Measuring correlations in quantum and statistical systems

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Overview

Research area: Quantum statistical physics, quantum information, and correlation measures in composite quantum and classical systems.
Technical level: Advanced (graduate-level physics; the paper uses operator algebra, density matrices, Hilbert spaces, and many-body formalism).
Scope: A survey of the correlation-index method for quantifying quantum and classical correlations in static and dynamic, bipartite and multipartite systems, with analytical examples from entanglement, Hartree-Fock states, spin systems, and a driven Bose-Einstein condensate.

The paper is theoretical and analytical. It reports no experimental datasets, dataset sizes, or numerical benchmarks.

What This Paper Is About

Correlations are central to quantum and statistical physics, but the tools used to measure them are fragmented: the Pearson coefficient for classical statistical data, the quantum Pearson coefficient for operator averages, and entropy-based measures such as entanglement entropy and quantum mutual information for bipartite quantum systems. These approaches do not provide one quantitative measure that works for quantum and classical correlations, static and dynamic processes, and bipartite and multipartite systems.

The paper’s goal is to present the correlation-index method as a general solution to that problem. The authors define the index, show how it is constructed for correlation operators and reduced density matrices, and demonstrate through worked examples that it captures entanglement as well as classical correlations.

Key Contributions

1. A general definition of the correlation index. The index compares an operator containing correlations with a constructed factorized, uncorrelated counterpart, using the logarithm of the ratio of their Hermitian operator norms. It can be applied to correlation operators of any order, reduced density matrices, statistical operators, and other operators acting on a composite Hilbert space.

2. A quantitative alternative to correlation functions. The authors distinguish correlation functions, which are qualitative—showing whether correlations are long-range or short-range—from correlation indices, which assign a numerical strength to correlations in both quantum and statistical systems.

3. Demonstration that classical correlations are included. For a separable multimode statistical operator with (M) modes and (N) parts, the correlation index is ((1-N)\sup_\mu n_\mu), generally nonzero. This shows that the index is not restricted to quantum entanglement and can quantify classical correlations in mixed and multipartite states.

4. Applications to entanglement, spin systems, and nonequilibrium dynamics. The index is evaluated for Einstein-Podolsky-Rosen states, generalized Bell states, multimode states, Hartree-Fock and reduced Hartree-Fock states, spin correlations on a lattice, and a trapped Bose-Einstein condensate subject to an alternating field.

Main Findings

• General formula: For an operator (\hat{A}), the correlation index is (\varkappa(\hat{A})=\ln(|\hat{A}|/|\hat{A}^{\otimes}|)), where (\hat{A}^{\otimes}) is the factorized uncorrelated counterpart built from reduced operators and normalized to have the same trace as (\hat{A}). The norm is the Hermitian operator norm, equal to the largest eigenvalue.

• Classical correlations are quantified: For a separable statistical operator describing (M) modes composed of (N) parts, the index is (\varkappa(\hat{\rho}{sep})=(1-N)\sup\mu n_\mu), where (n_\mu) is a fractional mode population. It is generally nonzero even though the state has no entanglement.

• Entanglement benchmarks agree with entropy where entropy is defined: For a generalized Einstein-Podolsky-Rosen state, (\varkappa(\hat{\rho}_{EPR})=-\ln\sup{|c_1|^2,|c_2|^2}). Its maximum, at (|c_i|^2=1/2), is (\ln 2), equal to the maximum entanglement entropy. For an (N)-partite Bell state, the maximum is ((N-1)\ln 2); for (N=2) this coincides with the bipartite entanglement entropy (\ln 2). For an (N)-partite multimode state, the maximum reported is ((N-1)\ln M), although the paper notes that entanglement entropy is not defined for multipartite states.

• Hartree-Fock states show strong scaling: For the Hartree-Fock state, (\varkappa(\hat{\rho}_{HF})=\ln(N^N/N!)). For (N=2), this gives (\ln 2); for large (N), it behaves as approximately (N). For the reduced Hartree-Fock state, (\varkappa(\hat{\rho}_n)=\ln[((N-n)!/N!)N^n]), which for large (N) and finite (n) becomes approximately (n(n-1)/(2N)).

• Spin correlations reveal an order-disorder effect: For spin-(1/2) operators (S_i^z) on (N) lattice sites, with mean-field decoupling at different sites, the second-order spin correlation index is (\varkappa(\hat{C}_2)=\ln[(6+6Ns^2+N

Authors’ abstract

The survey discusses the problem of measuring correlations in composite quantum and statistical systems. After briefly recalling the standard methods of describing correlations, the emphasis is on the method of correlation indices. The latter method for quantifying the strength of correlations in composite physical systems is sufficiently general allowing for measuring in a unique way the correlation strength in both quantum and statistical systems, for static as well as for dynamic processes. The correlation indices, can be defined for correlation operators, reduced density matrices, and other operators containing information on correlations in the studied composite systems. The correlation indices measure all types of correlations, quantum correlations, such as entanglement, as well as classical correlations. They are applicable for characterizing static as well as dynamical processes. Examples are given of correlation indices for several quantum states and for spin and pseudospin systems. The quantification of correlations in a nonequilibrium system is exemplified by calculating the correlation index for a trapped Bose-Einstein condensate subject to the action of an alternating field.