Symmetric logarithmic derivative explained

The symmetric logarithmic derivative is an important quantity in quantum metrology, and is related to the quantum Fisher information.

Definition

Let

\rho and A be two operators, where \rho is Hermitian and positive semi-definite. In most applications, \rho and A fulfill further properties, that also A is Hermitian and \rho is a density matrix (which is also trace-normalized), but these are not required for the definition.

The symmetric logarithmic derivative

L _{\varrho(A)} is defined implicitly by the equation^{[1] [2]} i[\varrho,A] = (1)/(2) \{\varrho, L _{\varrho(A)\}}

where

[X,Y] = XY - YX is the commutator and \{X,Y\} = XY + YX is the anticommutator. Explicitly, it is given by^[3] L _{\varrho(A)=2i\sum} _{k,l} (λ _{k-λ} _{l})/(λ _{k+λ} _{l}) \langle k \vert A \vert l\rangle \vert k\rangle \langle l \vert

where

λ _{k} and \vert k\rangle are the eigenvalues and eigenstates of \varrho , i.e. \varrho\vert k\rangle =λ _{k\vert} k\rangle and \varrho = \sum _{k} λ _{k} \vert k\rangle\langle k\vert .

Formally, the map from operator

A to operator L _{\varrho(A)} is a (linear) superoperator.

Properties

The symmetric logarithmic derivative is linear in

A :L _{\varrho(\mu} A) = \mu L _{\varrho(A)} L _{\varrho(A+B)=L} _{\varrho(A)+L} _{\varrho(B)}

The symmetric logarithmic derivative is Hermitian if its argument

A is Hermitian:A = A ^{\dagger = L} _{\varrho(A)}

The derivative of the expression

\exp( - i\theta A)\varrho\exp( + i\theta A) w.r.t. \theta at \theta =0 reads(\partial)/(\partial\theta) [ \exp( - i\theta A)\varrho\exp( + i\theta A) ] \ vert _{\theta=0} = i(\varrho A - A\varrho) = i[\varrho,A] = (1)/(2) \{\varrho, L _{\varrho(A)\}} where the last equality is per definition of L _{\varrho(A)} ; this relation is the origin of the name "symmetric logarithmic derivative". Further, we obtain the Taylor expansion\exp( - i\theta A)\varrho\exp( + i\theta A) = \varrho + \underbrace{ (1)/(2) \theta\{\varrho, L _{\varrho(A)\}}} _{=i\theta[\varrho,A]} + l{O}(\theta ^{2)} .

Notes and References

1. Braunstein . Samuel L. . Caves . Carlton M. . Carlton Caves . Statistical distance and the geometry of quantum states . . American Physical Society (APS) . 72 . 22 . 1994-05-30 . 0031-9007 . 10.1103/physrevlett.72.3439 . 10056200 . 3439–3443. 1994PhRvL..72.3439B .
2. Braunstein . Samuel L. . Caves . Carlton M. . Milburn . G.J. . Carlton Caves . Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance . . April 1996 . 247 . 1 . 135–173 . quant-ph/9507004 . 10.1006/aphy.1996.0040. 1996AnPhy.247..135B . 358923 .
3. Paris . Matteo G. A. . Quantum Estimation for Quantum Technology. . 21 November 2011 . 07 . supp01 . 125–137 . 10.1142/S0219749909004839. 0804.2981 . 2365312 .