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arXiv:0905.0510 · Journal of Modern Optics

How well can you know the edge of a quantum pyramid?

Berthold‐Georg Englert, Jaroslav Řeháček

Published Jul 31, 2009• 17 citations
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Abstract

We consider a symmetric quantum communication scenario in which the signal states are edges of a quantum pyramid of arbitrary dimension and arbitrary shape, and all edge states are transmitted with the same probability. The receiver could employ different decoding strategies: he could minimize the error probability, or discriminate without ambiguity, or extract the accessible information. We state the optimal measurement scheme for each strategy. For large parameter ranges, the standard square-root measurement does not extract the information optimally.

DOI
10.1080/09500340903094601

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We consider a symmetric quantum communication scenario in which the signal states are edges of a quantum pyramid of arbitrary dimension and arbitrary shape, and all edge states are transmitted with the same probability. The receiver could employ different decoding strategies: he could minimize the error probability, or discriminate without ambiguity, or extract the accessible information. We state the optimal measurement scheme for each strategy. For large parameter ranges, the standard square-root measurement does not extract the information optimally.
We introduce and analyze an information theoretical task that we call the quantum multiple-access one-time pad. Here, a number of senders initially share a correlated quantum state with a receiver and an eavesdropper. Each sender performs a local operation to encode a classical message and sends their system to the receiver, who subsequently performs a measurement to decode the messages. The receiver will be able to decode the messages almost perfectly, while the eavesdropper must not be able to extract information about the messages even if they have access to the quantum systems transmitted. We consider a “conditional” scenario in which a portion of the receiver’s side information is also accessible to the eavesdropper. We investigate the maximum amount of classical information that can be encoded by each of the senders. We derive a single-letter characterization for the achievable rate region in an asymptotic limit of infinitely many copies and vanishingly small error.
Distinguishing different quantum states is a fundamental task having practical applications in information processing. Despite the effort devoted so far, however, strategies for optimal discrimination are known only for specific examples. In this paper we consider the problem of minimum-error quantum state discrimination where one attempts to minimize the average error. We show the general structure of minimum-error state discrimination as well as useful properties to derive analytic solutions. Based on the general structure, we present a geometric formulation of the problem, which can be applied to cases where quantum state geometry is clear. We also introduce equivalent classes of sets of quantum states in terms of minimum-error discrimination: sets of quantum states in an equivalent class that share the same guessing probability. In particular, for qubit states where the state geometry is found with the Bloch sphere, we illustrate that for an arbitrary set of qubit states, the minimum-error state discrimination with equal prior probabilities can be analytically solved, that is, optimal measurement and the guessing probability are explicitly obtained.
Minimization of error probability is considered in entanglement-assisted quantum communication systems. It is shown that although quantum state signals being sent are not symmetric at a sender side, the square root measurement becomes optimum when they are made symmetric at the receiver side. For communication systems of coherent signals, where a two-mode squeezed-vacuum state is used as an entanglement resource, the quantum entanglement greatly reduces the average probability of error. The relation to the quantum dense coding of continuous variables is also discussed.
Standard protocols in optical communication systems employ coherent states of light as the information carriers. The quantum nature inherent to these states introduces a fundamental complexity into the decoding process. In particular, when dealing with attenuated signals, the overlap between different states precludes the possibility of perfectly discriminate them. In this study, we investigate the performance of conventional and nonconventional quantum discrimination strategies under different informational metrics. We focus our analysis on binary coherent states within the ideal case of a lossless channel. The usual homodyne receiver was studied in comparison with the Kennedy and the optimized displacement receiver. We find that nonconventional strategies employing non-Gaussian measurements surpass the conventional homodyne discrimination scheme according to measurement error probability and mutual information.