We study source compression with a helper in the fully quantum regime, extending our earlier result on classical source compression with a quantum helper [arXiv:1501.04366, 2015]. We characterise the quantum resources involved in this problem, and derive a single-letter expression of the achievable rate region when entanglement assistance is available. The direct coding proof is based on a combination of two fundamental protocols, namely the quantum state merging protocol and the quantum reverse Shannon theorem (QRST). This result connects distributed source compression with the QRST protocol, a quantum protocol that consumes noiseless resources to simulate a noisy quantum channel.
We study source compression with a helper in the fully quantum regime, extending our earlier result on classical source compression with a quantum helper [arXiv:1501.04366, 2015]. We characterise the quantum resources involved in this problem, and derive a single-letter expression of the achievable rate region when entanglement assistance is available. The direct coding proof is based on a combination of two fundamental protocols, namely the quantum state merging protocol and the quantum reverse Shannon theorem (QRST). This result connects distributed source compression with the QRST protocol, a quantum protocol that consumes noiseless resources to simulate a noisy quantum channel.
Coded source compression, also known as source compression with helpers, has been a major variant of distributed source compression, but has hitherto received little attention in the quantum regime. This letter treats and solves the corresponding quantum coded source compression through an observation that connects coded source compression with channel simulation. First, we consider classical source coding with quantum side information, where the quantum side information is observed by a helper and sent to the decoder via a classical channel. We derive a single-letter characterization of the achievable rate region for this problem. The direct coding theorem of our result is proved via the measurement compression theory of Winter, a quantum-to-classical channel simulation. Our result reveals that a helper's scheme which separately conducts a measurement and a compression is suboptimal, and measurement compression seems necessary to achieve the optimal rate region. We then study coded source compression in the fully quantum regime, where two different scenarios are considered depending on the types of communication channels between the legitimate source and the receiver. We further allow entanglement assistance from the quantum helper in both scenarios. We characterize the involved quantum resources and derive single-letter expressions of the achievable rate region. The direct coding proofs are based on well-known quantum protocols, the quantum state merging protocol, and the fully quantum Slepian-Wolf protocol, together with the quantum reverse Shannon theorem.
We study classical source coding with quantum side-information where the quantum side-information is observed by a helper and sent to the decoder via a classical channel. We derive a single-letter characterization of the achievable rate region for this problem. The direct part of our result is proved via the measurement compression theory by Winter. Our result reveals that a helper's scheme that separately conducts a measurement and a compression is suboptimal, and the measurement compression is fundamentally needed to achieve the optimal rate region.
We extend quantum rate-distortion theory by considering auxiliary resources that might be available to a sender and receiver performing lossy quantum data compression. The first setting we consider is that of quantum rate-distortion coding with the help of a classical side channel. Our result here is that the regularized entanglement of formation characterizes the quantum rate-distortion function, extending earlier work of Devetak and Berger. We also combine this bound with the entanglement-assisted bound from our prior work to obtain the best known bounds on the quantum rate-distortion function for an isotropic qubit source. The second setting we consider is that of quantum rate-distortion coding with quantum side information (QSI) available to the receiver. In order to prove results in this setting, we first state and prove a quantum reverse Shannon theorem with QSI (for tensor-power states), which extends the known tensor-power quantum reverse Shannon theorem. The achievability part of this theorem relies on the quantum state redistribution protocol, while the converse relies on the fact that the protocol can cause only a negligible disturbance to the joint state of the reference and the receiver's QSI. This quantum reverse Shannon theorem with QSI naturally leads to quantum rate-distortion theorems with QSI, with or without entanglement assistance.
Dual to the usual noisy channel coding problem, where a noisy (classical or quantum) channel is used to simulate a noiseless one, reverse Shannon theorems concern the use of noiseless channels to simulate noisy ones, and more generally the use of one noisy channel to simulate another. For channels of nonzero capacity, this simulation is always possible, but for it to be efficient, auxiliary resources of the proper kind and amount are generally required. In the classical case, shared randomness between sender and receiver is a sufficient auxiliary resource, regardless of the nature of the source, but in the quantum case, the requisite auxiliary resources for efficient simulation depend on both the channel being simulated, and the source from which the channel inputs are coming. For tensor power sources (the quantum generalization of classical memoryless sources), entanglement in the form of standard ebits (maximally entangled pairs of qubits) is sufficient, but for general sources, which may be arbitrarily correlated or entangled across channel inputs, additional resources, such as entanglement-embezzling states or backward communication, are generally needed. Combining existing and new results, we establish the amounts of communication and auxiliary resources needed in both the classical and quantum cases, the tradeoffs among them, and the loss of simulation efficiency when auxiliary resources are absent or insufficient. In particular, we find a new single-letter expression for the excess forward communication cost of coherent feedback simulations of quantum channels (i.e., simulations in which the sender retains what would escape into the environment in an ordinary simulation), on nontensor-power sources in the presence of unlimited ebits but no other auxiliary resource. Our results on tensor power sources establish a strong converse to the entanglement-assisted capacity theorem.
We resume the investigation of the problem of independent local compression of correlated quantum sources, the classical case of which is covered by the celebrated Slepian-Wolf theorem. We focus specifically on classical-quantum (cq) sources, for which one edge of the rate region, corresponding to the compression of the classical part, using the quantum part as side information at the decoder, was previously determined by Devetak and Winter [Phys. Rev. A 68, 042301 (2003)]. Whereas the Devetak-Winter protocol attains a rate-sum equal to the von Neumann entropy of the joint source, here we show that the full rate region is much more complex, due to the partially quantum nature of the source. In particular, in the opposite case of compressing the quantum part of the source, using the classical part as side information at the decoder, typically the rate sum is strictly larger than the von Neumann entropy of the total source.We determine the full rate region in the generic case, showing that, apart from the Devetak-Winter point, all other points in the achievable region have a rate sum strictly larger than the joint entropy. We can interpret the difference as the price paid for the quantum encoder being ignorant of the classical side information. In the general case, we give an achievable rate region, via protocols that are built on the decoupling principle, and the principles of quantum state merging and quantum state redistribution. Our achievable region is matched almost by a single-letter converse, which however still involves asymptotic errors and an unbounded auxiliary system.