37 research outputs found
On the curvature of the quantum state space with pull-back metrics
The aim of the paper is to extend the notion of -geometry in the
classical and in the noncommutative case by introducing a more general class of
pull-back metrics and to give concrete formulas for the scalar curvature of
these Riemannian manifolds. We introduce a more general class of pull-back
metrics of the noncommutative state spaces, we pull back the Euclidean
Riemannian metric of the space of self-adjoint matrices with functions which
have an analytic extension to a neighborhood of the interval and whose
derivative are nowhere zero. We compute the scalar curvature in this setting,
and as a corollary we have the scalar curvature of the classical probability
space when it is endowed with such a general pull-back metric. In the
noncommutative setting we consider real and complex state spaces too. We give a
simplification of Gibilisco's and Isola's conjecture for the first nontrivial
classical probability space and we present the result of a numerical
computation which indicate that the conjecture may be true for the space of
real and complex qubits.Comment: 18 page
Invariance of separability probability over reduced states in 4x4 bipartite systems
The geometric separability probability of composite quantum systems is
extensively studied in the last decades. One of most simple but strikingly
difficult problem is to compute the separability probability of qubit-qubit and
rebit-rebit quantum states with respect to the Hilbert-Schmidt measure. A lot
of numerical simulations confirm the P(rebit-rebit)=29/64 and
P(qubit-qubit)=8/33 conjectured probabilities. Milz and Strunz studied the
separability probability with respect to given subsystems. They conjectured
that the separability probability of qubit-qubit (and qubit-qutrit) depends on
sum of single qubit subsystems (D), moreover it depends just on the Bloch radii
(r) of D and it is constant in r. Using the Peres-Horodecki criterion for
separability we give mathematical proof for the P(rebit-rebit)=29/64
probability and we present an integral formula for the complex case which
hopefully will help to prove the P(qubit-qubit)=8/33 probability too. We prove
Milz and Strunz's conjecture for rebit-rebit and qubit-qubit states. The case,
when the state space is endowed with the volume form generated by the operator
monotone function f(x)=sqrt(x) is studied too in detail. We show, that even in
this setting the Milz and Strunz's conjecture holds and we give an integral
formula for separability probability according to this measure.Comment: 24 pages, 1 figur