Fully quantum source compression with a quantum helper

© 2015 IEEE. 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

Source compression with a quantum helper

© 2015 IEEE. 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

One-shot lossy quantum data compression

We provide a framework for one-shot quantum rate distortion coding, in which the goal is to determine the minimum number of qubits required to compress quantum information as a function of the probability that the distortion incurred upon decompression exceeds some specified level. We obtain a one-shot characterization of the minimum qubit compression size for an entanglement-assisted quantum rate-distortion code in terms of the smooth max-information, a quantity previously employed in the one-shot quantum reverse Shannon theorem. Next, we show how this characterization converges to the known expression for the entanglement-assisted quantum rate distortion function for asymptotically many copies of a memoryless quantum information source. Finally, we give a tight, finite blocklength characterization for the entanglement-assisted minimum qubit compression size of a memoryless isotropic qubit source subject to an average symbol-wise distortion constraint.Comment: 36 page

Quantum optimal transport with quantum channels

We propose a new generalization to quantum states of the Wasserstein distance, which is a fundamental distance between probability distributions given by the minimization of a transport cost. Our proposal is the first where the transport plans between quantum states are in natural correspondence with quantum channels, such that the transport can be interpreted as a physical operation on the system. Our main result is the proof of a modified triangle inequality for our transport distance. We also prove that the distance between a quantum state and itself is intimately connected with the Wigner-Yanase metric on the manifold of quantum states. We then specialize to quantum Gaussian systems, which provide the mathematical model for the electromagnetic radiation in the quantum regime. We prove that the noiseless quantum Gaussian attenuators and amplifiers are the optimal transport plans between thermal quantum Gaussian states, and that our distance recovers the classical Wasserstein distance in the semiclassical limit

Rate Reduction of Blind Quantum Data Compression with Local Approximations Based on Unstable Structure of Quantum States

In this paper, we propose a new protocol for a data compression task, blind quantum data compression, with finite local approximations. The rate of blind data compression is susceptible to approximations even when the approximations are diminutive. This instability originates from the sensitivity of a structure of quantum states against approximations, which makes the analysis of blind compression in the presence of approximations intractable. In this paper, we constructed a protocol that takes advantage of the instability to reduce the compression rate substantially. Our protocol shows a significant reduction in rate for specific examples we examined. Moreover, we apply our methods to diagonal states, and propose two types of approximation methods in this special case. We perform numerical experiments and observe that one of these two approximation methods performs significantly better than the other. Thus, our analysis makes a first step toward general investigation of blind quantum data compression with the allowance of approximations towards further investigation of approximation-rate trade-off of blind quantum data compression.Comment: Added Remark1 and updated numerical experiments; 17 pages, 4 figures; comments welcom

The Quantum Wasserstein Distance of Order 1

We propose a generalization of the Wasserstein distance of order 1 to the quantum states of $n$ qudits. The proposal recovers the Hamming distance for the vectors of the canonical basis, and more generally the classical Wasserstein distance for quantum states diagonal in the canonical basis. The proposed distance is invariant with respect to permutations of the qudits and unitary operations acting on one qudit and is additive with respect to the tensor product. Our main result is a continuity bound for the von Neumann entropy with respect to the proposed distance, which significantly strengthens the best continuity bound with respect to the trace distance. We also propose a generalization of the Lipschitz constant to quantum observables. The notion of quantum Lipschitz constant allows us to compute the proposed distance with a semidefinite program. We prove a quantum version of Marton's transportation inequality and a quantum Gaussian concentration inequality for the spectrum of quantum Lipschitz Moreover, we derive bounds on the contraction coefficients of shallow quantum circuits and of the tensor product of one-qudit quantum channels with respect to the proposed distance. We discuss other possible applications in quantum machine learning, quantum Shannon theory, and quantum many-body systems