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 fρθ gθ2Θ 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 fρθgθ2Θ, 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 fρθ gθ2Θ 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 y pared finite-dimensional states. In Section V we show that ρα,β = Dα ρβ Dα (4) every protocol achieving asymptotically faithful compression y where Dα = exp(αa^ − α¯a^) is the displacement operator, must use a quantum memory. Optimality of the protocols is defined in terms of a complex parameter α 2 C (the displace- proven later in Section VI. Finally, the conclusions are drawn ment), and a^ is the annihilation operator, satisfying the relation in Section VII. y thm [^a; a^ ] = 1, while ρβ is a thermal state, defined as 1 II. MAIN RESULT. thm X j ρβ := (1 − β) β jjihjj ; (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 2 [0; 1) is a real parameter, here called the thermal and infinite-dimensional displaced thermal states. parameter, and the basis fjjigj2N consists of the eigenvectors y Let us start from the first category. For a quantum system of a^ a^. For β = 0, the the displaced thermal states are pure. of dimension d < 1, 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] jαi := Dα j0i.