Applications of quantum technology often require fidelities to quantify performance. These provide a fundamental yardstick for the comparison of two quantum states. While this is straightforward in the case of pure states, it is much more subtle for the more general case of mixed quantum states often found in practice. A large number of different proposals exist. In this review, we summarize the required properties of a quantum fidelity measure, and compare them, to determine which properties each of the different measures has. We show that there are large classes of measures that satisfy all the required properties of a fidelity measure, just as there are many norms of Hilbert space operators, and many measures of entropy. We compare these fidelities, with detailed proofs of their properties. We also summarize briefly the applications of these measures in teleportation, quantum memories, quantum computers, quantum communications, and quantum phase-space simulations.
Three-qubit mixed states are used as a channel for controlled quantum teleportation (CQT) of single-qubit states. The connection between different channel parameters to achieve maximum controlled teleportation fidelity is investigated. We show that for a given multipartite entanglement and mixedness, a class of non-maximally entangled mixed $X$ states ($X-$NMEMS) achieves optimum controlled quantum teleportation fidelity, interestingly a class of maximally entangled mixed $X$ states ($X-$MEMS) fails to do so. This demonstrates, for a given spectrum and mixedness, that $X-$MEMS are not sufficie…
To explore the properties of a two-qubit mixed state, we consider quantum teleportation. The fidelity of a teleported state depends on the resource state purity and entanglement, as characterized by concurrence. Concurrence and purity are functions of state parameters. However, it turns out that a state with larger purity and concurrence, may have comparatively smaller fidelity. By computing teleportation fidelity, concurrence and purity for two-qubit X-states, we show it explicitly. We further show that fidelity changes monotonically with respect to functions of parameters - other than concur…
General semiclassical expression for quantum fidelity (Loschmidt echo) of arbitrary pure and mixed states is derived. It expresses fidelity as an interference sum of dephasing trajectories weighed by the Wigner function of the initial state, and does not require that the initial state be localized in position or momentum. This general dephasing representation is special in that, counterintuitively, all of fidelity decay is due to dephasing and none due to the decay of classical overlaps. Surprising accuracy of the approximation is justified by invoking the shadowing theorem: twice--both for ph…
A Necessary And Sufficient Condition of Distillability with unite fidelity from Finite Copies of a Mixed State: The Most Efficient Purification Protocol
We implement a two-qubit logic gate between a $^{43}\mathrm{Ca}^+\,$ hyperfine qubit and a $^{88}\mathrm{Sr}^+\,$ Zeeman qubit. For this pair of ion species, the S--P optical transitions are close enough that a single laser of wavelength $402\,\mathrm{nm}$ can be used to drive the gate, but sufficiently well separated to give good spectral isolation and low photon scattering errors. We characterize the gate by full randomized benchmarking, gate set tomography and Bell state analysis. The latter method gives a fidelity of $99.8(1)\%$, comparable to that of the best same-species gates and consis…
Polynomial-time certification of fidelity for many-body mixed states and mixed-state universality classes
Max Planck Institute of Quantum Optics（马克斯·普朗克量子光学研究所）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）
We show exactly that the $α$th power of Bures measure of entanglement and geometric measure of entanglement, as special case of entanglement measures based on fidelity, obey a class of general monogamy inequalities in an arbitrary multiqubit mixed state for $α\geq1$.
We show that $Ω(rd/ε)$ copies of an unknown rank-$r$, dimension-$d$ quantum mixed state are necessary in order to learn a classical description with $1 - ε$ fidelity. This improves upon the tomography lower bounds obtained by Haah, et al. and Wright (when closeness is measured with respect to the fidelity function).
In this article, we consider the heterodinuclear complex [Ni(dpt)(H2O)Cu(pba)].2H2O [pba =1,3-propylenebis(oxamato) and dpt = bis-(3-aminopropyl)amine] realized through the theoretical model of mixed spin-(1/2,1) coupled via Heisenberg interaction. We study the behaviors of thermal quantum correlations of the above material via Measurement-Induced Nonlocality (MIN) based on Hilbert-Schmidt norm and fidelity. We observe that the quantum correlation measures increase with the magnetic field in an unconventional way. The role of system parameters is also brought out at thermal equilibrium. The hi…
Two-body control of noiseless subsystems with mixed $SU(3)$ representations lifts teleportation above the classical fidelity limit
International Centre for Theory of Quantum Technologies (ICTQT), University of Gdańsk（格但斯克大学量子技术理论国际中心）；Department of Computer Science, University of Oxford（牛津大学计算机科学系）
The Eastin-Knill theorem is a central result of quantum error correction theory and states that a quantum code cannot correct errors exactly, possess continuous symmetries, and implement a universal set of gates transversely. As a way to circumvent this result, there are several approaches in which one gives up on either exact error correction or continuous symmetries. In this context, it is common to employ a complementary measure of fidelity as a way to quantify quantum state distinguishability and benchmark approximations in error correction. Despite having useful properties, evaluating fid…
We obtain the maximal average fidelity corresponding to the standard quantum teleportation protocol for an arbitrary isotropic distribution of input states and an arbitrary resource state. We extend this result to a family of von Neumann measurements, which includes the projections onto the computational and Bell basis, considering a Bell-diagonal resource state. We focus on three specific isotropic distributions of input states: 1) completely mixed input states, 2) states with a certain (fixed) degree of purity, and 3) quasi-pure input states. We show that the standard quantum teleportation p…
Variational quantum algorithms (VQAs) that estimate values of widely used physical quantities such as the rank, quantum entropies, the Bures fidelity and the quantum Fisher information of mixed quantum states are developed. In addition, variations of these VQAs are also adapted to perform other useful functions such as quantum state learning and approximate fractional inverses. The common theme shared by the proposed algorithms is that their cost functions are all based on minimizing the quantum purity of a quantum state. Strategies to mitigate or avoid the problem of exponentially vanishing c…
量化相干性是量子基础和量子技术的重要努力。本文根据[T. Baumgratz，M. Cramer和M. B. Plenio，Phys. Rev. Lett. 113，140401（2014）]。我们的措施是基于保真度和分析计算的任意状态的量子位。作为它的应用之一，我们证明了我们的测度可以用来检验一个纯量子比特态是否可以仅仅通过非相干操作转换成另一个纯量子比特态或混合量子比特态。
We investigate the discrimination of pure-mixed (quantum filtering) and mixed-mixed states and compare their optimal success probability with the one for discriminating other pairs of pure states superposed by the vectors included in the mixed states. We prove that under the equal-fidelity condition, the pure-pure state discrimination scheme is superior to the pure-mixed (mixed-mixed) one. With respect to quantum filtering, the coherence exists only in one pure state and is detrimental to the state discrimination for lower dimensional systems; while it is the opposite for the mixed-mixed case …
