944 research outputs found
Bell's Theorem from Moore's Theorem
It is shown that the restrictions of what can be inferred from
classically-recorded observational outcomes that are imposed by the no-cloning
theorem, the Kochen-Specker theorem and Bell's theorem also follow from
restrictions on inferences from observations formulated within classical
automata theory. Similarities between the assumptions underlying classical
automata theory and those underlying universally-unitary quantum theory are
discussed.Comment: 12 pages; to appear in Int. J. General System
Extremal covariant quantum operations and POVM's
We consider the convex sets of QO's (quantum operations) and POVM's (positive
operator valued measures) which are covariant under a general
finite-dimensional unitary representation of a group. We derive necessary and
sufficient conditions for extremality, and give general bounds for ranks of the
extremal POVM's and QO's. Results are illustrated on the basis of simple
examples.Comment: 18 pages, to appear on J. Math. Phy
Quantum Cloning and Distributed Measurements
We study measurements on various subsystems of the output of a universal 1 to
2 cloning machine, and establish a correspondence between these measurements at
the output and effective measurements on the original input. We show that one
can implement sharp effective measurement elements by measuring only two out of
the three output systems. Additionally, certain complete sets of sharp
measurements on the input can be realised by measurements on the two clones.
Furthermore, we introduce a scheme that allows to restore the original input in
one of the output bits, by using measurements and classical communication -- a
protocol that resembles teleportation.Comment: submitted to Phys. Rev.
Tight informationally complete quantum measurements
We introduce a class of informationally complete positive-operator-valued
measures which are, in analogy with a tight frame, "as close as possible" to
orthonormal bases for the space of quantum states. These measures are
distinguished by an exceptionally simple state-reconstruction formula which
allows "painless" quantum state tomography. Complete sets of mutually unbiased
bases and symmetric informationally complete positive-operator-valued measures
are both members of this class, the latter being the unique minimal rank-one
members. Recast as ensembles of pure quantum states, the rank-one members are
in fact equivalent to weighted 2-designs in complex projective space. These
measures are shown to be optimal for quantum cloning and linear quantum state
tomography.Comment: 20 pages. Final versio
Compatibility of quantum measurements and inclusion constants for the matrix jewel
In this work, we establish the connection between the study of free
spectrahedra and the compatibility of quantum measurements with an arbitrary
number of outcomes. This generalizes previous results by the authors for
measurements with two outcomes. Free spectrahedra arise from matricial
relaxations of linear matrix inequalities. A particular free spectrahedron
which we define in this work is the matrix jewel. We find that the
compatibility of arbitrary measurements corresponds to the inclusion of the
matrix jewel into a free spectrahedron defined by the effect operators of the
measurements under study. We subsequently use this connection to bound the set
of (asymmetric) inclusion constants for the matrix jewel using results from
quantum information theory and symmetrization. The latter translate to new
lower bounds on the compatibility of quantum measurements. Among the techniques
we employ are approximate quantum cloning and mutually unbiased bases.Comment: v5: section 3.3 has been expanded significantly to incorporate the
generalization of the Cartesian product and the direct sum to matrix convex
sets. Many other minor modifications. Closed to the published versio
Quantum state restoration and single-copy tomography
Given a single copy of an n qubit quantum state |psi>, the no-cloning theorem
greatly limits the amount of information which can be extracted from it.
Moreover, given only a procedure which verifies the state, for example a
procedure which measures the operator |psi> in
time polynomial in n . In this paper, we consider the scenario in which we are
given both a single copy of |psi> and the ability to verify it. We show that in
this setting, we can do several novel things efficiently. We present a new
algorithm that we call quantum state restoration which allows us to extend a
large subsystem of |psi> to the full state, and in turn this allows us to copy
small subsystems of |psi>. In addition, we present algorithms that can perform
tomography on small subsystems of |psi>, and we show how to use these
algorithms to estimate the statistics of any efficiently implementable POVM
acting on |psi> in time polynomial in the number of outcomes of the POVM.Comment: edited for clarity; 13 pages, 1 figur
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