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.
Similar content being viewed by others
References
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge: Cambridge University Press, 2010
L. Diósi, A Short Course in Quantum Information Theory: An Approach from Theoretical Physics, 2nd Ed., Berlin Heidelberg: Springer-Verlag, 2011
S. Massar and S. Popescu, Optimal extraction of information from finite quantum ensembles, Phys. Rev. Lett. 74(8), 1259 (1995)
C. A. Fuchs, Just two nonorthogonal quantum states, Quantum Communication, Computing, and Measurement 2, 11–16, Boston: Springer, 2002
C. A. Fuchs and M. Sasaki, Squeezing quantum information through a classical channel: Measuring the quantumness of a set of quantum states, Quantum Inf. Comput. 3, 377 (2003)
C. A. Fuchs and M. Sasaki, The quantumness of a set of quantum states, arXiv: quant-ph/0302108
M. Horodecki, P. Horodecki, R. Horodecki, and M. Piani, Quantumness of ensemble from no-broadcasting principle, Int. J. Quant. Inf. 04(01), 105 (2006)
O. Oreshkov and J. Calsamiglia, Distinguishability measures between ensembles of quantum states, Phys. Rev. A 79(3), 032336 (2009)
X. Zhu, S. Pang, S. Wu, and Q. Liu, The classicality and quantumness of a quantum ensemble, Phys. Lett. A 375(18), 1855 (2011)
T. Ma, M. J. Zhao, Y. K. Wang, and S. M. Fei, Noncommutativity and local indistinguishability of quantum states, Sci. Rep. 4, 6336 (2014)
S. Luo, N. Li, and X. Cao, Relative entropy between quantum ensembles, Period. Math. Hung. 59(2), 223 (2009)
S. Luo, N. Li, and W. Sun, How quantum is a quantum ensemble, Quantum Inform. Process. 9(6), 711 (2010)
S. Luo, N. Li, and S. Fu, Quantumness of quantum ensembles, Theor. Math. Phys. 169(3), 1724 (2011)
N. Li, S. Luo, and Y. Mao, Quantifying the quantumness of ensembles, Phys. Rev. A 96(2), 022132 (2017)
B. Dakić, V. Vedral, and Č. Brukner, Necessary and sufficient condition for nonzero quantum discord, Phys. Rev. Lett. 105(19), 190502 (2010)
S. Luo and S. Fu, Measurement-induced nonlocality, Phys. Rev. Lett. 106(12), 120401 (2011)
M. L. Hu and H. Fan, Measurement-induced nonlocality based on trace norm, New J. hys. 17(3), 033004 (2015)
X. Zhan, Matrix Theory, Graduate Studies in Mathematics Vol. 147, American Mathematical Society, Providence, Rhode Island, 2013
Y. Peng, Y. Jiang, and H. Fan, Maximally coherent states and coherence-preserving operations, Phys. Rev. A 93(3), 032326 (2016)
T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying coherence, Phys. Rev. Lett. 113(14), 140401 (2014)
X. Yuan, H. Zhou, Z. Cao, and X. Ma, Intrinsic randomness as a measure of quantum coherence, Phys. Rev. A 92(2), 022124 (2015)
X. F. Qi, T. Gao, and F. L. Yan, Measuring coherence with entanglement concurrence, J. Phys. A Math. Theor. 50(28), 285301 (2017)
S. Hill and W. K. Wootters, Entanglement of a pair of quantum bits, Phys. Rev. Lett. 78(26), 5022 (1997)
W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80(10), 2245 (1998)
H. Ollivier and W. H. Zurek, Quantum discord: A measure of the quantumness of correlations, Phys. Rev. Lett. 88(1), 017901 (2001)
L. Henderson and V. Vedral, Classical, quantum and total correlations, J. Phys. A Math. Gen. 34(35), 6899 (2001)
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant Nos. 11371005 and 11475054 and the Hebei Natural Science Foundation under Grant Nos. A2016205145 and A2018205125.
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Qi, XF., Gao, T. & Yan, FL. Quantifying the quantumness of ensembles via unitary similarity invariant norms. Front. Phys. 13, 130309 (2018). https://doi.org/10.1007/s11467-018-0773-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11467-018-0773-3
Keywords
- the quantumness of quantum ensemble
- measures of quantumness of quantum ensembles
- unitary similarity invariant norms