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arXiv:quant-ph/0608171 · International Journal of Quantum Information

PROPERTIES OF NONFREENESS: AN ENTROPY MEASURE OF ELECTRON CORRELATION

Alex D. Gottlieb, Norbert J. Mauser

Published Dec 1, 2007• 27 citations
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Abstract

The "nonfreeness" of a many-fermion state ω is the entropy of ω relative to the free state that has the same 1-particle statistics as ω. The nonfreeness of a pure state is the same as its "particle-hole symmetric correlation entropy", a variant of an established measure of electron correlation. However, nonfreeness is also defined for mixed states, and this allows one to compare the nonfreeness of subsystems to the nonfreeness of the whole. Nonfreeness of a part does not exceed that in the whole; nonfreeness is additive over independent subsystems; and nonfreeness is superadditive over subsystems that are independent on the 1-particle level.

DOI
10.1142/s0219749907003201

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The "nonfreeness" of a many-fermion state ω is the entropy of ω relative to the free state that has the same 1-particle statistics as ω. The nonfreeness of a pure state is the same as its "particle-hole symmetric correlation entropy", a variant of an established measure of electron correlation. However, nonfreeness is also defined for mixed states, and this allows one to compare the nonfreeness of subsystems to the nonfreeness of the whole. Nonfreeness of a part does not exceed that in the whole; nonfreeness is additive over independent subsystems; and nonfreeness is superadditive over subsystems that are independent on the 1-particle level.
In the context of many-fermion systems, "correlation" refers to the inadequacy of an independent-particle model. Using "free" states as archetypes of our independent-particle model, we have proposed a measure of correlation that we called "nonfreeness" [Int. J. Quant. Inf. 5, 815 (2007)]. The nonfreeness of a many-fermion state was defined to be its entropy relative to the unique free state with the same 1-matrix. In this article, we prove that the nonfreeness of a state is the minimum of its entropy relative to all free states. We also extend the definition of nonfreeness to many-boson states and discuss a couple of examples.
This article is a brief review of "nonfreeness" and related measures of "correlation" for many-fermion systems. The many-fermion states we deem "uncorrelated" are the gauge-invariant quasi-free states. Uncorrelated states of systems of finitely many fermions we call simply "free" states. Slater determinant states are free; all other free states are "substates" of Slater determinant states or limits of such. The nonfreeness of a many-fermion state equals the minimum of its entropy relative to all free states. Correlation functionals closely related to nonfreeness can be defined in terms of Rényi entropies; nonfreeness is the one that uses Shannon entropy. These correlation functionals all share desirable additivity and monotonicity properties, but nonfreeness has some additional attractive properties.
We analyze the problem of quantifying entanglement in pure and mixed states of fermionic systems with a fixed number parity yet not necessarily a fixed particle number. The mode entanglement between one single-particle level and its orthogonal complement is first considered, and an entanglement entropy for such a partition of a particular basis of the single-particle Hilbert space H\mathcal{H}H is defined. The sum over all single-particle modes of this entropy is introduced as a measure of the total entanglement of the system with respect to the chosen basis and it is shown that its minimum over all bases of H\mathcal{H}H is a function of the one-body density matrix. Furthermore, we show that if minimization is extended to all bases related through a Bogoliubov transformation, then the entanglement entropy is a function of the generalized one-body density matrix. These results are then used to quantify entanglement in fermion systems with four single-particle levels. For general pure states of such a system a closed expression for the fermionic concurrence is derived, which generalizes the Slater correlation measure defined by J. Schliemann et al. [Phys. Rev. A 64, 022303 (2001)], implying that particle entanglement may be seen as minimum mode entanglement. It is also shown that the entanglement entropy defined before is related to this concurrence by an expression analogous to that in the two-qubit case. For mixed states of this system the convex roof extension of the previous concurrence and entanglement entropy is evaluated analytically, extending the results in previous reference to general states.
Does the sum of correlations in subsystems constitute the correlation in the total system? Such a concept can be expressed by an additivity relationship of correlations. From a strong subadditivity condition of von Neumann entropy, four different additivity relations in total correlation are derived and quantified. Based upon the classification of the additivity in total correlation, we identify the corresponding additive relationships in entanglement. It is also discussed that similar relationships are satisfied for quantum discord of pure states, but it is not always true for mixed states.
We propose an entropic measure of nonclassical correlations in general mixed states of fermion systems, based on the loss of information due to the unread measurement of the occupancy of single-particle states of a given basis. When minimized over all possible single-particle bases, the measure reduces to an entanglement entropy for pure states and vanishes only for states which are diagonal in a Slater determinant basis. The approach is also suitable for states having definite number parity yet not necessarily a fixed particle number, in which case the minimization can be extended to all bases related through a Bogoliubov transformation if quasiparticle mode measurements are also considered. General stationary conditions for determining the optimizing basis are derived. For a mixture of a general pure state with the maximally mixed state, a general analytic evaluation of the present measure and optimizing basis is provided, which shows that nonentangled mixed states may nonetheless exhibit a nonzero information loss.
We study quantum coarse-grained entropy and demonstrate that the gap in entropy between local and global coarse-grainings is a natural generalization of entanglement entropy to mixed states and multipartite systems. This ``quantum correlation entropy'' Sqc{S}^{q}cSqc is additive over independent systems, is invariant under local unitary operations, measures total nonclassical correlations (vanishing on states with strictly classical correlation), and reduces to the entanglement entropy for bipartite pure states. It quantifies how well a quantum system can be understood via local measurements and ties directly to nonequilibrium thermodynamics, including representing a lower bound on the quantum part of thermodynamic entropy production. We discuss two other measures of nonclassical correlation to which this entropy is equivalent and argue that together they provide a unique thermodynamically distinguished measure.