43 research outputs found
Weak measurements are universal
It is well known that any projective measurement can be decomposed into a
sequence of weak measurements, which cause only small changes to the state.
Similar constructions for generalized measurements, however, have relied on the
use of an ancilla system. We show that any generalized measurement can be
decomposed into a sequence of weak measurements without the use of an ancilla,
and give an explicit construction for these weak measurements. The measurement
procedure has the structure of a random walk along a curve in state space, with
the measurement ending when one of the end points is reached. This shows that
any measurement can be generated by weak measurements, and hence that weak
measurements are universal. This may have important applications to the theory
of entanglement.Comment: 4 pages, RevTeX format, essentially the published version, reference
update
Non-Markovian dynamics of a qubit coupled to an Ising spin bath
We study the analytically solvable Ising model of a single qubit system
coupled to a spin bath. The purpose of this study is to analyze and elucidate
the performance of Markovian and non-Markovian master equations describing the
dynamics of the system qubit, in comparison to the exact solution. We find that
the time-convolutionless master equation performs particularly well up to
fourth order in the system-bath coupling constant, in comparison to the
Nakajima-Zwanzig master equation. Markovian approaches fare poorly due to the
infinite bath correlation time in this model. A recently proposed
post-Markovian master equation performs comparably to the time-convolutionless
master equation for a properly chosen memory kernel, and outperforms all the
approximation methods considered here at long times. Our findings shed light on
the applicability of master equations to the description of reduced system
dynamics in the presence of spin-baths.Comment: 17 pages, 16 figure
Causal structures and the classification of higher order quantum computations
Quantum operations are the most widely used tool in the theory of quantum
information processing, representing elementary transformations of quantum
states that are composed to form complex quantum circuits. The class of quantum
transformations can be extended by including transformations on quantum
operations, and transformations thereof, and so on up to the construction of a
potentially infinite hierarchy of transformations. In the last decade, a
sub-hierarchy, known as quantum combs, was exhaustively studied, and
characterised as the most general class of transformations that can be achieved
by quantum circuits with open slots hosting variable input elements, to form a
complete output quantum circuit. The theory of quantum combs proved to be
successful for the optimisation of information processing tasks otherwise
untreatable. In more recent years the study of maps from combs to combs has
increased, thanks to interesting examples showing how this next order of maps
requires entanglement of the causal order of operations with the state of a
control quantum system, or, even more radically, superpositions of alternate
causal orderings. Some of these non-circuital transformations are known to be
achievable and have even been achieved experimentally, and were proved to
provide some computational advantage in various information-processing tasks
with respect to quantum combs. Here we provide a formal language to form all
possible types of transformations, and use it to prove general structure
theorems for transformations in the hierarchy. We then provide a mathematical
characterisation of the set of maps from combs to combs, hinting at a route for
the complete characterisation of maps in the hierarchy. The classification is
strictly related to the way in which the maps manipulate the causal structure
of input circuits.Comment: 12 pages, revtex styl
Witnessing causal nonseparability
Our common understanding of the physical world deeply relies on the notion
that events are ordered with respect to some time parameter, with past events
serving as causes for future ones. Nonetheless, it was recently found that it
is possible to formulate quantum mechanics without any reference to a global
time or causal structure. The resulting framework includes new kinds of quantum
resources that allow performing tasks - in particular, the violation of causal
inequalities - which are impossible for events ordered according to a global
causal order. However, no physical implementation of such resources is known.
Here we show that a recently demonstrated resource for quantum computation -
the quantum switch - is a genuine example of "indefinite causal order". We do
this by introducing a new tool - the causal witness - which can detect the
causal nonseparability of any quantum resource that is incompatible with a
definite causal order. We show however that the quantum switch does not violate
any causal nequality.Comment: 15 + 12 pages, 5 figures. Published versio
Geometric Phase: a Diagnostic Tool for Entanglement
Using a kinematic approach we show that the non-adiabatic, non-cyclic,
geometric phase corresponding to the radiation emitted by a three level cascade
system provides a sensitive diagnostic tool for determining the entanglement
properties of the two modes of radiation. The nonunitary, noncyclic path in the
state space may be realized through the same control parameters which control
the purity/mixedness and entanglement. We show analytically that the geometric
phase is related to concurrence in certain region of the parameter space. We
further show that the rate of change of the geometric phase reveals its
resilience to fluctuations only for pure Bell type states. Lastly, the
derivative of the geometric phase carries information on both purity/mixedness
and entanglement/separability.Comment: 13 pages 6 figure
Bell Correlations and the Common Future
Reichenbach's principle states that in a causal structure, correlations of
classical information can stem from a common cause in the common past or a
direct influence from one of the events in correlation to the other. The
difficulty of explaining Bell correlations through a mechanism in that spirit
can be read as questioning either the principle or even its basis: causality.
In the former case, the principle can be replaced by its quantum version,
accepting as a common cause an entangled state, leaving the phenomenon as
mysterious as ever on the classical level (on which, after all, it occurs). If,
more radically, the causal structure is questioned in principle, closed
space-time curves may become possible that, as is argued in the present note,
can give rise to non-local correlations if to-be-correlated pieces of classical
information meet in the common future --- which they need to if the correlation
is to be detected in the first place. The result is a view resembling Brassard
and Raymond-Robichaud's parallel-lives variant of Hermann's and Everett's
relative-state formalism, avoiding "multiple realities."Comment: 8 pages, 5 figure
Distinguishability measures between ensembles of quantum states
A quantum ensemble is a set of quantum states each
occurring randomly with a given probability. Quantum ensembles are necessary to
describe situations with incomplete a priori information, such as the output of
a stochastic quantum channel (generalized measurement), and play a central role
in quantum communication. In this paper, we propose measures of distance and
fidelity between two quantum ensembles. We consider two approaches: the first
one is based on the ability to mimic one ensemble given the other one as a
resource and is closely related to the Monge-Kantorovich optimal transportation
problem, while the second one uses the idea of extended-Hilbert-space (EHS)
representations which introduce auxiliary pointer (or flag) states. Both types
of measures enjoy a number of desirable properties. The Kantorovich measures,
albeit monotonic under deterministic quantum operations, are not monotonic
under generalized measurements. In contrast, the EHS measures are. We present
operational interpretations for both types of measures. We also show that the
EHS fidelity between ensembles provides a novel interpretation of the fidelity
between mixed states--the latter is equal to the maximum of the fidelity
between all pure-state ensembles whose averages are equal to the mixed states
being compared. We finally use the new measures to define distance and fidelity
for stochastic quantum channels and positive operator-valued measures (POVMs).
These quantities may be useful in the context of tomography of stochastic
quantum channels and quantum detectors.Comment: 31 pages, typos correcte
Highly symmetric POVMs and their informational power
We discuss the dependence of the Shannon entropy of normalized finite rank-1
POVMs on the choice of the input state, looking for the states that minimize
this quantity. To distinguish the class of measurements where the problem can
be solved analytically, we introduce the notion of highly symmetric POVMs and
classify them in dimension two (for qubits). In this case we prove that the
entropy is minimal, and hence the relative entropy (informational power) is
maximal, if and only if the input state is orthogonal to one of the states
constituting a POVM. The method used in the proof, employing the Michel theory
of critical points for group action, the Hermite interpolation and the
structure of invariant polynomials for unitary-antiunitary groups, can also be
applied in higher dimensions and for other entropy-like functions. The links
between entropy minimization and entropic uncertainty relations, the Wehrl
entropy and the quantum dynamical entropy are described.Comment: 40 pages, 3 figure