Maximal correlation is a measure of correlation for bipartite distributions. This measure has two intriguing features: (1) it is monotone under local stochastic maps; (2) it gives the same number when computed on i.i.d. copies of a pair of random variables. This measure of correlation has recently been generalized for bipartite quantum states, for which the same properties have been proved. In this paper, based on maximal correlation, we define a new measure of entanglement which we call maximal entanglement. We show that this measure of entanglement is faithful (is zero on separable states and positive on entangled states), is monotone under local quantum operations, and gives the same number when computed on tensor powers of a bipartite state.
In this paper, a computable multipartite multimode Gaussian quantum correlation measure
M(k) is proposed for any
k-partite continuous-variable (CV) systems with
k≥2.
M(k) depends only on the covariance matrix of CV states, is invariant under any permutation of subsystems, is a quantification without ancilla problem, nonincreasing under
k-partite local Gaussian channels (particularly, invariant under
k-partite local Gaussian unitary operations), vanishes on
k-partite product states. For a
k-partite Gaussian state
ρ,
M(k)(ρ)=0 if and only if
ρ is a
k-partite product state. Thus, for the bipartite case,
M=M(2) is an accessible replacement of the Gaussian quantum discord and Gaussian geometric discord. Moreover,
M(k) satisfies the unification condition, hierarchy condition that a multipartite quantum correlation measure should obey.
M(k) is not bipartite like monogamous, but,
M(k) is complete monogamous and tight complete monogamous.
In this paper, definitions of the unification condition, the hierarchy condition, and three kinds of monogamy relations for multipartite quantum correlation measures are given and discussed. A computable multipartite multimode Gaussian quantum correlation measure
M(k) is proposed for any
k-partite multimode continuous-variable systems with $k\ensuremath{\ge}2$. The value of
M(k) only depends on the covariance matrices of continuous-variable states, is invariant under any permutation of subsystems, has no ancilla problem, is nonincreasing under
k-partite local Gaussian channels (particularly, invariant under
k-partite locally Gaussian unitary operations), and vanishes on
k-partite product states. For a
k-partite Gaussian state $\ensuremath{\rho}, {\mathcal{M}}^{(k)}(\ensuremath{\rho})=0$ if and only if $\ensuremath{\rho}$ is a
k-partite product state. Moreover,
M(k) satisfies the unification condition and the hierarchy condition that a multipartite quantum correlation measure should obey. We also show that
M(k) is not strongly monogamous but completely monogamous and tightly monogamous.
Quantum data processing inequality bounds the set of bipartite states that can be generated by two far apart parties under local operations; having access to a bipartite state as a resource, two parties cannot locally transform it to another bipartite state with a mutual information greater than that of the resource state. But due to the additivity of quantum mutual information under tensor product, the data processing inequality gives no bound when the parties are provided with arbitrary number of copies of the resource state. In this paper, we introduce a measure of correlation on bipartite quantum states, called maximal correlation, that is not additive and gives the same number when computed for multiple copies. Then by proving a data processing inequality for this measure, we find a bound on the set of states that can be generated under local operations even when an arbitrary number of copies of the resource state is available.
A quantum correlation N F G , A for ( n + m ) -mode continuous-variable systems is introduced in terms of local Gaussian unitary operations performed on Subsystem A based on Uhlmann fidelity F. This quantity is a remedy for the local ancilla problem associated with the geometric measurement-induced correlations; is local Gaussian unitary invariant; is non-increasing under any Gaussian quantum channel performed on Subsystem B;and is an entanglement monotone when restricted to pure Gaussian states in the ( 1 + m ) -mode case. A concrete formula for ( 1 + 1 ) -mode symmetric squeezed thermal states (SSTSs) is presented. We also compare N F G , A with other quantum correlations in scale, such as Gaussian quantum discord and Gaussian geometric discord, for two-mode SSTSs, which reveals that N F G , A has some advantage in detecting quantum correlations of Gaussian states.
How can we characterize different types of correlation between quantum systems? Since correlations cannot be generated locally, we take any real function of a multipartite state which cannot increase under local operations to measure a correlation. Correlation measures that can be expressed as an optimization of a linear combination of entropies are particularly useful, since they can often be interpreted operationally. We systematically study such optimized linear entropic functions, and by enforcing monotonicity under local processing we identify four cones of correlation measures for bipartite quantum states. This yields two new optimized measures of bipartite quantum correlation that are particularly simple, which have the additional property of being additive.