Bures metric

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In mathematics, in the area of quantum information geometry, the Bures metric (named after Donald Bures)[1] or Helstrom metric (named after Carl W. Helstrom)[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.


The metric may be defined as

[clarification needed]

where is Hermitian 1-form operator implicitly given by

which is a special case of a continuous Lyapunov equation.

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[4] and the use of the volume element as a candidate for the Jeffreys prior probability density[5] for mixed quantum states.

Bures distance[edit]

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

where the fidelity function is defined as[6]

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

which is a measure of the statistical distance[7] between quantum states.

Quantum Fisher information[edit]

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

which holds as long as and have the same rank. In cases where they do not have the same rank, there is an additional term on the right hand side.[8] is the Symmetric Logarithmic Derivative operator (SLD) defined from[9]

In this way, one has

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

The definition of the SLD implies that the quantum Fisher metric is 4 times the Bures metric. In other words, given that are components of the Bures metric tensor, one has

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[edit]

The actual computation of the Bures metric is not evident from the definition, so, some formulas were developed for that purpose. For 2x2 and 3x3 systems, respectively, the quadratic form of the Bures metric is calculated as[10]

For general systems, the Bures metric can be written in terms of the eigenvectors and eigenvalues of the density matrix as[11][12]

as an integral,[13]

or in terms of Kronecker product and vectorization,[14]

where the overbar denotes complex conjugate, and denotes conjugate transpose.

Two-level system[edit]

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

where is the vector of Pauli matrices and is the (three-dimensional) Bloch vector satisfying . The components of the Bures metric in this parametrization can be calculated as


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

which can be used to calculate the Bures volume as

Three-level system[edit]

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

where are the eight Gell-Mann matrices and the 8-dimensional Bloch vector satisfying certain constraints.

See also[edit]


  1. ^ Bures, Donald (1969). "An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite *-algebras" (PDF). Transactions of the American Mathematical Society. American Mathematical Society (AMS). 135: 199. doi:10.1090/s0002-9947-1969-0236719-2. ISSN 0002-9947.
  2. ^ Helstrom, C.W. (1967). "Minimum mean-squared error of estimates in quantum statistics". Physics Letters A. Elsevier BV. 25 (2): 101–102. doi:10.1016/0375-9601(67)90366-0. ISSN 0375-9601.
  3. ^ Facchi, Paolo; Kulkarni, Ravi; Man'ko, V.I.; Marmo, Giuseppe; Sudarshan, E.C.G.; Ventriglia, Franco (2010). "Classical and quantum Fisher information in the geometrical formulation of quantum mechanics". Physics Letters A. 374 (48): 4801–4803. arXiv:1009.5219. doi:10.1016/j.physleta.2010.10.005. ISSN 0375-9601.
  4. ^ Braunstein, Samuel L.; Caves, Carlton M. (1994-05-30). "Statistical distance and the geometry of quantum states". Physical Review Letters. American Physical Society (APS). 72 (22): 3439–3443. doi:10.1103/physrevlett.72.3439. ISSN 0031-9007.
  5. ^ Slater, Paul B. (1996). "Applications of quantum and classical Fisher information to two‐level complex and quaternionic and three‐level complex systems". Journal of Mathematical Physics. AIP Publishing. 37 (6): 2682–2693. doi:10.1063/1.531528. ISSN 0022-2488.
  6. ^ Unfortunately, some authors use a different definition,
  7. ^ Wootters, W. K. (1981-01-15). "Statistical distance and Hilbert space". Physical Review D. American Physical Society (APS). 23 (2): 357–362. doi:10.1103/physrevd.23.357. ISSN 0556-2821.
  8. ^ Šafránek, Dominik (2017-05-11). "Discontinuities of the quantum Fisher information and the Bures metric". Physical Review A. American Physical Society (APS). 95 (5): 052320. arXiv:1612.04581. doi:10.1103/physreva.95.052320. ISSN 2469-9926.
  9. ^ Paris, Matteo G. A. (2009). "Quantum estimation for quantum technology". International Journal of Quantum Information. World Scientific Pub Co Pte Lt. 07 (supp01): 125–137. arXiv:0804.2981. doi:10.1142/s0219749909004839. ISSN 0219-7499.
  10. ^ Dittmann, J (1999-01-01). "Explicit formulae for the Bures metric". Journal of Physics A: Mathematical and General. IOP Publishing. 32 (14): 2663–2670. arXiv:quant-ph/9808044. doi:10.1088/0305-4470/32/14/007. ISSN 0305-4470.
  11. ^ Hübner, Matthias (1992). "Explicit computation of the Bures distance for density matrices". Physics Letters A. Elsevier BV. 163 (4): 239–242. doi:10.1016/0375-9601(92)91004-b. ISSN 0375-9601.
  12. ^ Hübner, Matthias (1993). "Computation of Uhlmann's parallel transport for density matrices and the Bures metric on three-dimensional Hilbert space". Physics Letters A. Elsevier BV. 179 (4–5): 226–230. doi:10.1016/0375-9601(93)90668-p. ISSN 0375-9601.
  13. ^ PARIS, MATTEO G. A. (2009). "Quantum estimation for quantum technology". International Journal of Quantum Information. World Scientific Pub Co Pte Lt. 07 (supp01): 125–137. arXiv:0804.2981. doi:10.1142/s0219749909004839. ISSN 0219-7499.
  14. ^ Šafránek, Dominik (2018-04-12). "Simple expression for the quantum Fisher information matrix". Physical Review A. American Physical Society (APS). 97 (4): 042322. arXiv:1801.00945. doi:10.1103/physreva.97.042322. ISSN 2469-9926.

Further reading[edit]