The scalar curvature of the Bures metric on the space of density matrices

Dittmann, Jochen

doi:10.1016/s0393-0440(98)00068-0

Cited by 17 publications

(20 citation statements)

References 15 publications

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“…✷ This confirms the result obtained for the Kubo-Mori metric at ̺ = 1 n 1 in [15], Theorem 6.2. For the minimal metric this coincides with Corollary 3 of [23]. The differing factor is due to a factor 1/4 in the metric.…”

Section: A General Theoremsupporting

confidence: 80%

“…Proof: We set n = 3, λ 1 = λ 3 = x, λ 2 = y, X = b 13 and Y = b 23 . Then g(b 11 , b 11 ) = ||b 11 || 2 = ||b 33 || 2 = 4/x, ||X|| 2 = 2/x, ||b 12 || 2 = ||Y || 2 = 2c(x, y) and…”

Section: Lemmamentioning

confidence: 99%

“…All the terms indicated by dots in the above equation vanish. From (25), (23) and (20) we see that these terms are built by products of the four arguments of the curvature tensor and resolvents. Now, considering the indices of the arguments of the neglected terms it is not difficult to see, that these products (e.g.…”

Section: Lemmamentioning

confidence: 99%

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On the curvature of monotone metrics and a conjecture concerning the Kubo-Mori metric

Dittmann

2000

Linear Algebra and its Applications

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Monotone metrics on the space of positive matrices generalize the classical Fisher metric of statistical distinguishability of probability distributions to the quantum case. These metrics are in one to one correspondence with operator monotone functions. It is the aim of this article to determine curvature quantities of an arbitrary Riemannian monotone metric using the Riesz-Dunford operator calculus as the main technical tool. The resulting scalar curvature is explained in more detail for three examples. In particular, we show an important conjecture of Petz concerning the Kubo-Mori metric up to a formal proof of the concavity of a certain function on R 3 + . This concavity seems to be numerically evident. The conjecture asserts that the scalar curvature of the Kubo-Mori metric increases if one goes to more mixed states.

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Section: A General Theoremsupporting

confidence: 80%

“…Proof: We set n = 3, λ 1 = λ 3 = x, λ 2 = y, X = b 13 and Y = b 23 . Then g(b 11 , b 11 ) = ||b 11 || 2 = ||b 33 || 2 = 4/x, ||X|| 2 = 2/x, ||b 12 || 2 = ||Y || 2 = 2c(x, y) and…”

Section: Lemmamentioning

confidence: 99%

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On the curvature of monotone metrics and a conjecture concerning the Kubo-Mori metric

Dittmann

2000

Linear Algebra and its Applications

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“…It can be shown that the curvature of Bures metric space for single qubit states is constant [13], thus it can be isometric to a hypersphere in a 4-d space. A simple isometric mapping exist between the hemisphere of radius 1/2 and the Stokes parameters:…”

Section: Pacs Numbersmentioning

confidence: 99%

Experimental adaptive Bayesian tomography

et al. 2013

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We report an experimental realization of an adaptive quantum state tomography protocol. Our method takes advantage of a Bayesian approach to statistical inference and is naturally tailored for adaptive strategies. For pure states we observe close to 1/N scaling of infidelity with overall number of registered events, while best non-adaptive protocols allow for 1/ √ N scaling only. Experiments are performed for polarization qubits, but the approach is readily adapted to any dimension.

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“…The physical meaning of this fact seems to be an interesting open question" [18] (cf. [19,20]). ) Clarke and Barron [21,22] (cf.…”

Section: A Jeffreys' Priormentioning

confidence: 99%

A prioriprobabilities of separable quantum states

Slater

1999

J. Phys. A: Math. Gen.

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Zyczkowski, Horodecki, Sanpera, and Lewenstein (ZHSL) recently proposed a "natural measure" on the N -dimensional quantum systems, but expressed surprise when it led them to conclude that for N = 2 × 2, disentangled (separable) systems are more probable (0.632 ± 0.002) in nature than entangled ones. We contend, however, that ZHSL's (rejected) intuition has, in fact, a sound theoretical basis, and that the a priori probability of disentangled 2 × 2 systems should more properly be viewed as (considerably) less than 0.5. We arrive at this conclusion in two quite distinct ways, the first based on classical and the second, quantum considerations. Both approaches, however, replace (in whole or part) the ZHSL (product) measure by ones based on the volume elements of monotone metrics, which in the classical case amounts to adopting the Jeffreys' prior of Bayesian theory. Only the quantum-theoretic analysis -which yields the smallest probabilities of disentanglement -uses the minimum number of parameters possible, that is N 2 − 1, as opposed to N 2 + N − 1 (although this "over-parameterization", as recently indicated by Byrd, should be avoidable). However, despite substantial computation, we are not able to obtain precise estimates of these probabilities and the need for additional (possibly supercomputer) analyses is indicated -particularly so for higher-dimensional quantum systems (such as the 2 × 3 ones, we also study here). 89.70.+c, 02.40.Ky PACS Numbers

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The scalar curvature of the Bures metric on the space of density matrices

Cited by 17 publications

References 15 publications

On the curvature of monotone metrics and a conjecture concerning the Kubo-Mori metric

On the curvature of monotone metrics and a conjecture concerning the Kubo-Mori metric

Experimental adaptive Bayesian tomography

A prioriprobabilities of separable quantum states

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