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.
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.
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 [Winter, CMP 244 (157), 2004] 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. Our proof is based on quantum-proof randomness extractors and the post-selection technique for quantum channels [Christandl et al., PRL 102 (020504), 2009].
The Quantum Reverse Shannon Theorem states that any quantum channel can be simulated by an unlimited amount of shared entanglement and an amount of classical communication equal to the channel’s entanglement assisted classical capacity. In this paper, we provide a new proof of this theorem, which has previously been proved by Bennett, Devetak, Harrow, Shor, and Winter. Our proof has a clear structure being based on two recent information-theoretic results: one-shot Quantum State Merging and the Post-Selection Technique for quantum channels.
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.
Quantum theory is, in some sense, “non-classical.” For instance, the behavior of entangled systems (i.e., shared quantum information) under measurements cannot, in general, be explained by shared classical information. With classical communication, on the other hand, both the correlations entanglement leads to as well as quantum channels can be reproduced in principle. Here, crucial questions are whether the required communications is finite; if so, then its exact amount is related to the “degree of non-classicality” of the quantum primitive. We apply information-theoretic results such as the reverse Shannon theorem for determining the required communication in the asymptotic limit. The communication complexity of a quantum channel is the minimal amount of classical communication required for classically simulating the process of preparation, transmission through the channel, and subsequent measurement of a quantum state. At present, only little is known about this quantity. Our generic procedure allows for systematically evaluating the communication complexity of channels in any general probabilistic theory, in particular quantum theory. The procedure is constructive and provides the most efficient classical protocols. We illustrate it by evaluating the communication complexity of sending single qubits over a noiseless quantum channel with some finite sets of quantum states and measurements. As a second application, we determine the classical-communication rate required for the simulation of the behavior under measurements of entangled states. Here, the communication cost can be directly interpreted as the “non-classicality” of the correlation. A particular example is the simulation of non-maximally entangled pure qubit pairs, where we find the required communication rate to behave monotonically with the strength of the entanglement. For different measures of non-locality, such as the number of required non-local (PR) boxes, another behavior had been reported for the single-shot scenario.
We study the general framework of quantum channel simulation, that is, the ability of a quantum channel to simulate another one using different classes of codes. First, we show that the minimum error of simulation and the one-shot quantum simulation cost under no-signalling assisted codes are given by semidefinite programs. Second, we introduce the channel's smooth max-information, which can be seen as a one-shot generalization of the mutual information of a quantum channel. We provide an exact operational interpretation of the channel's smooth max-information as the one-shot quantum simulation cost under no-signalling assisted codes, which significantly simplifies the study of channel simulation and provides insights and bounds for the case under entanglement-assisted codes. Third, we derive the asymptotic equipartition property of the channel's smooth max-information; i.e., it converges to the quantum mutual information of the channel in the independent and identically distributed asymptotic limit. This implies the quantum reverse Shannon theorem in the presence of no-signalling correlations. Finally, we explore the simulation cost of various quantum channels.
We study and solve the problem of classical channel simulation with quantum side information at the receiver. This is a generalization of both the classical reverse Shannon theorem, and the classical-quantum Slepian-Wolf problem. The optimal noiseless communication rate is found to be reduced from the mutual information between the channel input and output by the Holevo information between the channel output and the quantum side information. Our main theorem has two important corollaries. The first is a quantum generalization of the Wyner-Ziv problem: rate-distortion theory with quantum side information. The second is an alternative proof of the tradeoff between classical communication and common randomness distilled from a quantum state. The fully quantum generalization of the problem considered isquantumstateredistribution. Here the sender and receiver share a mixed quantum state and the sender wants to transfer part of her state to the receiver using entanglement and quantum communication. We present outer and inner bounds on the achievable rate pairs.