Bures volume of the set of mixed quantum states

Sommers, Hans-Jürgen; Życzkowski, Karol

doi:10.1088/0305-4470/36/39/308

Cited by 164 publications

(309 citation statements)

References 45 publications

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“…In particular, analogous results presented by us in [33] for the measure [26,34] related to the Bures distance [35,36] allow us to investigate similarities and differences between the geometry of mixed states induced by different metrics.…”

Section: Discussionsupporting

confidence: 54%

Hilbert–Schmidt volume of the set of mixed quantum states

Życzkowski

Sommers

2003

J. Phys. A: Math. Gen.

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130

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We compute the volume of the convex N 2 − 1 dimensional set MN of density matrices of size N with respect to the Hilbert-Schmidt measure. The hyper-area of the boundary of this set is also found and its ratio to the volume provides an information about the complex structure of MN . Similar investigations are also performed for the smaller set of all real, symmetric density matrices.As an intermediate step we analyze volumes of the unitary and orthogonal groups and of the flag manifolds.

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Section: Discussionsupporting

confidence: 54%

Hilbert–Schmidt volume of the set of mixed quantum states

Życzkowski

Sommers

2003

J. Phys. A: Math. Gen.

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130

268

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“…It is generally believed that the first-order term disappears in fidelity, i.e., Tr(X ) = 0. However, this conclusion is only well established for pure states or full rank density matrices [20,21]. Below we will show that it also holds for density matrices with nonfull ranks.…”

Section: Fidelity Susceptibility and Quantum Fisher Information For Dmentioning

confidence: 71%

“…It is generally believed that the first-order term of δθ in fidelity is zero [20,21], thus FS is determined by the second-order term with the definition…”

Section: Introductionmentioning

confidence: 99%

“…Interestingly, the above two seemingly irrelevant concepts are in fact closely related to each other. Generally, people vaguely believe that for a given state, the first-order term in fidelity equals to zero and the expression of FS is proportional to that of QFI [20,21,24,25]. This is certainly inarguable for the cases with pure states or full-rank density matrices [20,21].…”

Section: Introductionmentioning

confidence: 99%

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Fidelity susceptibility and quantum Fisher information for density operators with arbitrary ranks

Liu

Xiong

Song

et al. 2014

Physica A: Statistical Mechanics and its Applications

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Abstract. Taking into account the density matrices with non-full ranks, we show that the fidelity susceptibility is determined by the support of the density matrix. Combining with the corresponding expression of the quantum Fisher information, we rigorously prove that the fidelity susceptibility is proportional to the quantum Fisher information. As this proof can be naturally extended to the full rank case, this proportional relation is generally established for density matrices with arbitrary ranks. Furthermore, we give an analytical expression of the quantum Fisher information matrix, and show that the quantum Fisher information matrix can also be represented in the density matrix's support.

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“…In this case, the relevant quantity is the so-called Bures infinitesimal distance between nearby point in the parameter space [31,32,33,34,35] …”

Section: The Physical Modelmentioning

confidence: 99%

Quantum probes for fractional Gaussian processes

Paris¹

2014

Physica A: Statistical Mechanics and its Applications

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We address the characterization of classical fractional random noise via quantum probes. In particular, we focus on estimation and discrimination problems involving the fractal dimension of the trajectories of a system subject to fractional Brownian noise. We assume that the classical degree of freedom exposed to the environmental noise is coupled to a quantum degree of freedom of the same system, e.g. its spin, and exploit quantum limited measurements on the spin part to characterize the classical fractional noise. More generally, our approach may be applied to any two-level system subject to dephasing perturbations described by fractional Brownian noise, in order to assess the precision of quantum limited measurements in the characterization of the external noise. In order to assess the performances of quantum probes we evaluate the Bures metric, as well as the Helstrom and the Chernoff bound, and optimize their values over the interaction time. We find that quantum probes may be successfully employed to obtain a reliable characterization of fractional Gaussian process when the coupling with the environment is weak or strong. In the first case decoherence is not much detrimental and for long interaction times the probe acquires information about the environmental parameters without being too much mixed. Conversely, for strong coupling information is quickly impinged on the quantum probe and can effectively retrieved by measurements performed in the early stage of the evolution. In the intermediate situation, none of the two above effects take place: information is flowing from the environment to the probe too slowly compared to decoherence, and no measurements can be effectively employed to extract it from the quantum probe. The two regimes of weak and strong coupling are defined in terms of a threshold value of the coupling, which itself increases with the fractional dimension.

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Bures volume of the set of mixed quantum states

Cited by 164 publications

References 45 publications

Hilbert–Schmidt volume of the set of mixed quantum states

Hilbert–Schmidt volume of the set of mixed quantum states

Fidelity susceptibility and quantum Fisher information for density operators with arbitrary ranks

Quantum probes for fractional Gaussian processes

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