8,966 research outputs found
Invariant and polynomial identities for higher rank matrices
We exhibit explicit expressions, in terms of components, of discriminants,
determinants, characteristic polynomials and polynomial identities for matrices
of higher rank. We define permutation tensors and in term of them we construct
discriminants and the determinant as the discriminant of order , where
is the dimension of the matrix. The characteristic polynomials and the
Cayley--Hamilton theorem for higher rank matrices are obtained there from
Measuring the purity of a qubit state: entanglement estimation with fully separable measurements
Given a finite number of copies of a qubit state we compute the maximum
fidelity that can be attained using joint-measurement protocols for estimating
its purity. We prove that in the asymptotic limit,
separable-measurement protocols can be as efficient as the optimal
joint-measurement one if classical communication is used. This in turn shows
that the optimal estimation of the entanglement of a two-qubit state can also
be achieved asymptotically with fully separable measurements. The relationship
between our global Bayesian approach and the quantum Cramer-Rao bound is also
discussed.Comment: 5 pages, 1 figure, RevTeX, improved versio
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 reverse-engineering and reference frame alignment without non-local correlations
Estimation of unknown qubit elementary gates and alignment of reference
frames are formally the same problem. Using quantum states made out of
qubits, we show that the theoretical precision limit for both problems, which
behaves as , can be asymptotically attained with a covariant protocol
that exploits the quantum correlation of internal degrees of freedom instead of
the more fragile entanglement between distant parties. This cuts by half the
number of qubits needed to achieve the precision of the dense covariant coding
protocol
Entanglement assisted alignment of reference frames using a dense covariant coding
We present a procedure inspired by dense coding, which enables a highly
efficient transmission of information of a continuous nature. The procedure
requires the sender and the recipient to share a maximally entangled state. We
deal with the concrete problem of aligning reference frames or trihedra by
means of a quantum system. We find the optimal covariant measurement and
compute the corresponding average error, which has a remarkably simple close
form. The connection of this procedure with that of estimating unitary
transformations on qubits is briefly discussed.Comment: 4 pages, RevTeX, Version to appear in PR
Optimal full estimation of qubit mixed states
We obtain the optimal scheme for estimating unknown qubit mixed states when
an arbitrary number N of identically prepared copies is available. We discuss
the case of states in the whole Bloch sphere as well as the restricted
situation where these states are known to lie on the equatorial plane. For the
former case we obtain that the optimal measurement does not depend on the prior
probability distribution provided it is isotropic. Although the
equatorial-plane case does not have this property for arbitrary N, we give a
prior-independent scheme which becomes optimal in the asymptotic limit of large
N. We compute the maximum mean fidelity in this asymptotic regime for the two
cases. We show that within the pointwise estimation approach these limits can
be obtained in a rather easy and rapid way. This derivation is based on
heuristic arguments that are made rigorous by using van Trees inequalities. The
interrelation between the estimation of the purity and the direction of the
state is also discussed. In the general case we show that they correspond to
independent estimations whereas for the equatorial-plane states this is only
true asymptotically.Comment: 19 pages, no figure
Phase-Covariant Quantum Benchmarks
We give a quantum benchmark for teleportation and quantum storage experiments
suited for pure and mixed test states. The benchmark is based on the average
fidelity over a family of phase-covariant states and certifies that an
experiment can not be emulated by a classical setup, i.e., by a
measure-and-prepare scheme. We give an analytical solution for qubits, which
shows important differences with standard state estimation approach, and
compute the value of the benchmark for coherent and squeezed states, both pure
and mixed.Comment: 4 pages, 2 figure
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