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Communications in Theoretical Physics

Some New Metrics Based on Partial Fidelity and Partial Super-Fidelity

Zhaoqi Wu, Chuanxi Zhu, Xiaozhi Zhang

Published Jan 1, 2014• 6 citations
Publisher

Abstract

In this paper, we introduce some new metrics between quantum states based on partial fidelity and partial super-fidelity and discuss their properties. We show that the new metrics are useful measures in quantum information processing.
DOI
10.1088/0253-6102/61/1/06

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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.
Fidelity plays an important role in quantum information theory. In this paper, two pairs of metrics of quantum states are introduced based on the Uhlmann-Jozsa fidelity and super-fidelity. They are proved to satisfy the axioms of metrics and return to the Sine metric and Bures metric for the qubit case. The CP expansive property and convex property of the metrics are studied. These metrics have deep connections with the known important metrics such as trace metric, Sine metric, Bures metric, and spectral metric, and are useful to quantify entanglement. I. Background, problem, and result Suppose one has two quantum states ρ and σ, then the Uhlmann-Jozsa fidelity [1] between ρ and σ is given by F(ρ, σ) = [Tr ρ 1 2 σρ 1 2] 2. Uhlmann-Jozsa fidelity plays an important role in quantum information theory and quantum computation [1], [2]. We know that for the case of qubit, Uhlmann-Jozsa fidelity has a simple form. From the Bloch sphere representation of quantum states, a qubit is described by a density matrix as([3]): ρ(u) = 1 (I + σ · u), 2 where I is the 2 × 2 unit matrix and σ = (σ1, σ2, σ3) are the Pauli matrices. Assume ρ(u) and ρ(v) are two states of one qubit, then they can represented by two vectors u and v in the Bloch sphere. The Uhlmann-Jozsa fidelity for qubits has an elegant form([4] and [5]): F(ρ(u), ρ(v)) = 1 2 [1 + u · v + √ 1 − |u | 2 √ 1 − |v | 2], where u · v is the inner product of two vectors u and v, and |u | is the magnitude of u. We know that for N × N quantum states, the Uhlmann-Jozsa fidelity has no simple form like the case of qubits. To use the simple form of fidelity, we note that in [6], the authors introduce a new fidelity, called super-fidelity, and proved that Uhlmann-Jozsa fidelity can be bounded above by the super-fidelity G(ρ1, ρ2): F(ρ1, ρ2) ≤ G(ρ1, ρ2)
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).