Winter's measurement compression theorem stands as one of the most penetrating insights of quantum information theory. In addition to making an original and profound statement about measurement in quantum theory, it also underlies several other general protocols used for entanglement distillation and local purity distillation. The theorem provides for an asymptotic decomposition of any quantum measurement into noise and information . This decomposition leads to an optimal protocol for having a sender simulate many independent instances of a quantum measurement and send the measurement outcomes to a receiver, using as little communication as possible. The protocol assumes that the parties have access to some amount of common randomness, which is a strictly weaker resource than classical communication. In this review, we provide a second look at Winter's measurement compression theorem, detailing the information processing task, giving examples for understanding it, reviewing Winter's achievability proof, and detailing a new approach to its single-letter converse theorem. We prove an extension of the theorem to the case in which the sender is not required to receive the outcomes of the simulated measurement. The total cost of common randomness and classical communication can be lower for such a 'non-feedback' simulation, and we prove a single-letter converse theorem demonstrating optimality. We then review the Devetak–Winter theorem on classical data compression with quantum side information, providing new proofs of its achievability and converse parts. From there, we outline a new protocol that we call 'measurement compression with quantum side information,' announced previously by two of us in our work on triple trade-offs in quantum Shannon theory. This protocol has several applications, including its part in the 'classically-assisted state redistribution' protocol, which is the most general protocol on the static side of the quantum information theory tree, and its role in reducing the classical communication cost in a task known as local purity distillation. We also outline a connection between measurement compression with quantum side information and recent work on entropic uncertainty relations in the presence of quantum memory. Finally, we prove a single-letter theorem characterizing measurement compression with quantum side information when the sender is not required to obtain the measurement outcome.
Winter's measurement compression theorem stands as one of the most penetrating insights of quantum information theory. In addition to making an original and profound statement about measurement in quantum theory, it also underlies several other general protocols used for entanglement distillation and local purity distillation. The theorem provides for an asymptotic decomposition of any quantum measurement into noise and information . This decomposition leads to an optimal protocol for having a sender simulate many independent instances of a quantum measurement and send the measurement outcomes to a receiver, using as little communication as possible. The protocol assumes that the parties have access to some amount of common randomness, which is a strictly weaker resource than classical communication. In this review, we provide a second look at Winter's measurement compression theorem, detailing the information processing task, giving examples for understanding it, reviewing Winter's achievability proof, and detailing a new approach to its single-letter converse theorem. We prove an extension of the theorem to the case in which the sender is not required to receive the outcomes of the simulated measurement. The total cost of common randomness and classical communication can be lower for such a 'non-feedback' simulation, and we prove a single-letter converse theorem demonstrating optimality. We then review the Devetak–Winter theorem on classical data compression with quantum side information, providing new proofs of its achievability and converse parts. From there, we outline a new protocol that we call 'measurement compression with quantum side information,' announced previously by two of us in our work on triple trade-offs in quantum Shannon theory. This protocol has several applications, including its part in the 'classically-assisted state redistribution' protocol, which is the most general protocol on the static side of the quantum information theory tree, and its role in reducing the classical communication cost in a task known as local purity distillation. We also outline a connection between measurement compression with quantum side information and recent work on entropic uncertainty relations in the presence of quantum memory. Finally, we prove a single-letter theorem characterizing measurement compression with quantum side information when the sender is not required to obtain the measurement outcome.
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 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 protocols 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.
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.
This paper concerns the problem of quantum measurement compression with side information in the one-shot setting with shared-randomness. In this problem, Alice shares a pure quantum state with Bob and the reference system. She performs a measurement on her registers and wishes to communicate the outcome to Bob using shared-randomness and classical communication. The outcome that Bob receives must be correctly correlated with the reference system and his own registers. Our goal is to concurrently minimize the classical communication and shared-randomness cost. The suggested protocol presented in this paper is based on convex-split and position based decoding. The communication is upper bounded in terms of smooth max and hypothesis testing relative entropies. A second protocol addresses the task of strong randomness extraction in the presence of quantum side information. The protocol provides an error guarantee in terms of relative entropy (as opposed to trace distance) and extracts close to the optimal number of uniform bits. As an application, we provide a new achievability result for the task of quantum measurement compression without feedback, in which Alice does not need to know the outcome of the measurement. The result achieves the optimal number of bits communicated and the required number of bits of shared-randomness, for the same task in the asymptotic and i.i.d. setting.
We use a Rényi entropy method to prove strong converse theorems for certain information-theoretic tasks which involve local operations and quantum (or classical) communication between two parties. These include state redistribution, coherent state merging, quantum state splitting, measurement compression with quantum side information, randomness extraction against quantum side information, and data compression with quantum side information. The method we employ in proving these results extends ideas developed by Sharma [preprint arXiv:1404.5940 [quant-ph] (2014)], which he used to give a new proof of the strong converse theorem for state merging. For state redistribution, we prove the strong converse property for the boundary of the entire achievable rate region in the (e, q)-plane, where e and q denote the entanglement cost and quantum communication cost, respectively. In the case of measurement compression with quantum side information, we prove a strong converse theorem for the classical communication cost, which is a new result extending the previously known weak converse. For the remaining tasks, we provide new proofs for strong converse theorems previously established using smooth entropies. For each task, we obtain the strong converse theorem from explicit bounds on the figure of merit of the task in terms of a Rényi generalization of the optimal rate. Hence, we identify candidates for the strong converse exponents for each task discussed in this paper. To prove our results, we establish various new entropic inequalities, which might be of independent interest. These involve conditional entropies and mutual information derived from the sandwiched Rényi divergence. In particular, we obtain novel bounds relating these quantities, as well as the Rényi conditional mutual information, to the fidelity of two quantum states.
We show that quantum-to-classical channels, i.e., quantum measurements, can be asymptotically simulated by an amount of classical communication equal to the quantum mutual information of the measurement, if sufficient shared randomness is available. This result generalizes Winter's measurement compression theorem for fixed independent and identically distributed inputs to arbitrary inputs, and more importantly, it identifies the quantum mutual information of a measurement as the information gained by performing it, independent of the input state on which it is performed. Our result is a generalization of the classical reverse Shannon theorem to quantum-to-classical channels. In this sense, it can be seen as a quantum reverse Shannon theorem for quantum-to-classical channels, but with the entanglement assistance and quantum communication replaced by shared randomness and classical communication, respectively. The proof is based on a novel one-shot state merging protocol for classically coherent states as well as the postselection technique for quantum channels, and it uses techniques developed for the quantum reverse Shannon theorem.