Classical data compression with quantum side information

The problem of classical data compression when the decoder has quantum side information at his disposal is considered. This is a quantum generalization of the classical Slepian-Wolf theorem. The optimal compression rate is found to be H(X)−χ where H(X) is the Shannon entropy of the source and χ is the Holevo quantity of the ensemble describing the classicalquantum correlations between the source and the side information. Generalizing classical information theory to the quantum setting has had varying success depending on the type of problem considered. Quantum problems hitherto solved (in the sense of Shannon theory) may be divided into three classes. The first comprises (pure) bipartite entanglement manipulation, such as Schumacher compression [1] and entanglement concentration/dilution [2, 3, 4], and their tractability is due to the formal similarities between random variables and the Schmidt decomposition of bipartite states. The second is the class of “partially quantum” problems, where only one of the “terminals” in the problem is quantum and the others are classical. The simplest example is the Holevo-Schumacher-Westmoreland (HSW) theorem [5], which deals with the capacity of a c-q (classical-quantum) channel. This carries over to the multiterminal case involving many classical senders and one quantum receiver [6]. Then we have Winter’s measurement compression theorem [7], and remote state preparation [8, 9, 10]. These two may be thought of as simulating q-c (quantum-classical) and c-q channels, respectively. Another recent discovery has been quantum data compression with classical side-information available to both the encoder and decoder [11], generalizing the “rate-entropy” curve of [10] to arbitrary pure state ensembles. The third class is that of entanglement assisted quantum communication, such as the entanglement-assisted capacity theorem [12] and its reverse – that of simulating quantum channels in the presence of entanglement [13]. These rely on methods of c-q channel coding combined with superdense coding [14], and q-c channel simulation combined with quantum teleportation [15], respectively. The problem addressed here belongs to the second class and concerns classical data compression with quantum side-information. We shall refer to it as the partially quantum Slepian-Wolf (PQSW) problem in analogy to its classical counterpart [16]. The PQSW problem is defined as follows. The decoder Bob has some quantum data correlated with the encoder Alice’s classical data. Such correlations, described by some ensemble E = (X, ρX), come about, for example, when Bob holds (generally mixed) quantum states resulting from a preparation or measurement done by Alice. A particular situation of interest is when Bob holds part of the purification of a state Alice is measuring. In this case his quantum data is correlated with her measurement outcome. Here X is a classical random variable defined on a set X of size a, with probability distribution {p(x) : x ∈ X}. The ρx are density operators on a d-dimensional Hilbert space H. With probability p(x) the classical index and quantum state take on values x and ρx, respectively. Define the average density operator ρ = EρX = ∑ x p(x)ρx, the average von Neumann entropy S = ES(ρX), where S(σ) = −trσ log σ, and the Holevo quantity χ = S(ρ)− S Electronic address: [email protected]