How to measure the fidelity between two quantum states?

KTH/Nordita/SU seminar in Theoretical Physics [before December 2013]

Wednesday 17 December 2008
from 11:00
to 12:00 at
FA31

Speaker :

Karol Zyczkowski (Krakow)

Abstract :

Quantum information processing is based on controlled transformations
of quantum states.
Assume that one laboratory designed a technique to produce quantum states
in a given state $\rho$. The other lab wants to generate exactly the
same state
and they produce states $\sigma$. If we want to know how well the
second lab is doing
we need to characterize the distance between $\sigma$ and $\rho$ by
some means,
e.g. by trying to measure their fidelity, which allows us to find the
Bures distance
between them.
The task is simple if the given state is pure, $\rho=|\psi\rangle
\langle \psi|$,
since then fidelity reduces to the expectation value,
$F=\langle\psi|\sigma | \psi\rangle$.
If $\rho$ is mixed the explicit formula for fidelity contains
the trace of an absolute value of an operator which is not simple to
compute nor to measure.
Therefore we provide lower and upper bounds for fidelity and propose schemes
to measure them. These experimental schemes
require much less effort than the full quantum tomography of both
states in question.
The bounds for fidelity are called {\sl sub-} and {\sl
super-fidelity}, respectively,
due to their properties: as fidelity is multiplicative with respect to
the tensor product,
the sub-fidelity is sub-multiplicative, while super-fidelity is shown
to be super-multiplicative.
In the case of any two states of a one qubit system the bounds are
strict and all three
quantities coincide. The super-fidelity allows us to define a modified
Bures distance which for larger systems induces an alternative geometry
of the space of quantum states.