Bures metric

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.

Definition

The Bures metric may be defined as

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

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

\rho G+G\rho =d\rho ,

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

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^[6]

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^[7] between quantum states.

Wootters distance

When both density operators are diagonal (so that they are just classical probability distributions), then let $\rho _{1}=diag(p_{1},...)$ and similarly $\rho _{2}=diag(q_{1},...)$ , then the fidelity is ${\sqrt {F}}=\sum _{i}{\sqrt {p_{i}q_{i}}}$ with the Bures length becoming the Wootters distance $\arccos \left(\sum _{i}{\sqrt {p_{i}q_{i}}}\right)$ . The Wootters distance is the geodesic distance between the probability distributions $p,q$ under the chi-squared metric $ds^{2}={\frac {1}{2}}\sum _{i}{\frac {dp_{i}^{2}}{p_{i}}}$ .^[8]

Perform a change of variables with $x_{i}:={\sqrt {p_{i}}}$ , then the chi-squared metric becomes $ds^{2}=\sum _{i}dx_{i}^{2}$ . Since $\sum _{i}x_{i}^{2}=\sum _{i}p_{i}=1$ , the points $x$ are restricted to move on the positive quadrant of a unit hypersphere. So, the geodesics are just the great circles on the hypersphere, and we also obtain the Wootters distance formula.

If both density operators are pure states, $\psi ,\phi$ , then the fidelity is ${\sqrt {F}}=|\langle \psi |\phi \rangle |$ , and we obtain the quantum version of Wootters distance

$\arccos(|\langle \psi |\phi \rangle |)$ .^[9]

In particular, the direct Bures distance between any two orthogonal states is ${\sqrt {2}}$ , while the Bures distance summed along the geodesic path connecting them is $\pi /2$ .

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_{B}(\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 },

which holds as long as $\rho$ and $\rho +d\rho$ have the same rank. In cases where they do not have the same rank, there is an additional term on the right hand side.^[10]^[11] $L_{\mu }$ is the Symmetric logarithmic derivative operator (SLD) defined from^[12]

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

In this way, one has

[D_{B}(\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 }^{}=4g_{\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. For 2x2 and 3x3 systems, respectively, the quadratic form of the Bures metric is calculated as^[13]

[D_{B}(\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_{B}(\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].

For general systems, the Bures metric can be written in terms of the eigenvectors and eigenvalues of the density matrix $\rho =\sum _{j=1}^{n}\lambda _{j}|j\rangle \langle j|$ as^[14]^[15]

[D_{B}(\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}}},

as an integral,^[16]

[D_{B}(\rho ,\rho +d\rho )]^{2}={\frac {1}{2}}\int _{0}^{\infty }{\text{tr}}[e^{-\rho t}d\rho e^{-\rho t}d\rho ]\ dt,

or in terms of Kronecker product and vectorization,^[17]

[D_{B}(\rho ,\rho +d\rho )]^{2}={\frac {1}{2}}{\text{vec}}[d\rho ]^{\dagger }{\big (}\rho ^{*}\otimes \mathbf {1} +\mathbf {1} \otimes \rho {\big )}^{-1}{\text{vec}}[d\rho ],

where $^{*}$ denotes complex conjugate, and $^{\dagger }$ denotes conjugate transpose. This formula holds for invertible density matrices. For non-invertible density matrices, the inverse above is substituted by the Moore-Penrose pseudoinverse. Alternatively, the expression can be also computed by performing a limit on a certain mixed and thus invertible state.

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 }}),

where ${\boldsymbol {\sigma }}$ is the vector of Pauli matrices and ${\boldsymbol {r}}$ is the (three-dimensional) Bloch vector satisfying $r^{2}{\stackrel {\mathrm {def} }{=}}{\boldsymbol {r\cdot r}}\leq 1$ . 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}}.

Three-level system

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

\rho ={\frac {1}{3}}(I+{\sqrt {3}}\sum _{\nu =1}^{8}\xi _{\nu }\lambda _{\nu }),

where $\lambda _{\nu }$ are the eight Gell-Mann matrices and ${\boldsymbol {\xi }}\in \mathbb {R} ^{8}$ the 8-dimensional Bloch vector satisfying certain constraints.

References

^ Bures, Donald (1969). "An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite $\omega$ *-algebras" (PDF). Transactions of the American Mathematical Society. 135. American Mathematical Society (AMS): 199. doi:10.1090/s0002-9947-1969-0236719-2. ISSN 0002-9947.
^ Helstrom, C.W. (1967). "Minimum mean-squared error of estimates in quantum statistics". Physics Letters A. 25 (2). Elsevier BV: 101–102. Bibcode:1967PhLA...25..101H. doi:10.1016/0375-9601(67)90366-0. ISSN 0375-9601.
^ 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. Bibcode:2010PhLA..374.4801F. doi:10.1016/j.physleta.2010.10.005. ISSN 0375-9601. S2CID 55558124.
^ Braunstein, Samuel L.; Caves, Carlton M. (1994-05-30). "Statistical distance and the geometry of quantum states". Physical Review Letters. 72 (22). American Physical Society (APS): 3439–3443. Bibcode:1994PhRvL..72.3439B. doi:10.1103/physrevlett.72.3439. ISSN 0031-9007. PMID 10056200.
^ 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. 37 (6). AIP Publishing: 2682–2693. Bibcode:1996JMP....37.2682S. doi:10.1063/1.531528. ISSN 0022-2488.
^ Some authors might instead use a different definition, $F(\rho _{1},\rho _{2})={\mbox{tr}}({\sqrt {{\sqrt {\rho _{1}}}\rho _{2}{\sqrt {\rho _{1}}}}})$
^ Wootters, W. K. (1981-01-15). "Statistical distance and Hilbert space". Physical Review D. 23 (2). American Physical Society (APS): 357–362. Bibcode:1981PhRvD..23..357W. doi:10.1103/physrevd.23.357. ISSN 0556-2821.
^ Braunstein, Samuel L.; Caves, Carlton M. (1994-05-30). "Statistical distance and the geometry of quantum states". Physical Review Letters. 72 (22): 3439–3443. Bibcode:1994PhRvL..72.3439B. doi:10.1103/PhysRevLett.72.3439. PMID 10056200.
^ Deffner, Sebastian; Campbell, Steve (2017-11-10). "Quantum speed limits: from Heisenberg's uncertainty principle to optimal quantum control". Journal of Physics A: Mathematical and Theoretical. 50 (45): 453001. arXiv:1705.08023. Bibcode:2017JPhA...50S3001D. doi:10.1088/1751-8121/aa86c6. hdl:11603/19391. ISSN 1751-8113. S2CID 3477317.
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^ Rezakhani, A. T.; Hassani, M.; Alipour, S. (2019-09-12). "Continuity of the quantum Fisher information". Physical Review A. 100 (3): 032317. arXiv:1507.01736. Bibcode:2019PhRvA.100c2317R. doi:10.1103/PhysRevA.100.032317. S2CID 51680508.
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^ Dittmann, J (1999-01-01). "Explicit formulae for the Bures metric". Journal of Physics A: Mathematical and General. 32 (14): 2663–2670. arXiv:quant-ph/9808044. Bibcode:1999JPhA...32.2663D. doi:10.1088/0305-4470/32/14/007. ISSN 0305-4470. S2CID 18298901.
^ Hübner, Matthias (1992). "Explicit computation of the Bures distance for density matrices". Physics Letters A. 163 (4). Elsevier BV: 239–242. Bibcode:1992PhLA..163..239H. doi:10.1016/0375-9601(92)91004-b. ISSN 0375-9601.
^ Hübner, Matthias (1993). "Computation of Uhlmann's parallel transport for density matrices and the Bures metric on three-dimensional Hilbert space". Physics Letters A. 179 (4–5). Elsevier BV: 226–230. Bibcode:1993PhLA..179..226H. doi:10.1016/0375-9601(93)90668-p. ISSN 0375-9601.
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