On the Riemannian metric on the space of density matrices

Dittmann, Jochen

doi:10.1016/0034-4877(96)83627-5

Cited by 32 publications

(25 citation statements)

References 6 publications

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“…Taking

, the tangent vectors

, and

, the Bures–Helstrom metric tensor

may be defined by (see [ 40 ] (Equation ( 3 )))

Then, it is possible to prove that (see [ 40 , 41 , 46 ]), given arbitrary tangent vectors

, the Bures–Helstrom metric tensor acts as

where

is the inverse of

(see Equation ( 126 )), which is the invertible (because

is invertible) linear operator on

given by

. Note that the expression of

given in Equation ( 131 ) crucially depends on the identification of the tangent space

with the space of traceless, self-adjoint elements in

.…”

Section: The Bures–helstrom Metric Tensormentioning

confidence: 99%

“…Then, in Section 6 , we will take inspiration from Uhlmann’s geometric construction of the Bures–Helstrom metric tensor (see [ 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 ]) to show that the Riemannian geometries on the states associated with the Jordan product may be realized as “projected shadows” of the Riemannian geometries of suitable spheres in suitable Hilbert spaces by means of the GNS construction.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

From the Jordan Product to Riemannian Geometries on Classical and Quantum States

Ciaglia

Jost

Schwachhöfer

2020

Entropy

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The Jordan product on the self-adjoint part of a finite-dimensional C * -algebra A is shown to give rise to Riemannian metric tensors on suitable manifolds of states on A , and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitly computed. In particular, it is proved that the Fisher–Rao metric tensor is recovered in the Abelian case, that the Fubini–Study metric tensor is recovered when we consider pure states on the algebra B ( H ) of linear operators on a finite-dimensional Hilbert space H , and that the Bures–Helstrom metric tensors is recovered when we consider faithful states on B ( H ) . Moreover, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction associated to a state is presented. In the case of pure and faithful states on B ( H ) , this alternative geometrical description clarifies the analogy between the Fubini–Study and the Bures–Helstrom metric tensor.

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“…Taking

, the tangent vectors

, and

, the Bures–Helstrom metric tensor

may be defined by (see [ 40 ] (Equation ( 3 )))

Then, it is possible to prove that (see [ 40 , 41 , 46 ]), given arbitrary tangent vectors

, the Bures–Helstrom metric tensor acts as

where

is the inverse of

(see Equation ( 126 )), which is the invertible (because

is invertible) linear operator on

given by

. Note that the expression of

given in Equation ( 131 ) crucially depends on the identification of the tangent space

with the space of traceless, self-adjoint elements in

.…”

Section: The Bures–helstrom Metric Tensormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

From the Jordan Product to Riemannian Geometries on Classical and Quantum States

Ciaglia

Jost

Schwachhöfer

2020

Entropy

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show abstract

“…That it is a distance of a Riemann metric has been seemingly overlooked for long, [31], [11]. A systematic way to find the metric tensor is in [17]. dim H = 2 is discussed in [8], dim H = 3 in [26].…”

Section: Parallelity and The Minimal Length Conditionmentioning

confidence: 99%

Transition Probability (Fidelity) and Its Relatives

Uhlmann

2010

Found Phys

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Transition Probability (fidelity) for pairs of density operators can be defined as a "functor" in the hierarchy of "all" quantum systems and also within any quantum system. The Introduction of "amplitudes" for density operators allows for a more intuitive treatment of these quantities, also pointing to a natural parallel transport. The latter is governed by a remarkable gauge theory with strong relations to the Riemann-Bures metric.

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“…In the case of the Bures metric these geodesics can be given because they are projections of large circles on a sphere in the purifying space ͑see Ref. 10, p. 311 and Refs. 12, 4, and 39͒.…”

Section: Remark 62mentioning

confidence: 99%

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

Gibilisco

Isola

2003

Journal of Mathematical Physics

103

123

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In the search of appropriate Riemannian metrics on quantum state space, the concept of statistical monotonicity, or contraction under coarse graining, has been proposed by Chentsov. The metrics with this property have been classified by Petz. All the elements of this family of geometries can be seen as quantum analogs of Fisher information. Although there exists a number of general theorems shedding light on this subject, many natural questions, also stemming from applications, are still open. In this paper we discuss a particular member of the family, the WignerYanase information. Using a well-known approach that mimics the classical pullback approach to Fisher information, we are able to give explicit formulas for the geodesic distance, the geodesic path, the sectional and scalar curvatures associated to Wigner-Yanase information. Moreover, we show that this is the only monotone metric for which such an approach is possible.

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On the Riemannian metric on the space of density matrices

Cited by 32 publications

References 6 publications

From the Jordan Product to Riemannian Geometries on Classical and Quantum States

From the Jordan Product to Riemannian Geometries on Classical and Quantum States

Transition Probability (Fidelity) and Its Relatives

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

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