Lossless quantum prefix compression for communication channels that are always open

Müller, Markus P.; Rogers, Caroline; Nagarajan, Rajagopal

doi:10.1103/physreva.79.012302

Cited by 5 publications

(11 citation statements)

References 16 publications

(48 reference statements)

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“…It is worth noticing that our approach provides an alternative to that of Müeller et al [18], where they have studied an analogous problem, but minimizing the average of the individual base lengths of the source. Our results are complementary to theirs.…”

Section: Discussionmentioning

confidence: 99%

“…Consequently, this allows us to provide a natural operational interpretation for the quantum Rényi entropy in relation with the problem of lossless quantum data compression. Finally, notice that this is an alternative approach to that of Müeller et al [18], where they have studied an analogous problem, but minimizing the average of the individual base lengths of the source instead of considering a penalization over large codewords.…”

Section: Source Coding and Quantum Rényi Entropy Boundsmentioning

confidence: 99%

“…Furthermore, they have provided the first quantum version of the Kraft-McMillan inequality and have found that the von Neumann entropy of the source ρ, S(ρ) = − Tr (ρ log 2 ρ) (binary logarithm for coding in qubits), plays an analogous role to that of the Shannon entropy in the classical source coding theorem. Several other authors have contributed to this subject proposing alternative or extended schemes [12][13][14][15][16][17][18][19][20]. In general, these approaches face the same disadvantage as in the classical case: namely they do not consider the fact that large codewords, even appearing with low probabilities, may have large impact in terms of resources needed for the encoding.…”

Section: Introductionmentioning

confidence: 99%

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Lossless quantum data compression with exponential penalization: an operational interpretation of the quantum Rényi entropy

Bellomo

Bosyk

Holik

et al. 2017

Sci Rep

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Based on the problem of quantum data compression in a lossless way, we present here an operational interpretation for the family of quantum Rényi entropies. In order to do this, we appeal to a very general quantum encoding scheme that satisfies a quantum version of the Kraft-McMillan inequality. Then, in the standard situation, where one is intended to minimize the usual average length of the quantum codewords, we recover the known results, namely that the von Neumann entropy of the source bounds the average length of the optimal codes. Otherwise, we show that by invoking an exponential average length, related to an exponential penalization over large codewords, the quantum Rényi entropies arise as the natural quantities relating the optimal encoding schemes with the source description, playing an analogous role to that of von Neumann entropy.

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Section: Discussionmentioning

confidence: 99%

Section: Source Coding and Quantum Rényi Entropy Boundsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Lossless quantum data compression with exponential penalization: an operational interpretation of the quantum Rényi entropy

Bellomo

Bosyk

Holik

et al. 2017

Sci Rep

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“…We also prove a related set of source-channel separation theorems that allow for some distortion in the reconstruction of the output of the information source. From these theorems we infer that it is best to search for the best quantum data compression protocols [16], [13], [9], [3], [42], [43], the best quantum error-correcting codes [51], [19], [18], [41], [44], [37], and the best entanglementassisted quantum error-correcting codes [17], [33], [36], [58] independently of each other whenever the source and channel are memoryless. The theorems then guarantee that combining these protocols in a two-stage encoding and decoding is optimal.…”

mentioning

confidence: 99%

Quantum Rate Distortion, Reverse Shannon Theorems, and Source-Channel Separation

Datta

Wilde

2013

IEEE Trans. Inform. Theory

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We derive quantum counterparts of two key theorems of classical information theory, namely, the rate distortion theorem and the source-channel separation theorem. The rate-distortion theorem gives the ultimate limits on lossy data compression, and the source-channel separation theorem implies that a two-stage protocol consisting of compression and channel coding is optimal for transmitting a memoryless source over a memoryless channel. In spite of their importance in the classical domain, there has been surprisingly little work in these areas for quantum information theory. In the present paper, we prove that the quantum rate distortion function is given in terms of the regularized entanglement of purification. We also determine a single-letter expression for the entanglement-assisted quantum rate distortion function, and we prove that it serves as a lower bound on the unassisted quantum rate distortion function. This implies that the unassisted quantum rate distortion function is non-negative and generally not equal to the coherent information between the source and distorted output (in spite of Barnum's conjecture that the coherent information would be relevant here). Moreover, we prove several quantum sourcechannel separation theorems. The strongest of these are in the entanglement-assisted setting, in which we establish a necessary and sufficient codition for transmitting a memoryless source over a memoryless quantum channel up to a given distortion.Index Terms-quantum rate distortion, reverse Shannon theorem, quantum Shannon theory, quantum data compression, source-channel separation

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“…Indeterminate-length quantum codes were considered by Schumacher and Westmoreland in [18], and later by Müller, Rogers and Nagarajan in [13,14]; and Bellomo, Bosyk, Holik and Zozor in [5]. In all three of these papers, the authors prove a version of the quantum Kraft-McMillan Theorem which states that every uniquely decodable quantum code must satisfy an inequality in terms of the lengths of its eigenstates.…”

Section: Introductionmentioning

confidence: 99%

Optimality in quantum data compression using dynamical entropy

Androulakis

Wright

2019

Phys. Rev. A

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In this article we study lossless compression of strings of pure quantum states of indeterminate-length quantum codes which were introduced by Schumacher and Westmoreland. Past work has assumed that the strings of quantum data are prepared to be encoded in an independent and identically distributed way. We introduce the notion of quantum stochastic ensembles, allowing us to consider strings of quantum states prepared in a general way. For any quantum stochastic ensemble we define an associated quantum dynamical system and prove that the optimal average codeword length via lossless coding is equal to the quantum dynamical entropy of the associated quantum dynamical system. Date: August 6, 2019. 2010 Mathematics Subject Classification. Primary: 81P45. Secondary: 81P70, 94A15, 37A35 . Key words and phrases. Kraft-McMillan inequality, quantum dynamical system, quantum Markov chain, quantum dynamical entropy.

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Lossless quantum prefix compression for communication channels that are always open

Cited by 5 publications

References 16 publications

Lossless quantum data compression with exponential penalization: an operational interpretation of the quantum Rényi entropy

Lossless quantum data compression with exponential penalization: an operational interpretation of the quantum Rényi entropy

Quantum Rate Distortion, Reverse Shannon Theorems, and Source-Channel Separation

Optimality in quantum data compression using dynamical entropy

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