Operational advantage of basis-independent quantum coherence

¹ Department of Mathematics, Shanghai Jiaotong University - Shanghai, 200240, China
² Institute for Complex Quantum Systems & Center for Integrated Quantum Science and Technology (IQST), Universität Ulm - Albert-Einstein-Allee 11, D-89075 Ulm, Germany
³ Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University Beijing 100084, China
⁴ School of Mathematical Sciences, Capital Normal University - Beijing 100048, China
⁵ Max-Planck-Institute for Mathematics in the Sciences - D-04103 Leipzig, Germany
⁶ Atomic and Laser Physics, University of Oxford, Clarendon Laboratory - Parks Road, Oxford, OX1 3PU UK
⁷ Centre for Quantum Technologies, National University of Singapore - 3 Science Drive 2, 117543 Singapore
⁸ New York University Shanghai, 1555 Century Ave, Pudong - Shanghai 200122, China
⁹ State Key Laboratory of Precision Spectroscopy, School of Physical and Material Sciences, East China Normal University - Shanghai 200062, China
¹⁰ NYU-ECNU Institute of Physics at NYU Shanghai - 3663 Zhongshan Road North, Shanghai 200062, China
¹¹ National Institute of Informatics - 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430, Japan
¹² Department of Physics, New York University - New York, NY, 10003, USA

^(a) tim.byrnes@nyu.edu

Received: 28 December 2018
Accepted: 7 March 2019

Abstract

In the quantitative theory of quantum coherence, the amount of coherence is defined as the distance between the given state to the closest incoherent state. The set of incoherent states is conventionally defined as any state with a diagonal density matrix. One of the objections to this formulation is that the incoherent states are intrinsically basis-dependent, which makes the amount of coherence also a basis-dependent quantity. Basis-independent measures have recently been proposed where the incoherent state is taken as the maximally mixed state. We show that this is the only possible choice of reference incoherent state, without modifying the original definition of coherence. We find a relation between the two formulations by defining a contribution to the coherence due to the basis choice. The hierarchical relationship between quantum coherence and the various quantum correlations is explored in detail. Finally, we illustrate some operational uses of the basis-independent quantum coherence in quantum information theory tasks.