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International Journal of Quantum Information

Fidelity, sub-fidelity, super-fidelity and their preservers

Li Wang, Jinchuan Hou, Kan He

Published Apr 1, 2015• 1 citation
Publisher

Abstract

Sub- and super-fidelity describe respectively the lower and super bound of fidelity of quantum states. In this paper, we obtain several properties of sub- and super-fidelity for both finite- and infinite-dimensional quantum systems. Furthermore, let H be a separable complex Hilbert space and ϕ : 𝒮(H) → 𝒮(H) a map, where 𝒮(H) denotes the convex set of all states on H. We show that, if dim H < ∞, or, if dim H = ∞ and ϕ is surjective, then the following statements are equivalent: (1) ϕ preserves the super-fidelity; (2) ϕ preserves the fidelity; (3) ϕ preserves the sub-fidelity; (4) there exists a unitary or an anti-unitary operator U on H such that ϕ(ρ) = UρU† for all ρ ∈ 𝒮(H).

DOI
10.1142/s0219749915500276

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•May 9, 2013•0 citations
Sub- and super-fidelity describe respectively the lower and super bound of fidelity of quantum states. In this paper, we obtain several properties of sub- and super-fidelity for both finite- and infinite-dimensional quantum systems. Furthermore, let H be a separable complex Hilbert space and ϕ : 𝒮(H) → 𝒮(H) a map, where 𝒮(H) denotes the convex set of all states on H. We show that, if dim H < ∞, or, if dim H = ∞ and ϕ is surjective, then the following statements are equivalent: (1) ϕ preserves the super-fidelity; (2) ϕ preserves the fidelity; (3) ϕ preserves the sub-fidelity; (4) there exists a unitary or an anti-unitary operator U on H such that ϕ(ρ) = UρU† for all ρ ∈ 𝒮(H).
One of the main recent achievements of quantum information theory [1] is the proof of the equivalence of several (sub-, super-) additivity conjectures for finite-dimensional quantum channels and systems [2] (see the survey in [3]). The main purpose of this paper is to give a generalization of this result to the infinite-dimensional case. Let H and H ′ be separable Hilbert spaces. A quantum channel is a linear completely positive trace-preserving map Φ: T(H ) 7→ T(H ′), where T(H ) is the ideal of all trace class operators on H . In particular, Φ generates an affine map of the convex set S(H ) of states (that is, density operators) on the space H to the set S(H ′) of states on the space H ′ [1]. Important characteristics of a quantum channel are the output entropy HΦ(ρ) = H(Φ(ρ)), a lower-semicontinuous concave function on the input state space with values in [0,+∞], and its convex closure (see [4]), denoted by b HΦ(ρ) and called the b H-function of the channel Φ. It was shown in [5] that the convex closure of the output entropy of an arbitrary quantum channel Φ is given by the expression b HΦ(ρ) = inf μ Z
In this paper, we give the definition of k th sub-fidelity and k th super-fidelity between quantum states ρ and ϱ, and investigate the relationship among k th partial fidelity ( F k (ρ, ϱ)), k th sub-fidelity ( E k (ρ, ϱ)) and k th super-fidelity ( G k (ρ, ϱ)). In particular, it is shown that E k (ρ, ϱ) ⩽ F k (ρ, ϱ) ⩽ G k (ρ, ϱ). In addition, we show that the partial super-fidelity is a super-multiplicative map, which is G kl (ρ 1 ⊗ϱ 1 , ρ 2 ⊗ϱ 2 ) ⩾ G k (ρ 1 , ρ 2 ) G l (ϱ 1 , ϱ 2 ), for 1 ⩽ k , l ⩽ d .
We derive several bounds on fidelity between quantum states. In particular we show that fidelity is bounded from above by a simple to compute quantity we call super--fidelity. It is analogous to another quantity called sub--fidelity. For any two states of a two--dimensional quantum system (N=2) all three quantities coincide. We demonstrate that sub-- and super--fidelity are concave functions. We also show that super--fidelity is super--multiplicative while sub--fidelity is sub--multiplicative and design feasible schemes to measure these quantities in an experiment. Super--fidelity can be used to define a distance between quantum states. With respect to this metric the set of quantum states forms a part of a N2−1N^2-1N2−1 dimensional hypersphere.
We derive several bounds on fidelity between quantum states. In particular we show that fidelity is bounded from above by a simple to compute quantity we call super--fidelity. It is analogous to another quantity called sub--fidelity. For any two states of a two--dimensional quantum system (N=2) all three quantities coincide. We demonstrate that sub-- and super--fidelity are concave functions. We also show that super--fidelity is super--multiplicative while sub--fidelity is sub--multiplicative and design feasible schemes to measure these quantities in an experiment.Super--fidelity can be used to define a distance between quantum states. With respect to this metric the set of quantum states forms a part of a N^2-1 dimensional hypersphere.
A conjecture – weak super-additivity inequality of relative entropy – was proposed in [1]: There exist three unitary operators UA ∈ U (HA) , UB ∈ U (HB), and UAB ∈ U (HA ⊗HB) such that S(UABρABU†AB||σAB)> S(UAρAU†A||σA) + S(UBρBU†B||σB), where the reference state σ is required to be full-ranked. A numerical study on the conjec-tured inequality is conducted in this notes. The results obtained indicates that weak super-additivity inequality of relative entropy seems to hold for all qubit pairs. Similarly, the weak sub-multiplicativity of fidelity is also considered. That is, there exist three unitary operators UA ∈ U (HA) , UB ∈ U (HB), and UAB ∈ U (HA ⊗HB) such that F(UABρABU†AB, σAB) 6 F(UAρAU†A, σA)F(UBρBU†B, σB). An attempt is made to give some potential applications in quantum information theory. 1