,R. Filip, J. Fiur´aˇsek,Phys. Lett. 59, 2044 (1987). A 310, 95 (2003). 6

[11] J. Beugnon, M. P. A. Jones, J. Dingjan, B. Darquié,G. [24]A. Cernoch,ˇ J. Soubusta, L. Bart˚uˇsková,M. Duˇsek,and Messin, A. Browaeys, and P. Grangier, Nature 440, 779 J. FiuráˇsekPhys. Rev. Lett. 100, 180501 (2008). (2006). [25] M. Kacprowicz, R. Demkowicz-Dobrzański, W. [12] J. Hofmann, M. Krug, N. Ortegel, L. Gérard, M. Weber, Wasilewski, K. Banaszek, and I. A. Walmsley, Na- W. Rosenfeld, and H. Weinfurter, Science 337, 72 (2012). ture Photonics 4, 357 (2010). [13] J. K. Thompson, J. Simon, H. Loh, and V. Vuletić,Sci- [26] A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, ence 313, 74 (2006). A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. [14] P. Maunz, D. L. Moehring, S. Olmschenk, K. C. Younge, Wörhoff,Y. Bromberg, Y. Silberberg, M. G. Thompson, D. N. Matsukevich, and C. Monroe, Nature Physics 3, and J. L. O’Brien, Science 329, 1500 (2010). 538 (2007). [27] V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, Science [15] R. B. Patel, A. J. Bennett, I. Farrer, C. A. Nicoll, D. A. 317, 1890 (2007). Ritchie, and A. J. Shields, Nature Photonics 4, 632 [28] A. Crespi, R. Ramponi, R. Osellame, L. Sansoni, I. Bon- (2010). gioanni, F. Sciarrino, G. Vallone, and P. Mataloni, Na- [16] A. J. Bennett, R. B. Patel, C. A. Nicoll, D. A. Ritchie, and ture Commun. 2, 566 (2011). A. J. Shields, Nature Physics 5, 715 (2009). [29] Z.-W. Wang, J. Li, Y.-F. Huang, Y.-S. Zhang, X.-F. Ren, [17] F. Boitier, A. Godard, E. Rosencher, and C. Fabre, Na- P. Zhang, G.-C. Guo Phys. Lett. A 372, 106 (2008). ture Physics 5, 267 (2009). [30] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. [18] Q. Zhang, A. Goebel, C. Wagenknecht, Y.-A. Chen, B. Dowling, and G. J. Milburn, Rev. Mod. Phys. 79, 135 Zhao, T. Yang, A. Mair, J. Schmiedmayer, and J.-W. (2007). Pan, Nature Physics 2, 678 (2006). [31] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. [19] M. Halder, A. Beveratos, N. Gisin, V. Scarani, Ch. Si- Mod. Phys. 74, 145 (2002). mon, and H. Zbinden, Nature Physics 3, 692 (2007). [32] J.-W. Pan, Z.-B. Chen, M. Zukowski,˙ H. Weinfurter, and [20] E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. A. Zeilinger, Rev. Mod. Phys. 84, 777 (2012). Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, [33] J.-P. Torres, K. Banaszek, I.-A. Walmsley, Progress in Nature Photonics 3, 720 (2009). Optics, Vol. 56, 227-331 (2011). [21] J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and [34] M. Miková, H. Fikerová, I. Straka, M. Miˇcuda, J. D. Branning, Nature 426, 264 (2003). Fiuráˇsek, M. Jeˇzek,and M. Duˇsek, Phys. Rev. A 85, [22] K. Lemr, A. Cernoch,ˇ J. Soubusta, K. Kieling, J. Eisert, 012305 (2012). and M. Duˇsek,Phys. Rev. Lett. 106, 013602 (2011). [35] M. G. A. Paris and J. Reháˇcek(Eds.),ˇ Quantum State [23] M. Miˇcuda,M. Jeˇzek,M. Duˇsek,J. Fiuráˇsek,Phys. Rev. Estimation, Lect. Notes Phys. 649, (Springer, Berlin, A 78, 062311 (2008). Heildeberg, 2004). [36] Z. Hradil, Phys. Rev. A 55, R1561 (1997). [37] M. Jeˇzek,J. Fiuráˇsek,and Z. Hradil, Phys. Rev. A 68, 012305 (2003).