PDF《Condensed Matter Physics, 2018,》

arXiv:1809.

09469v2 [quant-ph]

26 Sep

2018 Condensed Matter Physics, 2018, Vol. 21, No 3, 33003: 1C4 DOI: 10.5488/CMP.21.33003 http://www.icmp.lviv.ua/journal Geometric measure of mixing of quantum state H.P. Laba

1 , V.M. Tkachuk

2 1 Department of Applied Physics and Nanomaterials Science, Lviv Polytechnic National University,

5 Ustiyanovych St.,

79013 Lviv, Ukraine

2 Department for Theoretical Physics, Ivan Franko National University of Lviv,

12 Drahomanov St.,

79005 Lviv, Ukraine Received June 23,

2018 We de?ne the geometric measure of mixing of quantum state as a minimal Hilbert-Schmidt distance between the mixed state and a set of pure states. An explicit expression for the geometric measure is obtained. It is interesting that this expression corresponds to the squared Euclidian distance between the mixed state and the pure one in space of eigenvalues of the density matrix. As an example, geometric measure of mixing for spin-1/2 states is calculated. Key words: mixed states, density matrix, Hilbert-Schmidt distance PACS: 03.65.-w, 03.67.-a 1. Introduction Pure and mixed states are the key concept in quantum mechanics and in quantum information theory. Therefore, an important question arises regarding the degree of mixing of a quantum state. In the literature, von Neumann entropy is often used to answer this question: S = ? Tr ? ρ ln ? ρ = ? ln ? ρ , (1.1) which is zero for a pure state and has a maximal value for maximally mixed states. The entropy can be used as a measure of the degree of mixing of a quantum state. To explicitly calculate the von Neumann entropy, it is necessary to know the eigenvalue of density matrix which is a nontrivial problem. Therefore, the linear entropy as approximation of von Neumann entropy is also used ln ? ρ = ln [1 ? (1 ? ? ρ)] ? ?(1 ? ? ρ) . (1.2) In this approximation, the linear entropy reads SL = Tr ? ρ ? ? ρ2 =

1 ? Tr ? ρ2 . (1.3) Linear entropy does not satisfy the properties of von Neumann entropy. However, to calculate the linear entropy, it is not necessary to know the eigenvalues of a density matrix. In this case, we can directly calculate the trace of ? ρ2. Note that Tr ? ρ2 is called purity and is used for quantifying the degree of the purity of state. For pure state ? ρ2 = ? ρ, and purity takes a maximal value

1 and is less

1 for mixed states. A review on entropy in quantum information can be found in book [1] (see also [2]). Geometric ideas play an important role in quantum mechanics and in quantum information theory (for review see, for instance, [3]). In our previous paper [4], we use the geometric characteristics such as curvature and torsion to study the quantum evolution. The geometry of quantum states in the evolution of a spin system was studied in [5, 6]. In [7], the distance between quantum states was used for quantifying the entanglement of pure and mixed states. This work is licensed under a Creative Commons Attribution 4.0 International License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. 33003-1 H.P. Laba, V.M. Tkachuk In this paper, we use Hilbert-Schmidt distance in order to measure the degree of mixing of quantum state. We de?ne the geometric measure of mixing of quantum state as minimal Hilbert-Schmidt distance between the mixed state and a set of pure states. In section 2, using this de?nition, we ?nd an explicit expression for the geometric measure of mixing of quantum state. Conclusions are presented in section 3. 2. Hilbert-Schmidt distance and degree of mixing of quantum state To de?ne the geometric measure of degree of mixing of quantum state, we use the Hilbert-Schmidt distance between two mixed states. The squared Hilbert-Schmidt distance reads d2 ( ? ρ1, ? ρ2) = Tr ? ρ1 ? ? ρ2