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Physical Review A

Information theory, squeezing, and quantum correlations

Stephen Mark Barnett, Simon J. D. Phoenix

Published Jul 1, 1991• 148 citations
Publisher

Abstract

Information theory allows us to make quantitative statements about the strength and nature of the correlations between systems. Application of this theory to the quantized electromagnetic field reveals a special role for the two-mode squeezed states. The nonclassical properties of these states arise from the intermode correlations, and we apply information-theoretic methods to determine the strength of the correlation between specific pairs of observables. This analysis leads to the important general result that for any correlated pure state a given pair of single-system observables contains at most only half the information about the correlations. We discuss the implications of this result for the distinction between classical and quantum systems.

DOI
10.1103/physreva.44.535

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•Nov 16, 2020•1 citation
Information theory allows us to make quantitative statements about the strength and nature of the correlations between systems. Application of this theory to the quantized electromagnetic field reveals a special role for the two-mode squeezed states. The nonclassical properties of these states arise from the intermode correlations, and we apply information-theoretic methods to determine the strength of the correlation between specific pairs of observables. This analysis leads to the important general result that for any correlated pure state a given pair of single-system observables contains at most only half the information about the correlations. We discuss the implications of this result for the distinction between classical and quantum systems.
We discuss the connection between quantum correlations and squeezing in simple quantum optical systems. We illustrate this connection by a study of two-mode states of light produced by parametric down-conversion and similar two-photon processes. The intermode correlations in these systems are shown to be responsible for modifications in photon-number sum and difference operators, and for squeezing in the superpositions of the two modes. The disappearance of the diagonal coherent-state quasiprobability function P(α) when non-classical light properties are important is noted, and alternative and better-behaved Wigner functions and coherent-state expectation Q-functions for the two-mode system are developed.
The concept of photon antibunching as a manifestation of nonclassical character in single-mode radiation is extended to two-mode radiation. A simple criterion for the existence of nonclassical effects in two-mode radiation is established. It implies that the intermode photon bunching, rather than antibunching, can play a key role in rendering two-mode radiation nonclassical. The two-mode squeezed vacuum states are used as simple examples to illustrate the essential point of the criterion. The criterion is then used to study the nonclassical properties of two types of two-mode squeezed states. A state generated by applying the squeeze operator to the two-mode vacuum first, followed by the displacement operator, is called a two-mode coherent squeezed state. If the order of the two operators is reversed, a two-mode squeezed coherent state is obtained. It is found that the latter has a much stronger tendency to become a nonclassical state than the former, assuming that the parameters involved are all the same. A measure of nonclassical depth of radiation is adopted. It is found that, according to this measure, the effect of squeezing in making two-mode coherent squeezed states nonclassical is not monotonic. Such phenomena do not occur in squeezed coherent states.
The properties of the entangled pure states in phase space are analysed using the classical entropy introduced by Wehrl. The general entropy inequalities for non-entangled pure states are derived, which are violated by any entangled state . To measure the strength of intermode correlations in phase space the parameters related to these inequalities are proposed. As an example we study the correlations between amplitudes and phases for two-mode Fock states. We find that the amplitudes as well as phases of different modes are correlated. It is also shown that the degree of the intermode correlation strongly depends on the photon number difference in two-mode Fock states.
We examine the non-classical properties of two-mode coherent states based on different unitary irreducible representations of SU(1, 1). Such states are generated by the action of the two-mode squeezing operator on initial states of the field containing arbitrary numbers of photons in each of the two modes. If the initial state of the field is a two-mode vacuum state, the final state is of course the two-mode squeezed vacuum. An initial occupation generalizes the idea of a squeezed vacuum to the SU(1, 1) coherent states. We show that fields in such states have remarkable quantum properties such as sub-Poissonian statistics, violations of the Cauchy-Schwarz inequality, strong correlations in the photon number fluctuations and squeezing. Using information theory formalism, we show that these coherent states are less correlated than the usual two-mode squeezed vacuum. Moreover, we show that an initial coherent amplitude contribution, in a large amplitude limit, can result in the reduction of correlations between modes.
Quantum squeezing, a major resource for quantum information processing and quantum metrology, is best analyzed in terms of the field quadratures - the quantum optical analogues of position and momentum, which form the continuous-variable formalism of quantum light. Degenerate squeezing admits a very helpful and simple description in terms of the single-mode quadrature operators, but the non-degenerate case, i.e. when the squeezing involves pairs of modes, requires a more complicated treatment involving correlations between the quadratures of the different modes. We introduce a generalized set of complex quadrature operators that treats degenerate and non-degenerate squeezing on equal footing. We describe the mode-pairs (and photon-pairs) as a single entity, generalizing the concept of single-mode quadrature operators to two-mode fields of any bandwidth. These complex operators completely describe the SU(1,1) algebra of two-photon devices and directly relate to observable physical quantities, like power and visibility. Based on this formalism, we discuss the measurement of optically-broad squeezed signals with direct detection, and present a compact set of phase-dependent observables that completely and intuitively determine the two-mode squeezed state, and quantify the degree of inseparability and entanglement between the modes.