Entanglement measures induced by fidelity-based distances
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Entanglement measures induced by fidelity-based distances Yu Guo1,2
· Lin Zhang3,4 · Huting Yuan2
Received: 24 March 2020 / Accepted: 24 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We propose entanglement measures by means of the fidelity-based distances. The fidelity-based distance here is analogous to the relative entropy to the well-known entanglement measure, entanglement of formation. Our approach can be defined for any multipartite systems in a unified way. We also show that these fidelity-based measures are entanglement monotones and the bipartite ones are monogamous. Keywords Entanglement measure · Fidelity · Monogamy
1 Introduction A fundamental problem in characterizing quantum entanglement is to quantify entanglement [1–3]. Among the considerable entanglement measures, entanglement distillation [4], entanglement cost [4,5], and entanglement of formation (EoF) [5,6] are the most important ones since they are the first three measures of entanglement and in addition, entanglement distillation and entanglement cost are appeared in the context of manipulating entanglement with an operational meaning and the entanglement of formation has a close relation with the former ones [4,7–9].
B B
Yu Guo [email protected] Lin Zhang [email protected]
1
Institute of Quantum Information Science, Shanxi Datong University, Datong 037009, Shanxi, China
2
School of Mathematics and Statistics, Shanxi Datong University, Datong 037009, China
3
Institute of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, China
4
Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany 0123456789().: V,-vol
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EoF is defined as the von Neumann entropy of the reduced state for pure states and then by the convex-roof extension for the mixed states [5,6]: E f (ρ) := min
pi S (Tr B |ψi ψi |) ,
(1)
i
where the minimum is taken over all ensembles { pi , |ψi } for which ρ = i pi |ψi ψi |. Note that, S (Tr B |ψψ|) =
1 S |ψψ|ρ A ⊗ ρ B , 2
(2)
where ρ A,B = Tr B,A |ψψ| and S(ρσ ) := Tr[ρ(ln ρ − ln σ )] is the relative entropy. Namely, the entropy of reduced sate of pure state can also be regarded as the “distance” between the pure state and its product states of the marginal states measured by the relative entropy. Apart from the relative entropy, fidelity, as a measure of the degree of similarity of a pair of quantum states, is also a nice alternate of defining such a distance [10–20]. The most widely-employed fidelity that has been proposed in the literatures is the Uhlmann-Jozsa fidelity F [11,12], which is defined as √ √ 2 ρσ ρ . F(ρ, σ ) := Tr
(3)
It is shown that F is monotonic under any quantum operation and it admits many desirable properties [10]. The Uhlmann-Jozsa fidelity was also defined as
√ F(ρ, σ ) := F(ρ, σ )
(4)
in some studies [13–15]. Another alternative fidelity measure that also non-decreases under quantum operation is the square of the quantum affinity A(ρ, σ ) proposed in
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