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

Abstract

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.

Optimality in quantum data compression using dynamical entropy

“…Our main results, Theorems 3.4 and 3.8, give quantum dynamical entropy interpretations for the average minimum codeword length per symbol as the length of strings of symbol states tend to infinity when the coding is assumed to be lossless. These results extend the result of Schumacher [17] and Bellomo et al [5] which state that for an i.i.d. prepared quantum ensemble the optimal codeword length per symbol is equal to the von Neumann entropy of the initial ensemble state for asymptotically lossless coding.…”

“…We may think of the codes introduced in the previous section as being varying-length codes; the term indeterminate-length is used to draw attention to the fact that a quantum code must allow for superpositions of codewords, including those superpositions containing codewords with different lengths. We will follow mainly the formalisms in [5] as opposed to the zero-extended forms of [18]. A description of the connection between these two formalisms can be found in [6].…”

“…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.…”

Generalized entropies in quantum and classical statistical theories

“…The quantum counterpart of Shannon entropic measure is the von Neumann entropy [2][3][4]. Also other measures have been adapted to the quantum realm in different contexts [5][6][7][8][9][10]. Entropic measures are important in several fields of research.…”

“…• measures of mutual information [25][26][27][28]32] • the theory of quantum coding and quantum information transmission [4,10,29].…”