Quantum Coherence of Qubit States with respect to Mutually Unbiased Bases
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Quantum Coherence of Qubit States with respect to Mutually Unbiased Bases Ming-Yang Shen1 · Yi-Hao Sheng1 · Yuan-Hong Tao1 Shao-Ming Fei4,5
· Yao-Kun Wang2,3 ·
Received: 2 March 2020 / Accepted: 23 October 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study the coherence of qubit states with respect to any set of mutually unbiased bases (MUBs), in terms of the l1 norm of coherence, the modified trace norm of coherence and the geometry measure of coherence. We present the arbitrary expressions of a complete set of MUBs in C 2 and the forms of any qubit states under these complete sets of MUBs. Then we calculate coherence of qubit states with respect to any complete sets of MUBs based on different coherence measures. It is shown that the sum of the coherence or the squared coherence is upper or lower bounded. We derive analytically these bounds and establish tight relations between quantum coherence and mutually unbiased bases. Keywords l1 Norm of coherence · Modified trace norm of coherence · Geometry measure of coherence · Mutually unbiased bases
1 Introduction As the key feature of the quantum world, quantum coherence is a significant ingredient in quantum mechanics, describing the capability of a quantum state to exhibit quantum interference phenomena [1]. It is now clear that quantum coherence is very essential in studying low temperature thermodynamics [2, 3], quantum biology [4, 5] and nanoscale physics [6] etc. Although the theory of quantum coherence is developed well in physics, the rigorous framework to quantify the quantum coherence is recently proposed by Baumgratz et al. [7]. Yuan-Hong Tao
[email protected] 1
Department of Mathematics, College of Sciences, Yanbian University, Yanji 133002, China
2
College of Mathematics, Tonghua Normal University, Tonghua, Jilin 134001, China
3
Research Center for Mathematical, College of Mathematics, Tonghua Normal University, Tonghua, Jilin 134001, China
4
School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China
5
Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
International Journal of Theoretical Physics d For fixed computational basis {|i}d−1 i=0 of a d-dimensional Hilbert space C , the incoherd−1 d−1 pi |ii|, where 0 < pi < 1 and pi = 1. The set of ent states have the form: δ = i=0
i=0
incoherent states as I [7]. Let be a completely positive trace preserving (CPTP) is denoted map, (ρ) = Kn ρKn† , where Kn is a set of Kraus operators satisfying Kn† Kn = Id , with Id denoting the d × d identity operator. If Kn I Kn† ⊆ I for all n, then is called an incoherent operation [7]. A nonnegative function C defined on a space of quantum states can be used as a measure of coherence if the following four conditions is satisfied [7],
(C2b)
C(ρ) ≥ 0, C(ρ) = 0 if and only if ρ ∈ I ; C((ρ))≤ C(ρ), for any incoherent operation ; Ki ρKi† , where pi = T r(Ki ρKi† ), Ki is a set of incoherent Kraus i pi C pi
(C3)
operators; d−1 pi = 1.
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