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.
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 N2−1 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.
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).
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 .
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)
Abstract The quantum Fisher information (QFI) plays a crucial role in quantum information theory and in many practical applications such as quantum metrology. However, computing the QFI is generally a computationally demanding task. In this work we analyze a lower bound on the QFI which we call the sub-quantum Fisher information (sub-QFI). The bound can be efficiently estimated on a quantum computer for an n -qubit state using 2 n qubits. The sub-QFI is based on the super-fidelity, an upper bound on Uhlmann’s fidelity. We analyze the sub-QFI in the context of unitary families, where we derive several crucial properties including its geometrical interpretation. In particular, we prove that the QFI and the sub-QFI are maximized for the same optimal state, which implies that the sub-QFI is faithful to the QFI in the sense that both quantities share the same global extrema. Based on this faithfulness, the sub-QFI acts as an efficiently computable surrogate for the QFI for quantum sensing and quantum metrology applications. Finally, we provide additional meaning to the sub-QFI as a measure of coherence, asymmetry, and purity loss.
We define and study a fidelity criterion for quantum channels, which we term the minimax fidelity, through a noncommutative generalization of maximal Hellinger distance between two positive kernels in classical probability theory. Like other known fidelities for quantum channels, the minimax fidelity is well defined for channels between finite-dimensional algebras, but it also applies to a certain class of channels between infinite-dimensional algebras (explicitly, those channels that possess an operator-valued Radon-Nikodym density with respect to the trace in the sense of Belavkin-Staszewski) and induces a metric on the set of quantum channels that is topologically equivalent to the CB-norm distance between channels, precisely in the same way as the Bures metric on the density operators associated with statistical states of quantum-mechanical systems, derived from the well-known fidelity (“generalized transition probability”) of Uhlmann, is topologically equivalent to the trace-norm distance.