Statistical distance and the geometry of quantum states

Albuquerque Academy, Albuquerque, New Mexico, United States
Physical Review Letters (Impact Factor: 7.51). 06/1994; 72(22):3439-3443. DOI: 10.1103/PhysRevLett.72.3439
Source: PubMed


By finding measurements that optimally resolve neighboring quantum states, we use statistical distinguishability to define a natural Riemannian metric on the space of quantum-mechanical density operators and to formulate uncertainty principles that are more general and more stringent than standard uncertainty principles.

Download full-text


Available from: Samuel L. Braunstein, Apr 13, 2015
  • Source
    • "In a seminal work [24] , the authors pointed that both fidelity and QFI are highly related to the distinguishability of the states, which is measured by Bures distance [31] . "
    [Show abstract] [Hide abstract]
    ABSTRACT: An unknown quantum state cannot be copied on demand and broadcast freely due to the famous no-cloning theorem. Approximate cloning schemes have been proposed to achieve the optimal cloning characterized by the maximal fidelity between the original and its copies. Here, from the perspective of quantum Fisher information (QFI), we investigate the distribution of QFI in asymmetric cloning machines which produce two nonidentical copies. As one might expect, improving the QFI of one copy results in decreasing the QFI of the other copy, roughly the same as that of fidelity. It is perhaps also unsurprising that asymmetric phase-covariant cloning machine outperforms universal cloning machine in distributing QFI since a priori information of the input state has been utilized. However, interesting results appear when we compare the distributabilities of fidelity (which quantifies the full information of quantum states), and QFI (which only captures the information of relevant parameters) in asymmetric cloning machines. In contrast to the results of fidelity, where the distributability of symmetric cloning is always optimal for any d-dimensional cloning, we find that asymmetric cloning performs always better than symmetric cloning on the distribution of QFI for $d\leq18$, but this conclusion becomes invalid when $d>18$.
    Full-text · Article · Sep 2014 · Scientific Reports
  • Source
    • "The closeness of two pure quantum states ψ | 〉 1 and ψ | 〉 2 can be quantified in many different ways. One of the simplest and therefore most popular measures of the closeness is the absolute value of the scalar product between these states or some function of this quantity [32] [33] [34]. Following [35], we use the name fidelity for the square of the absolute value of this scalar product: "
    [Show abstract] [Hide abstract]
    ABSTRACT: We give analytical and numerical upper bounds for the relative energy difference of different superpositions of two coherent states ('cat' states) with a fixed fidelity between them.
    Full-text · Article · Jul 2014 · Journal of Physics A Mathematical and Theoretical
  • Source
    • "where A is found from solving the symmetric logarithmic derivative ∂ρ/∂φ = 1/2 [Aρ + ρA]. The precision in the phase measurement (more specifically the lower bound on the standard deviation) is given by the quantum Cramér-Rao bound (Braunstein and Caves, 1994): "
    [Show abstract] [Hide abstract]
    ABSTRACT: Quantum entanglement offers the possibility of making measurements beyond the classical limit, however some issues still need to be overcome before it can be applied in realistic lossy systems. Recent work has used the quantum Fisher information (QFI) to show that entangled coherent states (ECSs) may be useful for this purpose as they combine sub-classical phase precision capabilities with robustness (Joo et al., 2011). However, to date no effective scheme for measuring a phase in lossy systems using an ECS has been devised. Here we present a scheme that does just this. We show how one could measure a phase to a precision significantly better than that attainable by both unentangled "classical" states and highly-entangled NOON states over a wide range of different losses. This brings quantum metrology closer to being a realistic and practical technology.
    Full-text · Article · Jan 2014
Show more