Y. YANG, G. BAI, G. CHIRIBELLA, AND M. HAYASHI: COMPRESSION FOR QUANTUM POPULATION CODING 1
Compression for quantum population coding Yuxiang Yang, Ge Bai, Giulio Chiribella, and Masahito Hayashi Fellow, IEEE
Abstract—We study the compression of n quantum systems, It is important to stress that the problem of storing the n- each prepared in the same state belonging to a given parametric ⊗n copy states {ρθ }θ∈Θ in a quantum memory is different from family of quantum states. For a family of states with f indepen- the standard problem of quantum data compression [5], [6], dent parameters, we devise an asymptotically faithful protocol that requires a hybrid memory of size (f/2) log n, including both [7]. In our scenario, the mixed state ρθ is not regarded as quantum and classical bits. Our construction uses a quantum the average state of an information source, but, instead, as a version of local asymptotic normality and, as an intermediate physical encoding of the parameter θ. The goal of compression step, solves the problem of compressing displaced thermal states is to preserve the encoding of the parameter θ, by storing the of n identically prepared modes. In both cases, we show that state ρ⊗n into a memory and retrieving it with high fidelity (f/2) log n is the minimum amount of memory needed to achieve θ asymptotic faithfulness. In addition, we analyze how much of the for all possible values of θ. To stress the difference with memory needs to be quantum. We find that the ratio between standard quantum compression, we refer to our scenario as quantum and classical bits can be made arbitrarily small, but compression for quantum population coding. The expression cannot reach zero: unless all the quantum states in the family “quantum population coding” refers to the encoding of the commute, no protocol using only classical bits can be faithful, parameter θ into the many-particle state ρ⊗n, viewed as the even if it uses an arbitrarily large number of classical bits. θ state of a “population” of quantum systems. We choose this Index Terms—Population coding, compression, quantum sys- expression in analogy with the classical notion of population tem, local asymptotic normality, identically prepared state coding, where a parameter θ is encoded into the population of n individuals [8]. The typical example of population coding arises in computational neuroscience, where the population I.INTRODUCTION consists of neurons and the parameter θ represents an external Many problems in quantum information theory involve a stimulus. source that prepares multiple copies of the same quantum state. The compression for quantum population coding has been This is the case, for example, of quantum tomography [1], studied by Plesch and Buzekˇ [9] in the case where ρθ is a quantum cloning [2], [3], and quantum state discrimination pure qubit state and no error is tolerated (see also [10] for [4]. The state prepared by the source is generally unknown a prototype experimental implementation). A first extension to the agent who has to carry out the task. Instead, the agent to mixed states, higher dimensions, and non-zero error was knows that the state belongs to some parametric family of proposed by some of us in [11]. The protocol therein was density matrices {ρθ}θ∈Θ, with the parameter θ varying in proven to be optimal under the assumption that the decoding the set Θ. It is generally assumed that the source prepares operation must satisfy a suitable conservation law. Later, each particle identically and independently: when the source it was shown that, when the conservation law is lifted, a is used n times, it generates n quantum particles in the tensor new protocol can achieve a better compression, reaching the ⊗n product state ρθ . ultimate information-theoretic bound set by Holevo’s bound A fundamental question is how much information is con- [12]. This result applies to two-dimensional quantum systems ⊗n tained in the n-particle state ρθ . One way to address the with completely unknown Bloch vector and/or completely question is to quantify the minimum amount of memory unknown purity. The classical version of the compression for needed to store the state, or equivalently, the minimum amount population coding was addressed in [13]. However, finding of communication needed to transfer the state from a sender the optimal protocol for arbitrary parametric families and for arXiv:1701.03372v8 [quant-ph] 26 Jan 2019 to a receiver. Solving this problem requires an optimization quantum systems of arbitrary dimension has remained as an over all possible compression protocols. open problem so far. In this paper, we provide a general theory of compression Y. Yang (e-mail: [email protected]) and G. Bai (e-mail: for quantum states of the form ρ⊗n. We consider two cate- [email protected]) are with the Department of Computer Science, The θ University of Hong Kong, Pokfulam Road, Hong Kong, and with the HKU gories of states: (i) generic quantum states in finite dimensions, Shenzhen Institute of Research and Innovation, Kejizhong 2nd Road, Shen- and (ii) displaced thermal states in infinite dimension. These zhen, China. two categories of states are connected by the quantum version G. Chiribella (e-mail: [email protected]) is with the Department of Computer Science, The University of Oxford, Parks Road, Oxford, UK, of local asymptotic normality (Q-LAN) [14], [15], [16], [17], ⊗n with the Canadian Institute for Advanced Research, CIFAR Program in [18], which locally reduces the tensor product state ρθ to a Quantum Information Science, with the Department of Computer Science, displaced thermal state, regarded as the quantum version of The University of Hong Kong, Pokfulam Road, Hong Kong, and with the HKU Shenzhen Institute of Research and Innovation, Kejizhong 2nd Road, the normal distribution. Shenzhen, China. We will discuss first the compression of displaced thermal M. Hayashi is with the Graduate School of Mathematics, Nagoya Univer- states. Then, we will employ Q-LAN to reduce the problem of sity, Furocho, Chikusaku, Nagoya, 464-860, Japan, and Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore compressing generic finite-dimensional states to the problem 117542. of compressing displaced thermal states. In both cases, our Y. YANG, G. BAI, G. CHIRIBELLA, AND M. HAYASHI: COMPRESSION FOR QUANTUM POPULATION CODING 2 compression protocol uses a hybrid memory, consisting both We say that a parameter is independent if it can vary contin- of classical and quantum bits. For a family of quantum states uously while the other parameters are kept fixed. For a given described by f independent parameters, the total size of family of states, we denote by fc (fq) the maximum number of the memory is f/2 log n at the leading order, matching the independent classical (quantum) parameters describing states ultimate limit set by Holevo’s bound [19]. in the family. For example, the family of all diagonal density An intriguing feature of our compression protocol is that matrices ρ0(µ) in Eq. (2) has d − 1 independent parameters. the ratio between the number of quantum bits and the number The family of all quantum states in dimension d has d2 − 1 of classical bits can be made arbitrarily close to zero, but not independent parameters, of which d − 1 are classical and exactly equal to zero. Such a feature is not an accident: we d(d − 1) are quantum. In general, we will assume that the ⊗n show that, unless the states commute, every asymptotically family {ρθ }θ∈Θ is such that every component of the vector faithful protocol must use a non-zero amount of quantum θ is either independent or fixed to a specific value. memory. This result extends an observation made in [20] from Let us introduce now the second category of states that are certain families of pure states to generic families of states. relevant in this paper: the displaced thermal states [21], [22]. The paper is structured as follows. In section II we state Displaced thermal states are a type of infinite-dimensional the main results of the paper. In Section III we study the states frequently encountered in quantum optics [23]. Mathe- compression of displaced thermal states. In Section IV we matically, they have the form provide the protocol for the compression of identically pre- thm † pared finite-dimensional states. In Section V we show that ρα,β = Dα ρβ Dα (4) every protocol achieving asymptotically faithful compression † where Dα = exp(αaˆ − α¯aˆ) is the displacement operator, must use a quantum memory. Optimality of the protocols is defined in terms of a complex parameter α ∈ C (the displace- proven later in Section VI. Finally, the conclusions are drawn ment), and aˆ is the annihilation operator, satisfying the relation in Section VII. † thm [ˆa, aˆ ] = 1, while ρβ is a thermal state, defined as ∞ II.MAIN RESULT. thm X j ρβ := (1 − β) β |jihj| , (5) The main result of this work is the optimal compression j=0 of identically prepared quantum states. We consider two cat- egories of states: generic finite dimensional (i.e. qudit) states where β ∈ [0, 1) is a real parameter, here called the thermal and infinite-dimensional displaced thermal states. parameter, and the basis {|ji}j∈N consists of the eigenvectors † Let us start from the first category. For a quantum system of aˆ aˆ. For β = 0, the the displaced thermal states are pure. of dimension d < ∞, also known as qudit, we consider Specifically, the state ρα,β=0 is the projector on the coherent generic states described by density matrices with full rank state [24] |αi := Dα |0i. and non-degenerate spectrum. We parametrize the states of a In the context of quantum optics, infinite dimensional sys- d-dimensional quantum system as tems are often called modes. We will consider the compression of n modes, each prepared in the same displaced thermal † d(d−1) d−1 ⊗n ρθ = Uξ ρ0(µ) U , θ = (ξ, µ) ξ ∈ µ ∈ ξ R R state. We denote the n-mode states as {ρα,β}(α,β)∈Θ with (1) Θ = Θα × Θβ being the parameter space. There are three real parameters for the displaced thermal state family: the where ρ0(µ) is the fixed state thermal parameter β, the amount of displacement |α|, and the d d−1 X X phase ϕ = arg α. Here, β is a classical parameter, specifying ρ0(µ) = µj |jihj| µd := 1 − µk, (2) the eigenvalues, while |α| and ϕ are quantum parameters, j=1 k=1 determining the eigenstates. We will assume that each of the with spectrum ordered as µ1 > ··· > µd−1 > µd > 0, while three parameters β, |α| and ϕ is either independent, or fixed Uξ is the unitary matrix defined by to a determinate value. A compression protocol consists of two components: the I I R R X ξj,kTj,k + ξj,kTk,j encoder, which compresses the input state into a memory, Uξ = exp i √ (3) µj − µk and the decoder, which recovers the state from the memory. 1≤j
⊗n Theorem 1. Let {ρθ }θ=(ξ,µ)∈Θ be a generic family of n- where the displaced thermal states satisfy the rela- iϕ iϕ copy qudit states with fc independent classical parameters and tion α0 = |α0|e , α1 = |α1|e , and α2 = p 2 2 iϕ fq independent quantum parameters. For any δ ∈ (0, 2/9), |α0| + |α1| e . the states in the family can be compressed into [(1/2+δ)fc + • Heterodyne measurement. The heterodyne measurement (1/2)fq] log n classical bits and (fqδ) log n qubits with an (see e.g. Section 3.5.2 of [25]) is a common measurement error = O n−κ(δ) + O n−δ/2, where κ(δ) is the error in continuous variable quantum optics. Here, the mea- exponent of Q-LAN [17] (cf. Eq. (44) in the following). The surement outcome is a complex number αˆ and the cor- protocol is optimal, in the sense that any compression protocol responding measurement operator is the projector on the 0 0 d2αˆ using a memory of size [(fc + fq)/2 − δ ] log n with δ > 0 coherent state |αˆi. The heterodyne POVM { π |αˆihαˆ|} cannot be asymptotically faithful. is normalized in such a way that the integral over the ⊗n The same results hold for a family {ρα,β}(α,β)∈Θ of dis- complex plane gives the identity operator, namely placed thermal states, except that in this case the error is Z d2αˆ only = O n−δ/2. |αˆihαˆ| = I. (11) π Theorem 1 is a sort of “equipartition theorem”, stating that We will use heterodyne measurements to estimate the each independent parameter requires a memory of size (1/2+ amount of displacement of a displaced thermal state. For δ) log n. When the parameter is classical, the required memory a displaced thermal state ρα,β, the conditional probability is fully classical; when the parameter is quantum, a quantum density of finding the outcome αˆ is memory of δ log n qubits is required. 1 Q(ˆα|α, β) = hαˆ|ρα,β|αˆi III.COMPRESSION OF DISPLACED THERMAL STATES π 1 thm In this section, we focus on the compression of identically = hαˆ − α|ρ |αˆ − αi π β prepared displaced thermal states. We separately treat eight (1 − β) possible cases, corresponding to the possible combinations = exp[−(1 − β)|αˆ − α|2]. (12) π where the three parameter β, |α| and ϕ are either independent • Quantum amplifier. or fixed. The total memory cost for each case is determined by A quantum amplifier is a device that Theorem 1 and is summarized in Table I. On the other hand, increases the intensity of quantum light while preserving the errors for all cases satisfy the unified bound its phase information, namely a device which approxi- mately implements the process |αi → |γαi for γ > 1. = O n−δ/2 . (7) Quantum amplification is an analogue of approximate quantum cloning for finite-dimensional systems, since coherent or displaced thermal states can be merged and A. Quantum optical techniques used in the compression pro- split in a reversible fashion [cf. Eqs. (9), (10)]. In this tocols work, we use the following amplifier [26]: To construct compression protocols for the displaced ther- h −1 √ †ˆ† ˆ Aγ (ρ) = Tr ecosh ( γ)(ˆa b −aˆb)(ρ ⊗ |0ih0| ) mal states, we adopt several tools in quantum optics. As a B B −1 √ ˆ †ˆ† i preparation, we introduce three quantum optical tools which × ecosh ( γ)(ˆab−aˆ b ) (13) are key components of the compression protocols: the beam splitter, the heterodyne measurement, and the quantum ampli- where aˆ and ˆb are the annihilation operators of the input fier. mode and the ancillary mode B, and γ is the amplification • Beam splitter. A beam splitter is a linear optical device factor. implementing the unitary gate We now show details of the compression protocol for each h † † i case. Uτ = exp iτ(ˆa1aˆ2 +a ˆ1aˆ2) , (8)
where τ is a real parameter, and aˆ1 and aˆ2 are the B. Case 1: fixed α, independent β annihilation operators associated to the two systems. Let us start from Case 1, where the thermal parameter β is Beam splitters can be used to split or merge laser beams. the only independent parameter. Note that the input state can For example, one can use a beam splitter with τ = π/4 to be regarded as the state of n optical modes with each mode in merge two identical coherent states into a single coherent a displaced thermal state, and thus the compression protocol state with larger amplitude: √ can be regarded as a sequence of operations on the n-mode Uπ/4|αi1 ⊗ |αi2 = | 2αi1 ⊗ |0i2. (9) system. Since α is known, we can get rid of the displacement using We will use beam splitters to manipulate the information ⊗n a certain unitary and convert the displaced thermal states into about the parameter α in the n-copy state ρα,β. In (undisplaced) thermal states. For the compression of thermal particular, we will make frequent use of the beam splitter states, we have the following lemma: unitary Uτ that implements the transformation † thm Lemma 1 (Compression of identically prepared thermal Uτ (ρα0,β ⊗ ρα1,β) Uτ = ρα2,β ⊗ ρβ (10) thm ⊗n states). Let {(ρβ ) }β∈[βmin,βmax] be a family of n-copy Y. YANG, G. BAI, G. CHIRIBELLA, AND M. HAYASHI: COMPRESSION FOR QUANTUM POPULATION CODING 4
TABLE I COMPRESSION RATE FOR DIFFERENT STATE FAMILIES.HERE δ > 0 IS AN ARBITRARY POSITIVE CONSTANT.
Case displacement α = |α|eiϕ thermal parameter β quantum bits classical bits 0 fixed fixed 0 0 1 fixed independent 0 (1/2 + δ) log n 2 independent fixed 2δ log n log n 3 independent independent 2δ log n (3/2 + δ) log n 4 ϕ independent; |α| fixed fixed δ log n (1/2) log n 5 ϕ independent; |α| fixed independent δ log n (1 + δ) log n 6 |α| independent; ϕ fixed fixed δ log n (1/2) log n 7 |α| independent; ϕ fixed independent δ log n (1 + δ) log n thermal states. For any δ > 0, there exists a protocol where U is a suitable unitary gate, realizable with a circuit BS thm thm of beam splitter gates as in Eq. (10). Using Eq. (15), we can En,δ , Dn,δ that compresses n copies of a thermal state ρthm into (1/2 + δ) log n classical bits with error construct a protocol that separately processes the displaced β √ thermal state ρ nα,β and the n − 1 thermal modes, up to a −δ thm = O n . (14) gentle testing of the input state. The proof of the above lemma can be found in the appendix. For any δ > 0, the protocol for Case 2 (fixed β, independent Note that no quantum memory is required to encode thermal α) runs as follows (see also Fig 1 for a flowchart illustration): states, in agreement with the intuition that β is classical, • Preprocessing. A preprocessing procedure is needed in thm α α because all the states ρβ are diagonal in the same basis. order to store the estimate of : Divide the range of For any δ > 0, the compression protocol for Case 1 (fixed into n intervals, each labeled by a point αˆi in it, so that α, independent β) is constructed as follows: |α0 − α00| = O(n−1/2) (note that α is complex) for any 0 00 • Encoder. α , α in the same interval. 1) Transform each input copy with the displacement • Encoder. † † operation D−α, defined by D−α(·) := D−α · D−α, 1) Perform the unitary channel UBS(·) = UBS · UBS † where D−α = exp(−αaˆ +α ¯aˆ) is the displacement on the input state, where UBS is the unitary defined √ operator. The displacement operation transforms by Eq. (15). The output state has the form ρ nα,β ⊗ thm thm ⊗(n−1) each input copy ρα,β into the thermal state ρβ . (ρβ ) . thm 2) Apply the thermal state encoder En,δ in Lemma 1 2) Send the first and the last mode through a group of on the n-mode state and the outcome is encoded in beam splitters (10) that implements the transforma- tion ρ ⊗ρthm → ρ√ ⊗ρ√ . The a classical memory. α,β β n−n1−δ α,β n1−δ α,β • Decoder. n-mode state is now ρ√ ⊗(ρthm)⊗(n−2)⊗ n−n1−δ α,β β Dthm √ 1) Use the thermal state decoder n,δ in Lemma 1 to ρ n1−δ α,β. thm recover the n copies of the thermal state ρβ from 3) Estimate α by performing the heterodyne mea- d2α0 0 0 the classical memory. surement { π |α ihα |} on the last mode. If the 0 2) Perform the displacement operation Dα on each measurement√ outcome is α , use the value αˆ = mode. α0/ n1−δ as an estimate for the displacement α. Obviously, the memory cost and the error of the above protocol The conditional probability distribution of the esti- (1−β) 1−δ are given by Lemma 1. mate is Q(ˆα|α, β) = π exp[−(1 − β)n |αˆ − α|2] [ cf. Eq. (12)]. C. Case 2: fixed β, independent α 4) Encode the label αˆ∗ of the interval containing αˆ in Next we study the case when the displacement α is indepen- a classical memory. 5) Displace the first mode with D √ . dent, while the thermal parameter β is fixed. The heuristic idea − n−n1−δ αˆ∗ of the compression protocol is to gently test the input state, in order to extract information about the parameter α. The “gentle test” is based on a heterodyne measurement, performed on a small fraction of the n input copies. The information gained by the measurement is then used to perform suitable encoding operations on the remaining copies. Let us see the details of the compression protocol. The key observation is that n identically displaced thermal states are unitarily equivalent to a single displaced√ thermal state, with the displacement scaled up by a factor n, times the product of n − 1 undisplaced thermal states. In formula, one has [27], [28] ρ⊗n = U † ρ√ ⊗ (ρthm)⊗(n−1) U , (15) Fig. 1. Compression protocol for displaced thermal states with fixed β α,β BS nα,β β BS and independent α. Y. YANG, G. BAI, G. CHIRIBELLA, AND M. HAYASHI: COMPRESSION FOR QUANTUM POPULATION CODING 5
6) Truncate the state of the first mode in the photon having used Eq. (12) in the second equality. At this point, it number basis. The truncation is described by the is convenient to set f(n) = n−1/2+3δ/4, so that we obtain the channel Pn2δ : relation P (α, β, n) Pn2δ (ρ) = Pn2δ ρPn2δ + (1 − Tr[Pn2δ ρ])|0ih0| (16) Z d2αˆ = (1 − β) exp[−(1 − β)n1−δ|αˆ − α|2] |αˆ−α|>n−1/2+3δ/4 π where δ/2 = e−Ω(n ). (22) n2δ X Pn2δ = |mihm|. (17) Inserting this relation in Eq. (20), we obtain the bound m=0 1 n ≤ sup sup The output state on the first mode is encoded in a 2 α,β αˆ∗:|αˆ∗−α|≤f(n) quantum memory. γn A ◦ D√ ◦ P 2δ ρ√ n−n1−δ αˆ∗ n n−n1−δ (α−αˆ∗),β • Decoder. o δ/2 ∗ √ −Ω(n ) 1) Read αˆ and perform the displacement operation − ρ nα,β + e . (23) 1 D√ on the state of the quantum memory. n−n1−δ αˆ∗ Now, we have to bound the first term in the right hand side. γn 2) Apply a quantum amplifier A (13) with To this purpose, we split it into two terms, as follows 1 γn √ 2δ √ √ γn = (18) A ◦ D 1−δ ∗ ◦ Pn ρ 1−δ ∗ − ρ nα,β 1 − n−δ n−n αˆ n−n (α−αˆ ),β 1 γn ≤ A ◦ D√ ◦ P 2δ ρ√ to the output. n−n1−δ αˆ∗ n n−n1−δ (α−αˆ∗),β 3) Prepare (n − 1) modes in the thermal state ρthm, β γn √ √ − A ◦ D 1−δ ∗ ρ 1−δ ∗ and perform on all the n modes the unitary channel n−n αˆ n−n (α−αˆ ),β 1 −1 UBS : γn √ √ √ + A ◦ D 1−δ ∗ ρ 1−δ ∗ − ρ nα,β n−n αˆ n−n (α−αˆ ),β 1 −1 † UBS (ρ) = UBS ρ UBS. (19) 2δ √ √ ≤ Pn ρ 1−δ ∗ − ρ 1−δ ∗ n−n (α−αˆ ),β n−n (α−αˆ ),β 1 The total memory cost consists of two parts: log n bits for + Aγn ρ√ − ρ√ . (24) n−n1−δ α,β nα,β encoding the (rounded) value αˆ∗ of the estimate and 2δ log n 1 qubits for encoding the first mode (in a displaced thermal The two terms can be upper bounded individually. For the first state). term, we use the relations Let us analyze the error of the protocol. To upper bound the γ √ 0 β + γ − 1 A (ρ ) = ρ 0 β = (25) error, we first note that, with high probability, our estimate αˆ α,β γα,β γ is close to the correct value, say |αˆ − α| ≤ f(n) for some function f vanishing for large n. When this happens, we can and 2|β0 − β| bound the error introduced by the truncation Pn2δ and by thm thm 0 2 ρβ0 − ρβ ≤ + O(|β − β| ) , (26) the amplification Aγn . Otherwise, we just use the trivial error 1 (1 − β0)2
⊗n ⊗n bound kρα,β − Dn ◦ En(ρα,β) ≤ 2. In this way, we obtain proven in Appendices B and C, respectively. 1 the bound Using these two relations and Eq. (18), we obtain the bound 1 1 ⊗n ⊗n sup sup Aγn ρ√ − ρ√ = sup ρα,β − Dn ◦ En ρα,β n−n1−δ α,β nα,β α,β 2 1 2 α,β αˆ∗:|αˆ∗−α|≤n−1/2+3δ/4 1 1 n −δ ≤ sup sup = O(n ) . (27) 2 α,β αˆ∗:|αˆ∗−α|≤f(n) The first term in the right hand side of Eq. (24) can be γn A ◦ D√ ◦ P 2δ ρ√ bounded with the following lemma: n−n1−δ αˆ∗ n n−n1−δ (α−αˆ∗),β o √ Lemma 2 (Photon number truncation of displaced thermal − ρ nα,β + P (α, β, n) , (20) 1 states.). Define the channel PK as where P (α, β, n) is the probability that αˆ deviates from α by PK (ρ) = PK ρPK + (1 − Tr[PK ρ])|0ih0| (28) more than f(n), given by where P = PK |kihk|. When K = Ω |α|2+x, P Z K k=0 K P (α, β, n) = d2αˆ Q(ˆα|α, β) satisfies |αˆ−α|>f(n) 1 x/8 x/4 2 Ω(K ) −Ω(K ) Z d αˆ (ρα,β) := kPK (ρα,β) − ρα,βk1 = β + e = (1 − β) exp[−(1 − β)n1−δ|αˆ − α|2] , 2 |αˆ−α|>f(n) π (29) (21) for any 0 ≤ β < 1. Y. YANG, G. BAI, G. CHIRIBELLA, AND M. HAYASHI: COMPRESSION FOR QUANTUM POPULATION CODING 6
See Appendix D for the proof. n-mode state is now ρ√ ⊗(ρthm)⊗(n−2)⊗ n−n1−δ α,β β √ In our case, we are using√ the projector Pn2δ in Eq. ρ n1−δ α,β. (17), and the displacement is n − n1−δ(α − αˆ∗). Since 4) Estimate α by performing the heterodyne measure- √ 2 0 n − n1−δ|α − αˆ∗| = O n3δ/4, by Lemma 2 we obtain { d α |α0ihα0|} ment π on√ the last mode, which yields the bound an estimate αˆ = α0/ n1−δ with the probability 1 distribution Q(ˆα|α, β) as in Eq. (12). Encode the sup sup P 2δ (ρ√ ) n n−n1−δ (α−αˆ∗),β ∗ 2 α,β αˆ∗:|αˆ∗−α|≤n−1/2+3δ/4 label αˆ of the interval containing αˆ in a classical memory. − ρ√ n−n1−δ (α−αˆ∗),β 5) Displace the first mode with D √ . 1 − n−n1−δ αˆ∗ Ω(nδ/12) −Ω(nδ/6) 6) Prepare the n-th mode in the thermal state ρthm. = β + e . (30) βˆ The n-mode state is now ρ√ ⊗ Combining Eqs. (23), (27), and (30), we finally get the error n−n1−δ (α−αˆ∗),β (ρthm)⊗(n−2) ⊗ ρthm. bound β βˆ δ/2 δ/12 δ/6 7) Truncate the state of the first mode, using the ≤ e−Ω(n ) + O n−δ + βΩ(n ) + e−Ω(n ) channel Pn2δ defined by Eq. (28). The output state −δ = O n . (31) is encoded in a quantum memory. thm 8) Use the thermal state encoder En−1,δ (see Lemma 1) D. Case 3: independent α and β to compress the remaining n − 1 modes and encode Case 3 (independent α and β) can be treated in a similar the output state in a classical memory. way as Case 2. The main difference is that, since one mode • Decoder. is consumed in the estimation of α, we have to estimate also 1) Read αˆ∗ and perform the displacement β to reconstruct this mode. Luckily, the thermal parameter β D√ on the state of the quantum memory. can be estimated freely (i.e. without disturbing the input state), n−n1−δ αˆ∗ γn 2) Apply a quantum amplifier (13) A with γn = and thus its estimation strategy is simpler than that of α. −δ For any δ > 0 we can construct the protocol for Case 1/(1 − n ) to the state. Dthm 3 (independent α and β) as follows (see also Fig. 2 for a 3) Use the thermal state decoder n−1,δ to recover the thm flowchart illustration): other (n − 1) modes in the thermal state ρβ from the memory. • Preprocessing. Divide the range of α into n intervals, −1 0 00 4) Perform the channel UBS . each labeled by a point αˆi in it, so that |α − α | = O(n−1/2) for any α0, α00 in the same interval. The memory cost of the protocol consists of three parts: log n ∗ • Encoder. bits for encoding the (rounded) value αˆ of the estimate, † 2δ log n qubits for encoding the first mode (displaced thermal 1) Perform the unitary channel UBS(·) = UBS · U BS state), and (1/2 + δ) log n bits for encoding the other modes on the input state, where UBS is the unitary defined √ (thermal states). Overall, the protocol requires 2δ log n qubits by Eq. (15). The output state has the form ρ nα,β ⊗ thm ⊗(n−1) and (3/2 + δ) log n classical bits. (ρβ ) . 2) Estimate β with the von Neumann measurement of On the other hand, the error of the protocol can be analyzed the photon number on the n − 1 copies of ρthm and in a similar way as in Case 2, with the only difference that β an extra error is introduced by estimating and compressing the denote by βˆ the maximum likelihood estimate of β. thermal states. The state of the modes after the estimation step Note that the n − 1 copies will not be disturbed by is the photon number measurement because they are diagonal in the photon number basis. Z ρest = βˆ P (βˆ|β) ρthm ⊗ (ρthm)⊗(n−2) . (32) 3) Send the first and the last mode through a group of β d βˆ β beam splitters (10) that implements the transforma- thm √ √ ˆ ˆ tion ρ ⊗ρ → ρ ⊗ρ 1−δ . The P (β|β) β α,β β n−n1−δ α,β n α,β where is the probability density of estimating when the true value is β. Applying the thermal state compression to thm thm est this state, we obtain the output state Dn,δ ◦En,δ (ρβ ), whose distance from the initial state can be bounded as
1 thm thm est thm⊗(n−1) β := Dn,δ ◦ En,δ (ρβ ) − ρβ 2 1 1 n thm thm est ≤ sup Dn,δ ◦ En,δ ρβ 2 β thm thm h thm ⊗(n−1)i − Dn,δ ◦ En,δ (ρβ ) 1 thm thm h thm ⊗(n−1)i thm ⊗(n−1) o + Dn,δ ◦ En,δ (ρβ ) − (ρβ ) 1 1 est thm ⊗(n−1) −δ Fig. 2. Compression protocol for displaced thermal states with indepen- ≤ sup ρβ − (ρβ ) + O n , (33) dent α and β. 2 β 1 Y. YANG, G. BAI, G. CHIRIBELLA, AND M. HAYASHI: COMPRESSION FOR QUANTUM POPULATION CODING 7 having used Lemma 1 in the last inequality. The remaining 3) Estimate ϕ by the heterodyne measurement d2α0 0 0 term can be bounded as { π |α ihα |} on the last mode, which yields an Z estimate ϕˆ which is the phase of α0. Encode the est thm ⊗(n−1) thm thm ρ − (ρ ) ≤ βPˆ (βˆ|β) ρ − ρ . ∗ β β d βˆ β label ϕˆ of the interval containing ϕˆ in a classical 1 1 (34) memory. 4) Displace the first mode with D √ with − n−n1−δ/2αˆ∗ Now, we split the integral in the right hand side of Eq. (34) ∗ iϕˆ∗ into two terms, corresponding to the values of βˆ in regions αˆ := |α|e . ˆ ˆ −(1+δ)/2 5) Send the state of the first mode through a truncation R≤ := {β ∈ C | |β − β| ≤ n )} and R> = C \ R≤. In this way, we obtain the bound channel Pnδ defined in (28) and encode the output state in a quantum memory. Z βPˆ (βˆ|β) ρthm − ρthm • Decoder. d βˆ β 1 1) Read αˆ∗ and perform the displacement Z thm thm D√ ≤ sup sup ρ − ρ + 2 βPˆ (βˆ|β) , 1−δ/2 ∗ on the state of the quantum βˆ β d n−n αˆ β ˆ 1 β∈R≤ R> memory. γn 2) Apply the quantum amplifier A with γn = 1/(1− thm thm −δ/2 having used the elementary inequality ρ ˆ − ρβ ≤ n ). β 1 2. The first term in the right hand side is bounded by 3) Prepare the other (n−1) modes in the thermal state −(1+δ)/2 thm O(n ) using Eq. (26), while the second error term is ρβ . −1 thm bounded by the following property of the maximum likelihood 4) Perform UBS on the thermal state ρβ and the estimate [29] quantum memory. Z The protocol for Case 6 works in the same way except that ˆ ˆ l √ dβP (β|β) ≤ erfc √ (35) |α| is estimated instead of ϕ. For both cases the memory ˆ 2 |β−β|≥l/ nFβ cost consists of two parts: (1/2) log n bits for encoding the 2 3 ∗ where Fβ = (β + 1)/[β(1 − β) ] is the Fisher information (rounded) value αˆ of the estimate and δ log n qubits for R ∞ −s2 encoding the first mode (displaced thermal state). The error of β and erfc(x) := (2/π) x e ds is the complementary −δ/2p −δ/2 error function. Picking l = n Fβ, we have can be bounded as previous as = O n . Z −δ/2p ! n Fβ δ F. Case 5 (fixed |α|, independent ϕ and β) and Case 7 (fixed dβPˆ (βˆ|β) ≤ erfc √ = e−Ω(n ) . (36) R> 2 ϕ, independent |α| and β). Case 5 (fixed |α|, independent ϕ and β) and Case 7 (fixed ϕ, In conclusion, β can be bounded as independent |α| and β) can be treated in the same way as Case −δ −(1+δ)/2 −Ω(nδ ) β ≤ O(n ) + O(n ) + e 4 (independent ϕ, fixed |α| and β) and Case 6 (independent = O(n−δ) . (37) |α|, fixed ϕ and β), except that the thermal parameter β is now independent. We illustrate only the protocol for Case 5 (fixed The remaining contribution to the error can be bounded as |α|, independent ϕ and β) and the other naturally follows. The in Eq. (31), leading to an overall error of size O(n−δ). protocol runs as follows: 1/2 • Preprocessing. Divide the range of ϕ into n intervals, 0 00 E. Case 4 (independent ϕ, fixed |α| and β) and Case 6 each labeled by a point ϕˆi in it, so that |ϕ − ϕ | = (independent |α|, fixed ϕ and β). O(n−1/2) for any ϕ0, ϕ00 in the same interval. In Case 4 (independent ϕ, fixed |α| and β) and Case • Encoder. 6 (independent |α|, fixed ϕ and β), the displacement α is 1) Perform the unitary channel UBS(·) on the input √ thm ⊗(n−1) partially known. Such a knowledge allows us to reduce the state to transform it into ρ nα,β ⊗ (ρβ ) . amount of memory. 2) Estimate β with the von Neumann measurement of thm The protocols for these two cases 4 and 6 are very similar. the photon number on the n − 1 copies of ρβ . Let us start from Case 4, where the phase of the displacement Denote by βˆ the maximum likelihood estimate of is independent while the modulus is fixed. The protocol for β. Case 4 (independent ϕ, fixed |α| and β) runs as follows: 3) Send the first and the last mode through a group of 1/2 • Preprocessing. Divide the range of ϕ into n intervals, beam splitters (10) that implements the transforma- 0 00 tion ρ ⊗ ρthm → ρ√ ⊗ ρ√ . each labeled by a point ϕˆi in it, so that |ϕ − ϕ | = α,β β n−n1−δ/2α,β n1−δ/2α,β O(n−1/2) for any ϕ0, ϕ00 in the same interval. 4) Estimate ϕ by the heterodyne measurement d2α0 0 0 • Encoder. { π |α ihα |} on the last mode, which yields an estimate ϕˆ which is the phase of α0. Encode the 1) Perform the unitary channel UBS(·) on the input ∗ √ thm ⊗(n−1) label ϕˆ of the interval containing ϕˆ in a classical state to transform it into ρ nα,β ⊗ (ρβ ) . 2) Send the first and the last mode through a group of memory. 5) Displace the first mode with D √ with beam splitters (10) that implements the transforma- − n−n1−δ/2αˆ∗ tion ρ ⊗ ρthm → ρ√ ⊗ ρ√ . ∗ iϕˆ∗ α,β β n−n1−δ/2α,β n1−δ/2α,β αˆ := |α|e . Y. YANG, G. BAI, G. CHIRIBELLA, AND M. HAYASHI: COMPRESSION FOR QUANTUM POPULATION CODING 8
6) Prepare the n-th mode in the thermal state ρthm. can be approximated by a classical-quantum Gaussian βˆ The n-mode state is now ρ√ ⊗ state: n−n1−δ/2(α−αˆ∗),β (ρthm)⊗(n−2) ⊗ ρthm. β βˆ Gn,θ = N (δµ, Vµ0 ) ⊗ Φ(δξ, µ0) 7) Send the state of the first mode through a truncation O Φ(δξ, µ0) = ραj,k,βj,k , (39) channel Pnδ defined in (28) and encode the output state in a quantum memory. 1≤j
where kδθk∞ is the max vector norm kδθk∞ := using n copies of the state. maxi(δθ)i and x ∈ (0, 1). Q-LAN states that every n-fold Our compression protocol is illustrated in Fig. 3. For any ⊗n product state ρθ with θ in the neighborhood Θn,x(θ0) δ ∈ (0, 2/9), the protocol consists of the following steps: Y. YANG, G. BAI, G. CHIRIBELLA, AND M. HAYASHI: COMPRESSION FOR QUANTUM POPULATION CODING 9
• Preprocessing. Divide the parameter√ space Θ into a lattice 5) Gaussian state compression. Each quantum mode L := {θ ∈ Θ | θi = zi/(2 n), zi ∈ Z ∀ i}. The lattice of the amplified Gaussian state is then truncated by (fc+fq )/2 has approximately n points, which will be used Pnδ , defined by Eq. (28). The output state is then to store the outcome of tomography. stored in a quantum memory of size δ log n for each ⊗n • Encoder. The encoder of ρθ consists of five steps: mode. The classical mode is compressed by a map 1−δ/2 δ/2 1) Tomography. Use n copies of ρθ for quantum Pc that truncates the state into a O(n )-hypercube tomography, which yields an estimate θˆ of θ. centered around the mean of the Gaussian and 2) Storage of the estimate. Encode the estimate θˆ as a rounds the continuous variable u into a discrete point in the lattice lattice. Explicitly, we have √ L := {θ ∈ Θ | θ = z /(2 n), z ∈ ∀ i}. (46) X i i i Z Pc(ρ) = hu|ρ|ui |rn,δ(u)ihrn,δ(u)| δ/2 Choose the lattice point θ that is closest to θˆ, kuk∞≤n 0 namely X + 1 − hu0|ρ|u0i |0ih0|, ˆ 0 θ0 := argmin kθ − θ k∞. (47) 0 δ/2 θ0∈L ku k∞≤n (51) 3) Q-LAN. After the tomography step, we end up with n − n1−δ/2 copies of the state. Define where rn,δ(u) is the rounding function which fc 0 1 maps u ∈ R to the closest point on the lat- γ = . f n −δ/2 (48) δ/2 c 1 − n tice Z/n . The output of Pc is stored in 0 classical memory. The memory size is determined so that the number of remaining copies is n/γn. The (n/γ0 ) by the number of lattice points covered by the n/γ0 copies are sent through the channel T n n θ0 range of truncation. The separation between lattice (42) which outputs the Gaussian state G(n/γ0 ),θ n points is n−δ/2, and the range of truncation is defined by Eq. (39). [−nδ/2, nδ/2]fc , so it covers O(nfcδ) points on the 4) Amplification. To compensate the loss of copies in lattice. We therefore need f δ log n bits. tomography, the state ρ of each quantum c αj,k,βj,k The whole process for Gaussian state compression mode is amplified by the amplifier defined in Eq. 0 is described by the channel (13) with γ = γn. The Gaussian distribution on the classical register is rescaled by a constant factor: (n) O P = P ⊗ P P = P δ . (52) θ0 c q q n X p 0 p 0 Ac(ρ) := hu|ρ|ui | γnuih γnu| (49) j B. Error analysis. To bound the error of our protocol, we need to specify a small neighborhood for discussion, which should contain the true value θ with high probability. A proper choice is the neighborhood Θn,2δ/3(θ0) (38). Using the triangle inequality of trace distance, we split the overall error into four terms Fig. 3. Compression protocol for qudit states. ≤tomo + amp + G + Q−LAN, (54) Y. YANG, G. BAI, G. CHIRIBELLA, AND M. HAYASHI: COMPRESSION FOR QUANTUM POPULATION CODING 10 where The amplification error for the classical mode is: tomo = Prob θ 6∈ Θn,2δ/3(θ0) (55) 0 classical = kN (δµ, γnVµ0 ) − N (δµ, Vµ0 ) k1 1 (n/γ0 )→n n 0 amp = sup sup A Gn/γ0 ,θ = kN (0, γ Vµ0 ) − N (0,Vµ0 ) k1. (64) θ0 n n 2 θ0 θ∈Θn,2δ/3(θ0) Writing explicitly the probability density functions of the − Gn,θ (56) 1 Gaussian distributions, we have 1 (n) G = sup sup P (Gn,θ) − Gn,θ (57) T −1 ! θ0 Z x V x 2 1 1 µ0 θ0 θ∈Θn,2δ/3(θ0) classical = p exp − d 2 n 0 0 2π|Vµ0 | 1 (n/γn) ⊗(n/γn) R Q−LAN = sup sup T ρ θ0 θ T −1 ! 2 θ0 θ∈Θn,2δ/3(θ0) 1 x V x − exp − µ0 dx p2πγ0 |V | 2γ0 − Gn/γ0 ,θ n µ0 n n 1 ! o Z xT V −1x ⊗n (n) 1 µ0 + ρ − S (Gn,θ) ≤ exp − θ θ0 p 1 2π|V | d 2 (58) µ0 R ! xT V −1x are the error terms of tomography, amplification, truncation, − exp − µ0 dx 0 and Q-LAN, respectively. In the following, we will provide 2γn ! upper bounds for all four terms. 1 − (γ0 )−1/2 Z xT V −1x + n exp − µ0 dx Let us start from the tomography error. By definition of the p 0 2π|Vµ | d 2γn neighborhood Θn,2δ/3(θ0) (38), we have 0 R (1 − (γ0 )−1) Z 1 h −1/2+δ/3i ≤ n xT V −1x exp − xT V −1x dx tomo = Prob kθ − θ0k∞ > n p µ0 µ0 2 2π|Vµ | d 2 h i 0 R ˆ ˆ −1/2+δ/3 −δ/2 ≤ Prob kθ − θk∞ + kθ − θ0k∞ > n + O n h i ˆ −1/2+δ/3 −δ/3 −δ/2 ≤ Prob kθ − θk∞ > n (1 − O(n )) . = O n . (65) (59) 0 −1 0 −1/2 The first inequality comes from triangle inequality and the Note that (1 − (γn) ) and (1 − (γn) ) both have order ˆ √ O(n−δ/2). second inequality holds since kθ − θ0k∞ ≤ 1/(2 n) which is an immediate implication of Eq. (46) and Eq. (47). Now we check the quantum term. On the quantum register, To further bound the error, notice that the trace distance the amplifier acts independently on each mode as the displaced has a Euclidean expansion, of the form kρθ − ρθ0 k1 = Ckθ − thermal state amplifier defined by Eq. (13). From a similar 0 0 2 calculation as Eq. (25), we obtain the inequality θ k∞ + O(kθ − θ k∞), where C > 0 is a suitable constant. Then we have 1 X γ0 ≤ A n ρ − ρ , (66) 1 quantum αj,k,βj,k αj,k,βj,k ≤ Prob kρ − ρ k > (C/4)n−1/2+δ/3 . (60) 2 1 tomo 2 θ θˆ 1 j δ/3 can be bounded by noticing that kδµk∞ ≤ n , from which V. NECESSITYOFAQUANTUMMEMORY. we have In the previous section, we showed that the ratio between the quantum and the classical memory cost can be made arbitrarily kP [N(δµ, V )] − N (δµ, V )k c µ0 µ0 1 close to zero [see Eq. (75)]. It is then natural to ask whether the Z ratio can be exactly zero. The answer turns out to be negative. ≤ N (δµ, Vµ0 )(du) δ/2 kuk∞>n In fact, we prove an even stronger result: if a state family has Z at least one independent quantum parameter, then no protocol ≤ N (δµ, Vµ0 )(du) δ/2 δ/3 using a purely classical memory can be faithful, even if the ku−δµk∞>n −n δ amount of classical memory is arbitrarily large. = e−Ω(n ) (70) ⊗n Theorem 2. Let {ρθ } be a qudit state family with at least where N (δµ, Vµ )(du) denotes the probability density func- one independent quantum parameter, and let (En, Dn) be 0 ⊗n tion. For each of the quantum modes, employing Lemma 2, generic compression protocol for {ρθ }. If the protocol uses δ δ/3 with K substituted by n and |αj,k| = O(n ), we have solely a classical memory, then the compression error will not vanish in the large n limit, no matter how large the memory 1 Ω(nδ/8) −Ω(nδ/4) is. Pj,k ρα ,β − ρα ,β = β + e . 2 j,k j,k j,k j,k 1 j,k (71) The proof of Theorem 2 is based on the properties of two distance measures, known as the quantum Hellinger distance Substituting Eqs. (70) and (71) into Eq. (69), we have [31] (see also [32]) and the Bures distance [33], and defined as δ/8 Ω(n ) r −Ω(nδ/4) 1/2 1/2 G = max βj,k + e . (72) dH(ρ1, ρ2) := 2 − 2 Tr ρ1 ρ2 (76) j Proof. By definition, the condition dH(ρ1, ρ2) ≥ dB(ρ1, ρ2) Summarizing the above bounds (61), (68), (72), (73) on each is equivalent to the condition of the error terms, we conclude that the protocol generates an error which scales at most 1/2 1/2 1/2 1/2 Tr ρ1 ρ2 ≥ Tr ρ1 ρ2 . = O n−κ(δ) + O n−δ/2 . (74) The validity of this condition is immediate: for every square matrix A, one has Tr |A| ≥ Tr A. The equality holds if and only if A is equal to |A|, meaning that A is positive 1/2 1/2 C. Total memory cost. semidefinite. For A = ρ1 ρ2 , the Hermiticity requirement A = A† reads The total memory cost consists of three contributions: a † 1/2 1/2 1/2 1/2 1/2 1/2 classical memory of [(fc+fq)/2] log n bits for the tomography ρ1 ρ2 = ρ1 ρ2 = ρ2 ρ1 , outcome, a classical memory of f δ log n bits for the classical c which in turn is equivalent to the commutation relation part of the Gaussian state and a quantum memory of f δ log n q [ρ , ρ ] = 0. qubits for the quantum part of the Gaussian state. In short, it 1 2 takes (1/2 + δ) log n bits to encode a classical independent The second property used in the proof of Theorem 2 is parameter and (1/2) log n bits plus δ log n qubits to encode a quantum independent parameter. Lemma 4. Let E be a quantum channel sending states on H to states on K and let D be a quantum channel sending From the above discussion we can see that the ratio between states on K to states on H. Let ρ and ρ be two states on the quantum memory cost and the classical memory cost is 1 2 H, satisfying the conditions δfq Rq/c = , (75) [E(ρ1), E(ρ2)] = 0 , (78) (1/2 + δ)fc + (1/2)fq 1 and 2 kD◦E(ρi)−ρik1 ≤ for i ∈ {1, 2}. Then, the following which can be made close to zero by choosing δ close to zero. inequality holds: In conclusion, the size of the quantum memory can be made √ arbitrarily small compared to the classical memory. |dH(ρ1, ρ2) − dB(ρ1, ρ2)| ≤ 2 2. (79) Y. YANG, G. BAI, G. CHIRIBELLA, AND M. HAYASHI: COMPRESSION FOR QUANTUM POPULATION CODING 12 0 0 Proof. Using the the triangle inequality for the quantum Since the protocol (En, Dn) uses a purely classical memory, Hellinger distance, we obtain the upper bound Lemma 4 implies the bound p 0 dH(ρ1, ρ2) ≤ dH (ρ1, D ◦ E(ρ1)) + dH (D ◦ E(ρ1), D ◦ E(ρ2)) |dH (Gn,θ0 ,Gn,θ) − dB (Gn,θ0 ,Gn,θ)| ≤ 2 2n, (88) + dH (D ◦ E(ρ2), ρ2) . (80) 0 where n is the compression error for the states {Gn,θ0 ,Gn,θ}. Now, for every pair of states ρ and σ, the quantum Hellinger On the other hand, the errors from Q-LAN vanish as −κ(x) distance and the trace distance are related by inequality O n . Hence, we have the bound d (ρ, σ) ≤ pkρ − σk [31]. Using this fact, the upper bound H 1 0 ≤ + O n−κ(x) (89) (80) becomes n n √ Substituting Eqs. (85) and (86) into Eq. (88), we obtain the d (ρ , ρ ) ≤ d (D ◦ E(ρ ), D ◦ E(ρ )) + 2 2 . (81) H 1 2 H 1 2 expression At this point, we use the fact that the quantum Hellinger 0 −κ(x) (respectively, Bures) distance is non-increasing under the n ≥ n − O n action of quantum channels [32] (respectively, [33]). In this 1 ≥ |d (G ,G ) − d (G ,G )|2 − O n−κ(x) way, we obtain the inequality 8 H n,θ0 n,θ B n,θ0 n,θ 1 thm thm 2 −κ(x) = dH ρ , ρα(s),β − dB ρ , ρα(s),β − O n . dH (D ◦ E(ρ1), D ◦ E(ρ2)) ≤ dH (E(ρ1), E(ρ2)) 8 β β = dB (E(ρ1), E(ρ2)) (90) ≤ dB (ρ1, ρ2) , (82) The quantum Hellinger and Bures distances between displaced thermal states can be computed using previous results. Using in which we used the relation Eq. (16) of [34], we have d (E(ρ ), E(ρ )) = d (E(ρ ), E(ρ )) , (83) q H 1 2 B 1 2 |α|2 1 + β thm − 4γ dB ρ , ρα,β = 2 − 2e B γB = . (91) following from Eq. (78) and Lemma 3. Combining Eqs. (81) β 1 − β and (82), we finally obtain the bound Using Eq. (3.18) of [35], we have √ q √ dH(ρ1, ρ2) ≤ dB (ρ1, ρ2) + 2 2 . (84) |α|2 ( β + 1)2 thm − 4γ dH ρβ , ρα,β = 2 − 2e H γH = . Since the difference between the quantum Hellinger distance 2(1 − β) and the Bures distance is non-negative, the above inequality (92) is exactly Eq. (79). The difference between the two terms is strictly positive, except in the case when α = 0. Therefore, the right hand Now we give the proof of Theorem 2. side of Eq. (90) is strictly positive in the limit of n → ∞, and ⊗n Proof. Let {ρθ } be a qudit state family with at least one thus limn→∞ n > 0. This concludes the proof. independent quantum parameter. ⊗n ⊗n Pick two states ρ and ρ , with θ of the form θ = VI.OPTIMALITYOFTHECOMPRESSION √ θ0 θ θ0 + s/ n where all entries of the vector s are zero except Here we prove that our compression protocol is asymptoti- for the independent quantum parameter. Applying Q-LAN to cally optimal in terms of total memory cost. Specifically, we the neighborhood of θ , the two states ρ⊗n and ρ⊗n can be 0 θ0 θ show that every compression protocol with vanishing error converted into two multi-mode Gaussian states Gn,θ0 and Gn,θ must use an overall memory size of at least (f/2) log n, where that differ from each other only in one mode. Explicitly, the f is the number of independent parameters describing the input two Gaussian states can be written as states. (−) thm The proof idea is to construct a communication protocol Gn,θ = G ⊗ ρ (85) 0 n,θ0 β that transmits approximately (f/2) log n bits, using the com- G = G(−) ⊗ ρ . (86) n,θ n,θ0 α(s),β pression protocol. Once this is done, the Holevo bound [19] implies that the overall amount of memory must be of at least where the thermal parameter β and the displacement α(s) are (f/2) log n qubits/bits. non-zero quantities depending only on s and θ0 via Eq. (39), (−) To construct the communication protocol, we define a mesh while Gn,θ is the state of the remaining modes. 0 Mn on the parameter space Θ, by choosing a set of equally Now, let (En, Dn) be a compression protocol that uses a ⊗n spaced points starting from a fixed point θ0 ∈ Θ. Specifically, purely classical memory to compress the state family {ρθ }. 0 0 we define the mesh as By Q-LAN, there exists a compression protocol (En, Dn) √ that uses a purely classical memory to compress the states Mn = θ ∈ Θ | |(θ − θ0)i| = zi · log n/ n, zi ∈ Z ∀ i . {Gn,θ0 ,Gn,θ}. Explicitly, the encoder and the decoder are (93) described by the channels The number of points in the mesh Mn satisfies the bound 0 (n) 0 (n) E := En ◦ S D := T ◦ Dn , (87) √ f n θ0 n θ0 n |Mn| ≥ TΘ , (94) where T (n) and S(n) are the channels used for Q-LAN. log n θ0 θ0 Y. YANG, G. BAI, G. CHIRIBELLA, AND M. HAYASHI: COMPRESSION FOR QUANTUM POPULATION CODING 13 where TΘ > 0 is a constant independent of n. The next step is to define a finite set of states ⊗n Sn = ρθ | θ ∈ Mn , (95) Fig. 4. A protocol to communicate log |Mn| bits of information. Here En and to observe that they are almost perfectly distinguishable and Dn are the encoder and the decoder, Qn(θˆ) is the POVM to recover the in the large n limit. One way to distinguish between the message, and Mn denotes the memory. states in Sn is to use quantum tomography. Intuitively, since tomography provides an estimate of the state with error of √ 4) The receiver applies the decoder D . order 1/ n, the distance between two states in the set is large n 5) The receiver measures the output state with the POVM enough to make the states almost perfectly distinguishable. To {Q (θ)} ∪ {Q (rest)}. make this argument rigorous, we describe the tomographic n θ∈Mn n The protocol is illustrated in Fig. 4. A protocol can be protocol using a POVM Pn(dρˆ), where ρˆ is the estimate of the state. In particular, we use the POVM defined in Eq. (45), constructed for displaced thermal states, following the steps which has the property [30] 1) - 4) and replacing the POVM in step 5) of the above Z protocol by the heterodyne measurement of α and maximum ⊗n 3d2 −nε2 Tr[Pn(dρˆ) ρ ] ≤ (n + 1) e . (96) likelihood estimation of β [27]. In this way, the proof here can kρˆ−ρk1>ε be converted to a proof of optimality for displaced thermal The continuous POVM Pn(dρˆ) can be used to distinguish states, which we omit for simplicity. between the states in the set Sn. To this purpose, we con- It is not hard to see that the protocol can communicate no less than (f/2) log n bits, with an error probability struct the discrete POVM with operators {Qn(θ)}θ∈Mn ∪ {Qn(rest)}, where the operator Qn(rest) is defined as ∗ Perr,n ≤ Perr,n + n (102) ⊗n X Qn(rest) = I − Qn(θ) (97) where Perr,n := maxθ∈Mn Perr,n(θ) and n is the error of the θ∈Mn compression protocol (En, Dn). Consider the case when the while the operator Qn(θ) is defined as messages are uniformly distributed. In this case, the number Z of transmitted bits can be bounded through Fano’s inequality, Qn(θ) = Pn(dρˆ) , θ ∈ Mn . (98) which yields the bound εmin kρˆ−ρθ k1≤ 2 ˆ ∗ ∗ I(Θ : Θ) ≥ (1 − Perr,n) log |Mn| − h(Perr,n) , (103) Here εmin is the minimum distance between two distinct states in Sn, which can be quantified as with h(x) = −x log x and h(0) := 0. When the compression 1 C log n log2 n protocol has vanishing error, i.e. limn→∞ n = 0, using Eqs. εmin = min kρθ − ρθ0 k1 = √ + O , 0 0 (94), (101), and (102) we obtain the lower bound θ,θ ∈Mn ,θ6=θ 2 2 n n (99) f I(Θ : Θ)ˆ ≥ log n − f log log n + o(1) . (104) having used the Euclidean expansion of trace distance, given 2 by Using the monotonicity of mutual information and the upper bound of entropy, the total number n of memory bits/qubits 0 0 2 enc kρθ − ρθ0 k1 = Ckθ − θ k∞ + O(kθ − θ k∞) , (100) is lower bounded as where C > 0 is a suitable constant. Hence, for any θ ∈ M " !# n −1 X ⊗n ⊗n nenc ≥ H En |Mn| ρ the probability of error for the state ρθ can be bounded as θ Z θ∈Mn ⊗n Perr,n(θ) ≤ Tr Pn(dρˆ)ρθ ≥ I(Θ : Mn) kρˆ−ρ k > εmin θ 1 2 ˆ 2 ≥ I(Θ : Θ). (105) 3d2 − C log n ≤ (n + 1) e 16 Combining Eq. (105) with Eq. (104), we obtain that nenc must 3d2 − C log n = (n + 1) n 16 ln 2 be at least − C log n ≤ n 16 , (101) f n ≥ log n − f log log n + o(1) . (106) enc 2 where the last inequality holds for large enough n. Using the results above, we can construct a communication This proves that f/2 log n bits/qubits are necessary to achieve protocol that communicates (f/2) log n bits given any com- compression with vanishing error. pression protocol (En, Dn). The protocol is defined as follows: 1) Both parties agree on a code that associates messages VII.CONCLUSION with points in the mesh Mn. In this work we addressed the problem of compressing iden- 2) To communicate a certain message, the sender picks the tically prepared states of finite-dimensional quantum systems ⊗n corresponding point θ ∈ Mn and prepares the state ρθ . and identically prepared displaced thermal states. We showed 3) The sender applies the encoder En and transmits that the total size of the required memory is approximately ⊗n En(ρθ ) to the receiver. (f/2) log n, where f is the number of independent parameters Y. YANG, G. BAI, G. CHIRIBELLA, AND M. 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What remains thm ⊗n To compress the n-fold thermal state (ρβ ) into classical is to bound the error of the protocol. First, we notice that the memory, we can first do measurements on the state, and recovered state is then encode the outcome using the smallest possible classical memory. The state ρthm is fully described by the thermal X β Dthm ◦ Ethm (ρthm)⊗n = (ρ ) |~mih~m| (108) parameter β, or equivalently, the average photon number n,δ n,δ β β,n ~m β/(1 − β). An estimate of β/(1 − β) can be obtained by ~m βm−| ~m| n+m−1 n | ~m| P ( m ) measuring the photon number of the n modes independently (1 − β) β m∈L n+| ~m|−1 i(|~m|) < t i(| ~m|) |Li(| ~m|)|( ) and by computing the sum. The sum can be encoded into | ~m| a classical memory. To this purpose, we divide the set of (ρ ) = 0 β,n ~m P (1 − β)nβ| ~m | ~m = ~t nonnegative integers into intervals, and only store the index ~m0:i(| ~m0|)=t of the interval the sum lies in. For any δ > 0, we define a series of t + 1 = bn1/2+δc 0 else. (109) intervals L0,... Lt as L = {0} 0 We choose S as the minimal set satisfying i) S is a union L = {(i − 1) bn(1−δ)/2c + 1, . . . , i bn(1−δ)/2c} 0 < i < t i of several intervals chosen from the set {Li} and ii) (1−δ)/2 Lt = {(t − 1) bn c + 1,... }. βn βn For any non-negative integer m, we denote by i(m) the index S ⊃ − n(1−δ)/2, + n(1−δ)/2 . (110) 1 − β 1 − β of the interval containing m, i.e. m ∈ Li(m). Y. YANG, G. BAI, G. CHIRIBELLA, AND M. HAYASHI: COMPRESSION FOR QUANTUM POPULATION CODING 16 Apparently, S ⊂ [0, n1+δ/2] for large enough n. Then the error Note that the amplifier of Eq. (13) can be represented as can be bounded as γ h † i −1 √ √ A (ρ) = TrB Scosh ( γ)(ρ ⊗ |0ih0|B)Scosh−1( γ) (111) 1 thm thm thm ⊗n thm ⊗n thm = D ◦ E (ρ ) − (ρ ) n,δ n,δ β β 1 † † 2 with S := er(ˆa ˆb −aˆˆb). The unitary S satisfies S aSˆ † = 1 X r r r r = (1 − β)nβ| ~m| − (ρ ) (cosh r)ˆa−(sinh r)ˆb† (cf. Eq. (B8) of [?]), which immediately 2 β,n ~m ~m implies the relation SrDα = D(cosh r)αSr. Hence, we have γ √ γ X 1 X A ◦ Dα = D γα ◦ A (112) ≤ (1 − β)nβ| ~m| + (1 − β)nβ| ~m| 2 ~m:| ~m|6∈S ~m:| ~m|∈S for any α ∈ C and γ ≥ 1. In particular, when the amplifier is m n+m−1 applied to displaced thermal states, one has the relation X β × max m − 1 0 n+m0−1 γ √ γ thm m0 ∈ L m A (ρα,β) = D γα ◦ A (ρ ) . (113) i |Li|β 0 β Li ∩ S 6= ∅ m∈Li m X n | ~m0| To prove Eq. (25), it only remains to show the identity + (1 − β) β γ thm thm 0 A (ρ ) = ρ 0 with β as in Eq. (25). This equality, which ~m0:i(| ~m0|)=t β β is standard in quantum optics, can be proven by observing that X n | ~m| 1 X n | ~m| every thermal state can be generated from the vacuum through ≤ (1 − β) β + (1 − β) β 2 the action of the Gaussian additive noise channel Nx, defined ~m:| ~m|6∈S ~m:| ~m|∈S as m n+m−1 β Z x 2 × max m − 1 N (ρ) = 2µ e−x|µ| D ρD† . (114) 0 n+m0−1 x d µ µ m, m0 ∈ L m π i β 0 Li ∩ S 6= ∅ m + O ((1 − β)nβn) Specifically, it is easy to verify the relation X n | ~m| thm ρβ = N 1−β (|0ih0|), (115) ≤ (1 − β) β β ~m:| ~m|6∈S valid for every β ∈ (0, 1]. Then, using Eq. (113), we have m n+m−1 1 β m γ thm γ γ + max − 1 A (ρ ) = A ◦ N 1−β (|0ih0|) = N 1−β ◦ A (|0ih0|). 0 n+m0−1 β 2 m, m0 ∈ L m β βγ i β m0 Li ∩ S 6= ∅ (116) + O ((1 − β)nβn) . It is straightforward to verify that Aγ (|0ih0|) is a thermal state; On one hand, we notice that |~m| is the sum of n i.i.d. random specifically, one has i ∞ variables with geometric distribution {(1−β)β }i=0 and thus, N 1 (|0ih0|) . (117) by Central Limit Theorem, the first error term scales as γ−1 δ/2 Hence, we have X n | ~m| n (1 − β) −Ω(nδ ) (1 − β) β = O erfc √ = e γ thm 2β A (ρβ ) = N 1−β ◦ N 1 (|0ih0|) ~m:| ~m|6∈S βγ γ−1 R ∞ −s2 = N 1−β (|0ih0|) where erfc(δ) := (2/π) δ e ds is the complementary γ−1+β error function. On the other hand, in the second error term, m thm 0 β + γ − 1 = ρ 0 β = , (118) and m0 are in the same order as n, so the second term can be β γ bounded as the second equality following from the composition property βm n+m−1 of the additive Gaussian noise m max 0 0 − 1 m, m0 ∈ L m n+m −1 i β 0 N ◦ N = N xy ∀ x, y > 0. Li ∩ S 6= ∅ m x y x+y 0 0 (n + m − 1) ··· (n + m ) = max βm−m − 1 Indeed, the above equation can be derived using the definition 0 0 m, m ∈ Li m . . . (m + 1) of the additive Gaussian noise (114): Li ∩ S 6= ∅ 0 Z Z 2 2 m−m 2 2 2 xy −x|µ| −y|ν| † βm + βn |L | Nx ◦ Ny(ρ) = d µ d ν e Dµ+ν ρD = max 1 + O i − 1 π2 µ+ν m, m0 ∈ L m n i Z Z 2 2 Li ∩ S 6= ∅ 2 2 xy −x|µ| −y|α−µ| † = d µ d α e · e Dα ρD −δ −δ 2 α = 1 + O n 1 + O n − 1 π α := µ + ν = O n−δ . Z 2 2 x + y −(x+y)|µ− y α| = d µ e x+y Therefore, we have proved Eq. (14). π Z 2 xy − xy |α|2 † B. Derivation of Eq. (25) × d α e x+y Dα ρD (x + y)π α Eq. (25) is a standard result in quantum optics. Here we = N xy (ρ). provide its derivation for the benefit of those readers who may x+y be less familiar with this area. Combining Eqs. (113) and (118) gives Eq. (25) as desired. Y. YANG, G. BAI, G. CHIRIBELLA, AND M. HAYASHI: COMPRESSION FOR QUANTUM POPULATION CODING 17 x/8 2 C. Proof of Eq. (26) Here we set l0 = K . Notice that |hl|Dα|ki| is the photon number distribution of a displaced number state [49], which The distance between can be bounded as 0 2 thm thm 2|β − β| 0 2 −|α|2 2 k+l min{k,l} 2 −i ρ 0 − ρ ≤ + O(|β − β| ) . e (|α| ) k!l!(−|α| ) β β 1 0 2 2 X (1 − β ) |hl|Dα|ki| = k!l! i!(k − i)!(l − i)! i=0 By definition 2 2 k i e−|α| (|α|2)k+l X k l ≤ ∞ k!l! i |α|2 thm thm X j 0 0 j i=0 ρβ0 − ρβ = |(1 − β)β − (1 − β )(β ) | 1 −|α|2 2l 2 2k j=0 e |α| l + |α| ∞ = . X k!l! |α| ≤ |βj − (β0)j| + |βj+1 − (β0)j+1| Then we can bound the first term in (120) as j=0 ∞ 2l −|α|2 2 2k X j 0 j 0 2 X 2 |α| X e l + |α| = 2 |β − (β ) | + O(|β − β | ). max |hl|Dα|ki| ≤ max l≤l0 l≤l0 l! k! |α| j=0 k>K k>K 2 2l+ l |α|2 2l Extracting |β − β0| from the first term on the r.h.s. of the last e |α| X = max Poisλα (k), l≤l0 l! equality and focusing on the first order, we get k>K ∞ where Poisλα (k) is the Poisson distribution with mean thm thm 0 X j 0 2 2 2 2 1/2+x/2 ρβ0 − ρβ ≤ 2|β − β | (j + 1)β + O(|β − β | ) λα = (l + |α| ) /|α| . Notice that [λα − K , λα + 1 1/2+x/2 j=0 K ] ⊆ [0,K] and thus we have 0 2 2 2|β − β | 0 2 2l+ l = + O(|β − β | ) |α|2le |α|2 (1 − β)2 X 2 X max |hl|Dα|ki| ≤ max Poisλα (k) l≤l0 l≤l0 l! 1/2+x/2 k>K |k−λα|>K as desired. l2 2l 2l+ 2 |α| e |α| x/2 = max e−Ω(K ) l≤|α|x/4 l! x/4 D. Proof of Lemma 2. =e−Ω(K ). For any input state ρ we have having used the tail bound for Poisson distribution and l0 = Kx/8. Substituting the above bound into Eq. (120), we get 1 1 (ρ) ≤ kPK ρPK − ρk1 + [1 − Tr(ρPK )] r 2 2 h thm † i Ω(Kx/8) −Ω(Kx/4) 1 − Tr D ρ DαP ≤β + e . 1 h p i α β α ≤ 2 [1 − Tr(ρPK )] + 1 − Tr(ρPK ) 2 Finally, substituting the above inequality into Eq. (119), we 3 ≤ p1 − Tr(ρP ). can bound the error as 2 K Ω(Kx/8) −Ω(Kx/4) (ρα,β) = β + e . The second inequality came from the gentle measurement thm † lemma [47], [48]. Substituting ρα,β = Dαρβ Dα into the above bound, we express the truncation error for ρα,β as 3r h i (ρ ) ≤ 1 − Tr D ρthmD† P . (119) α,β 2 α β α K We now bound the trace part in the right hand side of the last inequality as ∞ thm † X X l 2 1 − Tr Dαρβ DαPα = (1 − β)β |hl|Dα|ki| k≤K l=0 ∞ X 2 X l ≤ max |hl|Dα|ki| + (1 − β)β l≤l0 k>K l≥l0 X 2 l0 ≤ max |hl|Dα|ki| + β l≤l0 k>K (120)