Reverse Shannon theorems concern the use of noiseless channels to simulate noisy ones. This is dual to the usual noisy channel coding problem, where a noisy (classical or quantum) channel is used to simulate a noiseless one. The Quantum Reverse Shannon Theorem is extensively studied by Bennett and co-authors in [IEEE Trans. Inf. Theory, 2014]. They present two distinct theorems, each tailored to classical and quantum channel simulations respectively, explaining the fact that these theorems remain incomparable due to the fundamentally different nature of correlations they address. The authors leave as an open question the challenge of formulating a unified theorem that could encompass the principles of both and unify them. We unify these two theorems into a single, comprehensive theorem, extending it to the most general case by considering correlations with a general mixed-state reference system. Furthermore, we unify feedback and non-feedback theorems by simulating a general side information system at the encoder side.
Reverse Shannon theorems concern the use of noiseless channels to simulate noisy ones. This is dual to the usual noisy channel coding problem, where a noisy (classical or quantum) channel is used to simulate a noiseless one. The Quantum Reverse Shannon Theorem is extensively studied by Bennett and co-authors in [IEEE Trans. Inf. Theory, 2014]. They present two distinct theorems, each tailored to classical and quantum channel simulations respectively, explaining the fact that these theorems remain incomparable due to the fundamentally different nature of correlations they address. The authors leave as an open question the challenge of formulating a unified theorem that could encompass the principles of both and unify them. We unify these two theorems into a single, comprehensive theorem, extending it to the most general case by considering correlations with a general mixed-state reference system. Furthermore, we unify feedback and non-feedback theorems by simulating a general side information system at the encoder side.
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 IID 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 non-tensor-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.
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.
In the first part of this thesis, we discuss the algebraic approach to classical and quantum physics and develop information theoretic concepts within this setup. In the second part, we discuss the uncertainty principle in quantum mechanics. The principle states that even if we have full classical information about the state of a quantum system, it is impossible to deterministically predict the outcomes of all possible measurements. In comparison, the perspective of a quantum observer allows to have quantum information about the state of a quantum system. This then leads to an interplay between uncertainty and quantum correlations. We provide an information theoretic analysis by discussing entropic uncertainty relations with quantum side information. In the third part, we discuss the concept of randomness extractors. Classical and quantum randomness are an essential resource in information theory, cryptography, and computation. However, most sources of randomness exhibit only weak forms of unpredictability, and the goal of randomness extraction is to convert such weak randomness into (almost) perfect randomness. We discuss various constructions for classical and quantum randomness extractors, and we examine especially the performance of these constructions relative to an observer with quantum side information. In the fourth part, we discuss channel simulations. Shannon's noisy channel theorem can be understood as the use of a noisy channel to simulate a noiseless one. Channel simulations as we want to consider them here are about the reverse problem: simulating noisy channels from noiseless ones. Starting from the purely classical case (the classical reverse Shannon theorem), we develop various kinds of quantum channel simulation results. We achieve this by using classical and quantum randomness extractors that also work with respect to quantum side information.
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.
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.
The theory of zero-error communication is re-examined in the broader setting of using one classical channel to simulate another exactly in the presence of various classes of nonsignalling correlations between sender and receiver i.e., shared randomness, shared entanglement and arbitrary nonsignalling correlations. When the channel being simulated is noiseless, this is zero-error coding assisted by correlations. When the resource channel is noiseless, it is the reverse problem of simulating a noisy channel exactly by a noiseless one, assisted by correlations. In both cases, separations between the power of the different classes of assisting correlations are exhibited for finite block lengths. The most striking result here is that entanglement can assist in zero-error communication. In the large block length limit, shared randomness is shown to be just as powerful as arbitrary nonsignalling correlations for exact simulation, but not for asymptotic zero-error coding. For assistance by arbitrary nonsignalling correlations, linear programming formulas for the asymptotic capacity and simulation rates are derived, the former being equal (for channels with nonzero unassisted capacity) to the feedback-assisted zero-error capacity derived by Shannon. Finally, a kind of reversibility between nonsignalling-assisted zero-error capacity and exact simulation is observed, mirroring the usual reverse Shannon theorem.