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Compression of Pure and Mixed States in Quantum Detection

Gianfranco Cariolaro, Roberto Corvaja, Gianfranco L. Pierobon

Published Dec 1, 2011• 10 citations

Abstract

Quantum detection in an N-dimensional Hilbert space H involves quantum states and corresponding measure ment operators which span an r-dimensional subspace U of H, with r ≤ N. Quantum detection could be restricted to this subspace, but the operations in U are still redundant, since the kets have N components. By applying the singular-value decomposition to the state matrix, it is possible to perform a compression from the subspace U onto a "compressed" space U̅, where the redundancy is removed and kets are represented by r components. The detection can be perfectly reformulated in the "compressed" space, without loss of information, with a greatly reduced complexity. The compression is particularly attractive when r ≪ N, as shown with an example of application to quantum optical communications.

DOI: 10.1109/glocom.2011.6134027

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Quantum detection in an N-dimensional Hilbert space H involves quantum states and corresponding measure ment operators which span an r-dimensional subspace U of H, with r ≤ N. Quantum detection could be restricted to this subspace, but the operations in U are still redundant, since the kets have N components. By applying the singular-value decomposition to the state matrix, it is possible to perform a compression from the subspace U onto a "compressed" space U̅, where the redundancy is removed and kets are represented by r components. The detection can be perfectly reformulated in the "compressed" space, without loss of information, with a greatly reduced complexity. The compression is particularly attractive when r ≪ N, as shown with an example of application to quantum optical communications.

We study an exact local compression of a quantum bipartite state; that is, applying local quantum operations to the state to reduce the dimensions of Hilbert spaces while perfectly maintaining the correlation. We provide a closed formula for calculating the minimal achievable dimensions, provided as a minimization of the Schmidt rank of a particular pure state constructed from the given state. Numerically more tractable upper and lower bounds of the rank are also obtained. Subsequently, we consider the exact compression of quantum channels as an application. Using this method, a post-processing step that can reduce the output dimensions while retaining information on the output of the original channel can be analyzed.

We consider the most general (finite-dimensional) quantum mechanical information source, which is given by a quantum system A that is correlated with a reference system R. The task is to compress A in such a way as to reproduce the joint source state ρARat the decoder with asymptotically high fidelity. This includes Schumacher's original quantum source coding problem of a pure state ensemble and that of a single pure entangled state, as well as general mixed state ensembles. Here, we determine the optimal compression rate (in qubits per source system) in terms of the Koashi-Imoto decomposition of the source into a classical, a quantum, and a redundant part. The same decomposition yields the optimal rate in the presence of unlimited entanglement between compressor and decoder, and indeed the full region of feasible qubitebit rate pairs. Full version at arXiv:1912.08506 [1].

The pixel values of an image can be casted into a real ket of a Hilbert space using an appropriate block structured addressing. The resulting state can then be rewritten in terms of its matrix product state representation in such a way that quantum entanglement corresponds to classical correlations between different coarse-grained textures. A truncation of the MPS representation is tantamount to a compression of the original image. The resulting algorithm can be improved adding a discrete Fourier transform preprocessing and a further entropic lossless compression.

We consider the most general finite-dimensional quantum mechanical information source, which is given by a quantum system

A

that is correlated with a reference system

R

. The task is to compress

A

in such a way as to reproduce the joint source state

\rho ^{AR}

at the decoder with asymptotically high fidelity. This includes Schumacher’s original quantum source coding problem of a pure state ensemble and that of a single pure entangled state, as well as general mixed state ensembles. Here, we determine the optimal compression rate (in qubits per source system) in terms of the Koashi-Imoto decomposition of the source into a classical, a quantum, and a redundant part. The same decomposition yields the optimal rate in the presence of unlimited entanglement between compressor and decoder, and indeed the full region of feasible qubit-ebit rate pairs.