Wigner–Yanase information on quantum state space: The geometric approach

Gibilisco, Paolo; Isola, Tommaso

doi:10.1063/1.1598279

Cited by 100 publications

(128 citation statements)

References 36 publications

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“…(The upper value appeared to be unbounded. We also applied the formula (26) to the 8-dimensional convex set of 3 × 3 density matrices and found, through [52].) But there appears to be no "hint" in the literature as to how one might formally derive simply the separable -as opposed to separable plus nonseparable -volumes for any of the monotone metrics.…”

Section: Levy-gromov Isoperimetric Inequalitymentioning

confidence: 99%

Silver mean conjectures for 15-dimensional volumes and 14-dimensional hyperareas of the separable two-qubit systems

Slater

2005

Journal of Geometry and Physics

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Extensive numerical integration results lead us to conjecture that the silver mean, that is, σ Ag = √ 2− 1 ≈ .414214 plays a fundamental role in certain geometries (those given by monotone metrics) imposable on the 15-dimensional convex set of two-qubit systems. For example, we hypothesize that the volume of separable two-qubit states, as measured in terms of (four times) the minimal monotone or Bures metric is σ Ag 3 , and 10σ Ag in terms of (four times) the Kubo-Mori monotone metric. Also, we conjecture, in terms of (four times) the Bures metric, that that part of the 14-dimensional boundary of separable states consisting generically of rank-four 4 × 4 density matrices has volume ("hyperarea") 55σ Ag 39 , and that part composed of rank-three density matrices,

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Section: Levy-gromov Isoperimetric Inequalitymentioning

confidence: 99%

Silver mean conjectures for 15-dimensional volumes and 14-dimensional hyperareas of the separable two-qubit systems

Slater

2005

Journal of Geometry and Physics

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“…Theorem 2.5. We exhibit the measure µ c in the canonical representation (10) for a number of Morozova-Chentsov functions.…”

Section: λ − T Dµ(−λ)mentioning

confidence: 99%

Metric adjusted skew information

Hansen

2008

Proc. Natl. Acad. Sci. U.S.A.

115

161

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We extend the concept of Wigner-Yanase-Dyson skew information to something we call "metric adjusted skew information" (of a state with respect to a conserved observable). This "skew information" is intended to be a non-negative quantity bounded by the variance (of an observable in a state) that vanishes for observables commuting with the state. We show that the skew information is a convex function on the manifold of states. It also satisfies other requirements, proposed by Wigner and Yanase, for an effective measure-of-information content of a state relative to a conserved observable. We establish a connection between the geometrical formulation of quantum statistics as proposed by Chentsov and Morozova and measures of quantum information as introduced by Wigner and Yanase and extended in this article. We show that the set of normalized Morozova-Chentsov functions describing the possible quantum statistics is a Bauer simplex and determine its extreme points. We determine a particularly simple skew information, the "λ-skew information," parametrized by a λ ∈ (0, 1], and show that the convex cone this family generates coincides with the set of all metric adjusted skew informations.convexity | monotone metric | Morozova-Chentsov function | λ-skew information I n the mathematical model for a quantum mechanical system, the physical observables are represented by self-adjoint operators on a Hilbert space. The "states" (that is, the "expectation functionals" associated with the states) of the physical system are often "modeled" by the unit vectors in the underlying Hilbert space. So, if A represents an observable and x ∈ H corresponds to a state of the system, the expectation of A in that state is (Ax | x). For what we shall be proving, it will suffice to assume that our Hilbert space is finite dimensional and that the observables are self-adjoint operators, or the matrices that represent them, on that finite dimensional space. In this case, the states can be realized with the aid of the trace (functional) on matrices and an associated "density matrix." We denote by Tr(B) the usual trace of a matrix B [that is, Tr(B) is the sum of the diagonal entries of B]. The expectation functional of a state can be expressed as Tr(ρA), where ρ is a matrix, the density matrix associated with the state, and "Tr(ρA)" is the trace of the product ρA of the two matrices ρ and A. (Henceforth, we write "Tr ρA" omitting the parentheses when they are clearly understood.)In ref. 1, Wigner noticed that in the presence of a conservation law the obtainable accuracy of the measurement of a physical observable is limited if the operator representing the observable does not commute with (the operator representing) the conserved quantity (observable). Wigner proved it in the simple case where the physical observable is the x-component of the spin of a spin one-half particle and the z-component of the angular momentum is conserved. Araki and Yanase (2) demonstrated that this is a general phenomenon and pointed out, following Wigner's example, that u...

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“…For instance, there is no general formula for geodesic paths of manifolds of density operators endowed with an arbitrary given monotone Riemannian metric [43]. Explicit expressions are known only for two special metrics: the Bures metric [44] and the Wigner-Yanase metric [27]. In [44], geodesic paths on manifolds of density operators are obtained as projections of large circles on a large sphere within the purifying Hilbert-Schmidt space.…”

Section: Concluding Remarks and Open Issuesmentioning

confidence: 99%

“…(25) implies that α ≃ 2 √ N . Although Grover's algorithm evolves with discrete m, in the limit of N ≫ 1 the output state (27) can be approximated by a quantum wave-state |ψ (θ) depending on a continuous parameter θ. Indeed, considering the following formal substitutions,…”

Section: B Grover's Algorithmmentioning

confidence: 99%

“…In [44], geodesic paths on manifolds of density operators are obtained as projections of large circles on a large sphere within the purifying Hilbert-Schmidt space. In [27], geodesic paths are obtained exploiting the classical pull-back approach to Fisher information metric via sphere geometry. If diagonalization procedures are used, it is possible to obtain an explicit expression for the Bures distance of two arbitrary density matrices in the two-dimensional case [45].…”

Section: Concluding Remarks and Open Issuesmentioning

confidence: 99%

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On Grover’s search algorithm from a quantum information geometry viewpoint

Cafaro

Mancini

2012

Physica A: Statistical Mechanics and its Applications

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We present an information geometric characterization of Grover's quantum search algorithm. First, we quantify the notion of quantum distinguishability between parametric density operators by means of the Wigner-Yanase quantum information metric. We then show that the quantum searching problem can be recast in an information geometric framework where Grover's dynamics is characterized by a geodesic on the manifold of the parametric density operators of pure quantum states constructed from the continuous approximation of the parametric quantum output state in Grover's algorithm. We also discuss possible deviations from Grover's algorithm within this quantum information geometric setting.

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Wigner–Yanase information on quantum state space: The geometric approach

Cited by 100 publications

References 36 publications

Silver mean conjectures for 15-dimensional volumes and 14-dimensional hyperareas of the separable two-qubit systems

Silver mean conjectures for 15-dimensional volumes and 14-dimensional hyperareas of the separable two-qubit systems

Metric adjusted skew information

On Grover’s search algorithm from a quantum information geometry viewpoint

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