We describe a method for lossless quantum compression if the output of the information source is not known. We compute the best possible compression rate, minimizing the expected base length of the output quantum bit string (the base length of a quantum string is the maximal length in the superposition). This complements work by Schumacher and Westmoreland who calculated the corresponding rate for minimizing the output's average length. Our compressed code words are prefix-free indeterminate-length quantum bit strings which can be concatenated in the case of multiple sources. Therefore, we generalize the known theory of prefix-free quantum codes to the case where strings have indeterminate length. Moreover, we describe a communication model which allows the lossless transmission of the compressed code words. The benefit of compression is then the reduction of transmission errors in the presence of noise.
Consider a general quantum stochastic source that emits at discrete time steps quantum pure states which are chosen from a finite alphabet according to some probability distribution which may depend on the whole history. Also, fix two positive integers
m and
l. We encode any tensor product of
m many states emitted by the quantum stochastic source by breaking the tensor product into
m many blocks where each block has length
l, and considering sequences of
m many isometries so that each isometry encodes one of these blocks into the Fock space, followed by the concatenation of their images. We only consider certain sequences of such isometries that we call \lq\lq special block codes\rq\rq\ in order to ensure that the string of encoded states is uniquely decodable. We compute the minimum average codeword length of these encodings which depends on the quantum source and the integers
m,
l, among all possible special block codes. Our result extends the result of [Bellomo, Bosyk, Holik and Zozor, Scientific Reports 7.1 (2017): 14765] where the minimum was computed for one block, i.e.\ for
m=1. Lastly, we give a simplified non-adaptive compression technique based on constrained special block codes for general quantum stochastic sources. For quantum stationary sources in particular, we show that the minimum average codeword length per symbol computed over all constrained special block codes is equal to the von-Neumann entropy rate of the source for an asymptotically long block size.
After K. Bostro/spl uml/m and T. Felbinger observed that lossless quantum data compression does not exist unless decoders know the lengths of codewords, they introduced a classical noiseless channel to inform the decoder of a quantum source about the lengths of codewords. In this paper we analyze their codes and present: 1) a sufficient and necessary condition for the existence of such codes for given lists of lengths of codes; 2) a characterization of the optimal compression rate for their codes. However our main contribution is a more efficient way to use the classical channel. We propose a more general coding scheme. It turned out that the optimal compression can always be achieved by a code obtained by this scheme. A von Neumann entropy lower bound to rates of our codes and a necessary and sufficient condition to achieve the bound are obtained. The gap between this lower bound and the compression rates is also well analyzed. For a special family of quantum sources we provide a sharper lower bound in terms of Shannon entropy. Finally, we propose some problems for further research.
In order to compress quantum messages without loss of information it is necessary to allow the length of the encoded messages to vary. We develop a general framework for variable-length quantum messages in close analogy to the classical case and show that lossless compression is only possible if the message to be compressed is known to the sender. The lossless compression of an ensemble of messages is bounded from below by its von-Neumann entropy. We show that it is possible to reduce the number of qbits passing through a quantum channel even below the von Neumann entropy by adding a classical side channel. We give an explicit communication protocol that realizes lossless and instantaneous quantum data compression and apply it to a simple example. This protocol can be used for both online quantum communication and storage of quantum data.
Based on the concept of many-letter theory a general characterization of quantum codes using the Kraus representation is given. Compression codes are defined by their property of decreasing the average information content of a given a priori message ensemble. Lossless quantum codes, in contrast to lossy codes, provide the retrieval of the original input states with perfect fidelity. A general lossless coding scheme is given that translates between two quantum alphabets. It is shown that this scheme is never compressive. Furthermore, a lossless quantum coding scheme, analog to the classical Huffman scheme but different from the Braunstein scheme, is implemented, which provides optimal compression. Motivated by the concept of lossless quantum compression, an observable is defined that measures the amount of compressible quantum information contained in a particular message with respect to a given \\emph{a priori} message ensemble. The average of this observable yields the von Neumann entropy, which is finally established as the amount of compressible quantum information of any message ensemble, confirming its commonly assigned meaning.
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.