Bures metric

In mathematics, in the area of quantum information geometry, the Bures metric^[1] or Helstrom metric^[2] defines an infinitesimal distance between density matrix operators defining quantum states. It is a quantum generalization of the Fisher information metric, and is identical to the Fubini–Study metric^[3] when restricted to the pure states alone.

Definition

The metric may be defined as

[d(\rho, \rho+d\rho)]^2 = \frac{1}{2}\mbox{tr}( d \rho G ),

where $G$ is Hermitian 1-form operator implicitly given by

\rho G + G \rho = d \rho^{\,}

Some of the applications of the Bures metric include that given a target error, it allows the calculation of the minimum number of measurements to distinguish two different states and the use of the volume element as a candidate for the Jeffreys prior probability density for mixed quantum states.

Bures distance

The Bures distance is the finite version of the infinitesimal square distance described above and is given by

D_B(\rho_1,\rho_2)^2 = 2(1-\sqrt{F(\rho_1,\rho_2)}),

where the fidelity function is defined as

F(\rho_1,\rho_2) = \left[ \mbox{tr}( \sqrt{ \sqrt{\rho_1}\rho_2\sqrt{\rho_1}})\right]^2

Another associated function is the Bures arc also known as Bures angle, Bures length or quantum angle, defined as

D_A(\rho_1,\rho_2) = \arccos \sqrt{F(\rho_1,\rho_2)},

which is a measure of the statistical distance between the quantum states.

Quantum Fisher information

The Bures metric can be seen as the quantum equivalent of the Fisher information metric and can be rewritten in terms of the variation of coordinate parameters as

[d(\rho, \rho+d\rho)]^2 = \frac{1}{2} \mbox{tr}\left( \frac{d \rho}{d \theta^{\mu}} L_{\nu} \right) d \theta^{\mu} d\theta^{\nu},

where $L_\mu$ is the Symmetric Logarithmic Derivative operator (SLD) defined from

\frac{\rho L_{\mu} + L_{\mu} \rho}{2} = \frac{d \rho^{\,}}{d \theta^{\mu}}.

In this way, one has

[d(\rho, \rho+d\rho)]^2 = \frac{1}{2} \mbox{tr}\left[ \rho \frac{L_{\mu} L_{\nu} + L_{\nu} L_{\mu}}{2} \right] d \theta^{\mu} d\theta^{\nu}

where the quantum Fisher metric (tensor components) is identified as

J_{\mu \nu} = \mbox{tr}\left[ \rho \frac{L_{\mu} L_{\nu} + L_{\nu} L_{\mu}}{2}\right].

The definition of the SLD implies that the quantum Fisher metric is 4 times the Bures metric. In other words, given that $g_{\mu\nu}$ are components of the Bures metric tensor, one has

J_{\mu\nu}^{ } = 4 g_{\mu \nu}

As it happens with the classical Fisher information metric, the quantum Fisher metric can be used to find the Cramér–Rao bound of the covariance.

Explicit formulas

The actual computation of the Bures metric is not evident from the definition, so, some formulas were developed for that purpose. Dittmann obtained the following formulas for the quadratic form of the Bures metric, valid for 2x2 and 3x3 systems, respectively

[d(\rho, \rho+d\rho)]^2 = \frac{1}{4}\mbox{tr}\left[ d \rho d \rho + \frac{1}{\det(\rho)}(\mathbf{1}-\rho)d\rho (\mathbf{1}-\rho)d\rho \right]

[d(\rho, \rho+d\rho)]^2 = \frac{1}{4}\mbox{tr}\left[ d \rho d \rho + \frac{3}{1-\mbox{tr} \rho^3} (\mathbf{1}-\rho)d\rho (\mathbf{1}-\rho)d\rho + \frac{3 \det{\rho} }{1-\mbox{tr} \rho^3} (\mathbf{1}-\rho^{-1})d\rho (\mathbf{1}-\rho^{-1})d\rho \right]

Another important formula is the one found by Hübner. This formula is written in terms of the eigenvectors and eigenvalues of the density matrix and reads

[d(\rho, \rho+d\rho)]^2 = \frac{1}{2} \sum_{j,k=1}^{n} \frac{|\langle j| d\rho | k\rangle |^2}{\lambda_j+\lambda_k}.

Two-level system

The state of a two-level system can be parametrized with three variables as

\rho = \frac{1}{2}( I + \boldsymbol{r\cdot\sigma} )

with $r^2\stackrel{\mathrm{def}}{=} \boldsymbol{r\cdot r} \le 1$ . The components of the Bures metric in this parametrization can be calculated as

\mathsf{g} = \frac{\mathsf{I}}{4}+\frac{\boldsymbol{r\otimes r}}{4(1-r^2)}

The Bures measure can be calculated by taking the square root of the determinant to find

dV_B = \frac{d^3\boldsymbol{r}}{8\sqrt{ 1 - r^2}},

which can be used to calculate the Bures volume as

V_B = \iiint_{r^2\leq 1}\frac{d^3\boldsymbol{r}}{8\sqrt{1-r^2}} = \frac{\pi^2}{8}

References

↑ D. Bures, (1969) Trans. Am. Math. Soc. 135, p.199.
↑ C.W. Helstrom, (1967) "Minimum mean-squared error of estimates in quantum statistics", Phys. Lett. A 25 pp.101-102.
↑ Paolo Facchi, Ravi Kulkarni, V. I. Man'ko, Giuseppe Marmo, E. C. G. Sudarshan, Franco Ventriglia "Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics" (2010), Physics Letters A 374 pp. 4801. DOI: 10.1016/j.physleta.2010.10.005

Uhlmann, A., The Metric of Bures and the Geometric Phase, Quantum Groups and Related Topics: Proceedings of the First Max Born Symposium, Gielerak, R.; Lukierski, J.; Popowicz, Z. (Eds.), pp 267, 1992.

^ Braunstein, Samuel L. and Caves, Carlton M., Statistical distance and the geometry of quantum states, Phys. Rev. Lett.,72, 3439, 1994.

Sommers, H.J. and Zyczkowski, K., Bures volume of the set of mixed quantum states, Journal of Physics A,36, 39, 2003.

Dittmann, J., On the Riemannian Geometry of Finite Dimensional Mixed States, Seminar Sophus Lie, 73, 1993.

Dittmann J., Explicit formulae for the Bures metric, Journal of Physics A, 32, 14, 1999.

Hübner, M., Computation of Uhlmann's parallel transport for density matrices and the Bures metric on three-dimensional Hilbert space, Phys. Lett. A, 179, 4-5, 1993.

^ Paul B Slater, Applications of quantum and classical Fisher information to two-level complex and quaternionic and three-level complex systems , J. Math. Phys. 37, 2682, 1996.

Paul B Slater, Quantum Fisher-Bures information of two-level systems and a three-level extension, J. Phys. A: Math. Gen., 29, No 10, 1995.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.

^ Unfortunately, some authors use a different definition as $F(\rho_1,\rho_2) = \mbox{tr}( \sqrt{ \sqrt{\rho_1}\rho_2\sqrt{\rho_1}})$

^ Wootters WK., Statistical distance and Hilbert space, Phys. Rev. D, 23, 357 (1981)

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