Quantifying the quantumness of ensembles via unitary similarity invariant norms

Abstract

The quantification of the quantumness of a quantum ensemble has theoretical and practical significance in quantum information theory. We propose herein a class of measures of the quantumness of quantum ensembles using the unitary similarity invariant norms of the commutators of the constituent density operators of an ensemble. Rigorous proof shows that they share desirable properties for a measure of quantumness, such as positivity, unitary invariance, concavity under probabilistic union, convexity under state decomposition, decreasing under coarse graining, and increasing under fine graining. Several specific examples illustrate the applications of these measures of quantumness in studying quantum information.

Optimal approximations of available states and a triple uncertainty relation

“…The above definition is actually a mathematical distinction between quantum states. The use of trace norm is due to its significance in quantum coherence and entanglement measures [12][13][14][15][16][17][18]. Now we study the optimal approximation of a given qubit state to the set of eigenstates of arbitrarily two different real quantum logic gates.…”

“…For a given state ρ, the problem is recast to look for the least distinguishable states from {|Ψ i } n i=1 such that distance between ρ and the convex set n i p i |Ψ i Ψ i | is minimized [7], whose solution is beneficial to the selection of available quantum resources [9][10][11]. Similar to the choice of distance measure for quantum coherence and quantum entanglement, we adopt here the trace norm as the measure of distance [12][13][14][15][16][17][18].…”

Analytically Computable Symmetric Quantum Correlations

“…[26][27][28][29][30][31][32][33][34][35][36][37][38][39] The same is true for its counterpart quantity, [40][41][42] that is, the measurementinduced nonlocality (MIN) which characterizes the global disturbance in a composite state caused by a local nondisturbing measurement on one subsystem. In particular, with the development of the resource theory of quantum coherence, [53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70] it was shown that the coherence of a system could be converted through incoherent operations into the SQD, instead of the asymmetric QD, of a composite system. [42][43][44][45][46][47][48][49] However, neither the QD nor the MIN of a bipartite system is symmetric in the sense that different results will be obtained (especially for some zero-discord states) if the two subsystems are swapped, which implies that the QD and MIN cannot completely quantify all of the quantum correlations present in a state.…”

“…To overcome this shortcoming, symmetric versions of the QD and the MIN are defined, i.e., the symmetric quantum discord (SQD) [50][51][52] and the symmetric measurement-induced nonlocality (SMIN). In particular, with the development of the resource theory of quantum coherence [53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70], it was shown that the coherence of a system could be converted through incoherent operations into the SQD, instead of the asymmetric QD, of a composite system [54,65,66]. By considering different bases, the coherence can also be related to the SMIN.…”

Characterizing quantum ensemble using geometric measure of quantum coherence

“…Hence, the quantum ensembles are both fundamental and practical objects in quantum information theory. Owing to the quantumness of ensemble as a physical resource, measures have been proposed from different perspectives such as relative entropy [27], commutators [26], unitary similarity invariant norms [28], coherence [29], quantum correlations, accessible information, security of information transmission, and quantum cloning [30,31].…”