117 research outputs found
Extremal covariant POVM's
We consider the convex set of positive operator valued measures (POVM) which
are covariant under a finite dimensional unitary projective representation of a
group. We derive a general characterization for the extremal points, and
provide bounds for the ranks of the corresponding POVM densities, also relating
extremality to uniqueness and stability of optimized measurements. Examples of
applications are given.Comment: 15 pages, no figure
Quantum information becomes classical when distributed to many users
Any physical transformation that equally distributes quantum information over
a large number M of users can be approximated by a classical broadcasting of
measurement outcomes. The accuracy of the approximation is at least of the
order 1/M. In particular, quantum cloning of pure and mixed states can be
approximated via quantum state estimation. As an example, for optimal qubit
cloning with 10 output copies, a single user has error probability p > 0.45 in
distinguishing classical from quantum output--a value close to the error
probability of the random guess.Comment: 4 pages, no figures, published versio
Extremal covariant measurements
We characterize the extremal points of the convex set of quantum measurements
that are covariant under a finite-dimensional projective representation of a
compact group, with action of the group on the measurement probability space
which is generally non-transitive. In this case the POVM density is made of
multiple orbits of positive operators, and, in the case of extremal
measurements, we provide a bound for the number of orbits and for the rank of
POVM elements. Two relevant applications are considered, concerning state
discrimination with mutually unbiased bases and the maximization of the mutual
information.Comment: 11 pages, no figure
How continuous quantum measurements in finite dimension are actually discrete
We show that in finite dimension a quantum measurement with continuous set of
outcomes is always equivalent to a continuous random choice of measurements
with only finite outcomes.Comment: 4 pages, 1 figur
On defining the Hamiltonian beyond quantum theory
Energy is a crucial concept within classical and quantum physics. An
essential tool to quantify energy is the Hamiltonian. Here, we consider how to
define a Hamiltonian in general probabilistic theories, a framework in which
quantum theory is a special case. We list desiderata which the definition
should meet. For 3-dimensional systems, we provide a fully-defined recipe which
satisfies these desiderata. We discuss the higher dimensional case where some
freedom of choice is left remaining. We apply the definition to example toy
theories, and discuss how the quantum notion of time evolution as a phase
between energy eigenstates generalises to other theories.Comment: Authors' accepted manuscript for inclusion in the Foundations of
Physics topical collection on Foundational Aspects of Quantum Informatio
Optimal cloning of unitary transformations
After proving a general no-cloning theorem for black boxes, we derive the
optimal universal cloning of unitary transformations, from one to two copies.
The optimal cloner is realized by quantum channels with memory, and greately
outperforms the optimal measure-and-reprepare cloning strategy. Applications
are outlined, including two-way quantum cryptographic protocols.Comment: 4 pages, 1 figure, published versio
Quantum Circuits for the Unitary Permutation Problem
We consider the Unitary Permutation problem which consists, given unitary
gates and a permutation of , in
applying the unitary gates in the order specified by , i.e. in
performing . This problem has been
introduced and investigated by Colnaghi et al. where two models of computations
are considered. This first is the (standard) model of query complexity: the
complexity measure is the number of calls to any of the unitary gates in
a quantum circuit which solves the problem. The second model provides quantum
switches and treats unitary transformations as inputs of second order. In that
case the complexity measure is the number of quantum switches. In their paper,
Colnaghi et al. have shown that the problem can be solved within calls in
the query model and quantum switches in the new model. We
refine these results by proving that quantum switches
are necessary and sufficient to solve this problem, whereas calls
are sufficient to solve this problem in the standard quantum circuit model. We
prove, with an additional assumption on the family of gates used in the
circuits, that queries are required, for any
. The upper and lower bounds for the standard quantum circuit
model are established by pointing out connections with the permutation as
substring problem introduced by Karp.Comment: 8 pages, 5 figure
Quantum thermodynamics with missing reference frames: Decompositions of free energy into non-increasing components
If an absolute reference frame with respect to time, position, or orientation
is missing one can only implement quantum operations which are covariant with
respect to the corresponding unitary symmetry group G. Extending observations
of Vaccaro et al., I argue that the free energy of a quantum system with
G-invariant Hamiltonian then splits up into the Holevo information of the orbit
of the state under the action of G and the free energy of its orbit average.
These two kinds of free energy cannot be converted into each other. The first
component is subadditive and the second superadditive; in the limit of
infinitely many copies only the usual free energy matters.
Refined splittings of free energy into more than two independent
(non-increasing) terms can be defined by averaging over probability measures on
G that differ from the Haar measure.
Even in the presence of a reference frame, these results provide lower bounds
on the amount of free energy that is lost after applying a covariant channel.
If the channel properly decreases one of these quantities, it decreases the
free energy necessarily at least by the same amount, since it is unable to
convert the different forms of free energies into each other.Comment: 17 pages, latex, 1 figur
Optimal estimation of group transformations using entanglement
We derive the optimal input states and the optimal quantum measurements for
estimating the unitary action of a given symmetry group, showing how the
optimal performance is obtained with a suitable use of entanglement. Optimality
is defined in a Bayesian sense, as minimization of the average value of a given
cost function. We introduce a class of cost functions that generalizes the
Holevo class for phase estimation, and show that for states of the optimal form
all functions in such a class lead to the same optimal measurement. A first
application of the main result is the complete proof of the optimal efficiency
in the transmission of a Cartesian reference frame. As a second application, we
derive the optimal estimation of a completely unknown two-qubit maximally
entangled state, provided that N copies of the state are available. In the
limit of large N, the fidelity of the optimal estimation is shown to be
1-3/(4N).Comment: 11 pages, no figure
Extremal quantum cloning machines
We investigate the problem of cloning a set of states that is invariant under
the action of an irreducible group representation. We then characterize the
cloners that are "extremal" in the convex set of group covariant cloning
machines, among which one can restrict the search for optimal cloners. For a
set of states that is invariant under the discrete Weyl-Heisenberg group, we
show that all extremal cloners can be unitarily realized using the so-called
"double-Bell states", whence providing a general proof of the popular ansatz
used in the literature for finding optimal cloners in a variety of settings.
Our result can also be generalized to continuous-variable optimal cloning in
infinite dimensions, where the covariance group is the customary
Weyl-Heisenberg group of displacements.Comment: revised version accepted for publicatio
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