15 research outputs found
A Geometric Approach to Quantum State Separation
Probabilistic quantum state transformations can be characterized by the
degree of state separation they provide. This, in turn, sets limits on the
success rate of these transformations. We consider optimum state separation of
two known pure states in the general case where the known states have arbitrary
a priori probabilities. The problem is formulated from a geometric perspective
and shown to be equivalent to the problem of finding tangent curves within two
families of conics that represent the unitarity constraints and the objective
functions to be optimized, respectively. We present the corresponding
analytical solutions in various forms. In the limit of perfect state
separation, which is equivalent to unambiguous state discrimination, the
solution exhibits a phenomenon analogous to a second order symmetry breaking
phase transition. We also propose a linear optics implementation of separation
which is based on the dual rail representation of qubits and single-photon
multiport interferometry
Quantum change point
Sudden changes are ubiquitous in nature. Identifying them is of crucial
importance for a number of applications in medicine, biology, geophysics, and
social sciences. Here we investigate the problem in the quantum domain,
considering a source that emits particles in a default state, until a point
where it switches to another state. Given a sequence of particles emitted by
the source, the problem is to find out where the change occurred. For large
sequences, we obtain an analytical expression for the maximum probability of
correctly identifying the change point when joint measurements on the whole
sequence are allowed. We also construct strategies that measure the particles
individually and provide an online answer as soon as a new particle is emitted
by the source. We show that these strategies substantially underperform the
optimal strategy, indicating that quantum sudden changes, although happening
locally, are better detected globally.Comment: 4+8 pages, published version. New results added, including a theorem
applicable to general multihypothesis discrimination problem
Beating noise with abstention in state estimation
We address the problem of estimating pure qubit states with non-ideal (noisy)
measurements in the multiple-copy scenario, where the data consists of a number
N of identically prepared qubits. We show that the average fidelity of the
estimates can increase significantly if the estimation protocol allows for
inconclusive answers, or abstentions. We present the optimal such protocol and
compute its fidelity for a given probability of abstention. The improvement
over standard estimation, without abstention, can be viewed as an effective
noise reduction. These and other results are exemplified for small values of N.
For asymptotically large N, we derive analytical expressions of the fidelity
and the probability of abstention, and show that for a fixed fidelity gain the
latter decreases with N at an exponential rate given by a Kulback-Leibler
(relative) entropy. As a byproduct, we obtain an asymptotic expression in terms
of this very entropy of the probability that a system of N qubits, all prepared
in the same state, has a given total angular momentum. We also discuss an
extreme situation where noise increases with N and where estimation with
abstention provides a most significant improvement as compared to the standard
approach
Quantum-limited metrology with product states
We study the performance of initial product states of n-body systems in
generalized quantum metrology protocols that involve estimating an unknown
coupling constant in a nonlinear k-body (k << n) Hamiltonian. We obtain the
theoretical lower bound on the uncertainty in the estimate of the parameter.
For arbitrary initial states, the lower bound scales as 1/n^k, and for initial
product states, it scales as 1/n^(k-1/2). We show that the latter scaling can
be achieved using simple, separable measurements. We analyze in detail the case
of a quadratic Hamiltonian (k = 2), implementable with Bose-Einstein
condensates. We formulate a simple model, based on the evolution of
angular-momentum coherent states, which explains the O(n^(-3/2)) scaling for k
= 2; the model shows that the entanglement generated by the quadratic
Hamiltonian does not play a role in the enhanced sensitivity scaling. We show
that phase decoherence does not affect the O(n^(-3/2)) sensitivity scaling for
initial product states.Comment: 15 pages, 6 figure