300 research outputs found
Connections and Metrics Respecting Standard Purification
Standard purification interlaces Hermitian and Riemannian metrics on the
space of density operators with metrics and connections on the purifying
Hilbert-Schmidt space. We discuss connections and metrics which are well
adopted to purification, and present a selected set of relations between them.
A connection, as well as a metric on state space, can be obtained from a metric
on the purification space. We include a condition, with which this
correspondence becomes one-to-one. Our methods are borrowed from elementary
*-representation and fibre space theory. We lift, as an example, solutions of a
von Neumann equation, write down holonomy invariants for cyclic ones, and ``add
noise'' to a curve of pure states.Comment: Latex, 27 page
Lower and upper bounds on the fidelity susceptibility
We derive upper and lower bounds on the fidelity susceptibility in terms of
macroscopic thermodynamical quantities, like susceptibilities and thermal
average values. The quality of the bounds is checked by the exact expressions
for a single spin in an external magnetic field. Their usefulness is
illustrated by two examples of many-particle models which are exactly solved in
the thermodynamic limit: the Dicke superradiance model and the single impurity
Kondo model. It is shown that as far as divergent behavior is considered, the
fidelity susceptibility and the thermodynamic susceptibility are equivalent for
a large class of models exhibiting critical behavior.Comment: 19 page
Transition probabilities between quasifree states
We obtain a general formula for the transition probabilities between any
state of the algebra of the canonical commutation relations (CCR-algebra) and a
squeezed quasifree state. Applications of this formula are made for the case of
multimode thermal squeezed states of quantum optics using a general canonical
decomposition of the correlation matrix valid for any quasifree state. In the
particular case of a one mode CCR-algebra we show that the transition
probability between two quasifree squeezed states is a decreasing function of
the geodesic distance between the points of the upper half plane representing
these states. In the special case of the purification map it is shown that the
transition probability between the state of the enlarged system and the product
state of real and fictitious subsystems can be a measure for the entanglement.Comment: 13 pages, REVTeX, no figure
Berry phase and fidelity susceptibility of the three-qubit Lipkin-Meshkov-Glick ground state
Berry phases and quantum fidelities for interacting spins have attracted
considerable attention, in particular in relation to entanglement properties of
spin systems and quantum phase transitions. These efforts mainly focus either
on spin pairs or the thermodynamic infinite spin limit, while studies of the
multipartite case of a finite number of spins are rare. Here, we analyze Berry
phases and quantum fidelities of the energetic ground state of a
Lipkin-Meshkov-Glick (LMG) model consisting of three spin-1/2 particles
(qubits). We find explicit expressions for the Berry phase and fidelity
susceptibility of the full system as well as the mixed state Berry phase and
partial-state fidelity susceptibility of its one- and two-qubit subsystems. We
demonstrate a realization of a nontrivial magnetic monopole structure
associated with local, coordinated rotations of the three-qubit system around
the external magnetic field.Comment: The title of the paper has been changed in this versio
Generating random density matrices
We study various methods to generate ensembles of random density matrices of
a fixed size N, obtained by partial trace of pure states on composite systems.
Structured ensembles of random pure states, invariant with respect to local
unitary transformations are introduced. To analyze statistical properties of
quantum entanglement in bi-partite systems we analyze the distribution of
Schmidt coefficients of random pure states. Such a distribution is derived in
the case of a superposition of k random maximally entangled states. For another
ensemble, obtained by performing selective measurements in a maximally
entangled basis on a multi--partite system, we show that this distribution is
given by the Fuss-Catalan law and find the average entanglement entropy. A more
general class of structured ensembles proposed, containing also the case of
Bures, forms an extension of the standard ensemble of structureless random pure
states, described asymptotically, as N \to \infty, by the Marchenko-Pastur
distribution.Comment: 13 pages in latex with 8 figures include
Fidelity approach to quantum phase transitions
We review briefly the quantum fidelity approach to quantum phase transitions
in a pedagogical manner. We try to relate all established but scattered results
on the leading term of the fidelity into a systematic theoretical framework,
which might provide an alternative paradigm for understanding quantum critical
phenomena. The definition of the fidelity and the scaling behavior of its
leading term, as well as their explicit applications to the one-dimensional
transverse-field Ising model and the Lipkin-Meshkov-Glick model, are introduced
at the graduate-student level. In addition, we survey also other types of
fidelity approach, such as the fidelity per site, reduced fidelity,
thermal-state fidelity, operator fidelity, etc; as well as relevant works on
the fidelity approach to quantum phase transitions occurring in various
many-body systems.Comment: 41 pages, 31 figures. We apologize if we omit acknowledging your
relevant works. Do tell. An updated version with clearer figures can be found
at: http://www.phy.cuhk.edu.hk/~sjgu/fidelitynote.pd
Statistical distinguishability between unitary operations
The problem of distinguishing two unitary transformations, or quantum gates,
is analyzed and a function reflecting their statistical distinguishability is
found. Given two unitary operations, and , it is proved that there
always exists a finite number such that and are perfectly distinguishable, although they were not in the single-copy
case. This result can be extended to any finite set of unitary transformations.
Finally, a fidelity for one-qubit gates, which satisfies many useful properties
from the point of view of quantum information theory, is presented.Comment: 6 pages, REVTEX. The perfect distinguishability result is extended to
any finite set of gate
The geometry of entanglement: metrics, connections and the geometric phase
Using the natural connection equivalent to the SU(2) Yang-Mills instanton on
the quaternionic Hopf fibration of over the quaternionic projective space
with an fiber the geometry of
entanglement for two qubits is investigated. The relationship between base and
fiber i.e. the twisting of the bundle corresponds to the entanglement of the
qubits. The measure of entanglement can be related to the length of the
shortest geodesic with respect to the Mannoury-Fubini-Study metric on between an arbitrary entangled state, and the separable state nearest to
it. Using this result an interpretation of the standard Schmidt decomposition
in geometric terms is given. Schmidt states are the nearest and furthest
separable ones lying on, or the ones obtained by parallel transport along the
geodesic passing through the entangled state. Some examples showing the
correspondence between the anolonomy of the connection and entanglement via the
geometric phase is shown. Connections with important notions like the
Bures-metric, Uhlmann's connection, the hyperbolic structure for density
matrices and anholonomic quantum computation are also pointed out.Comment: 42 page
Statistical properties of random density matrices
Statistical properties of ensembles of random density matrices are
investigated. We compute traces and von Neumann entropies averaged over
ensembles of random density matrices distributed according to the Bures
measure. The eigenvalues of the random density matrices are analyzed: we derive
the eigenvalue distribution for the Bures ensemble which is shown to be broader
then the quarter--circle distribution characteristic of the Hilbert--Schmidt
ensemble. For measures induced by partial tracing over the environment we
compute exactly the two-point eigenvalue correlation function.Comment: 8 revtex pages with one eps file included, ver. 2 - minor misprints
correcte
Adiabatic Elimination in Compound Quantum Systems with Feedback
Feedback in compound quantum systems is effected by using the output from one
sub-system (``the system'') to control the evolution of a second sub-system
(``the ancilla'') which is reversibly coupled to the system. In the limit where
the ancilla responds to fluctuations on a much shorter time scale than does the
system, we show that it can be adiabatically eliminated, yielding a master
equation for the system alone. This is very significant as it decreases the
necessary basis size for numerical simulation and allows the effect of the
ancilla to be understood more easily. We consider two types of ancilla: a
two-level ancilla (e.g. a two-level atom) and an infinite-level ancilla (e.g.
an optical mode). For each, we consider two forms of feedback: coherent (for
which a quantum mechanical description of the feedback loop is required) and
incoherent (for which a classical description is sufficient). We test the
master equations we obtain using numerical simulation of the full dynamics of
the compound system. For the system (a parametric oscillator) and feedback
(intensity-dependent detuning) we choose, good agreement is found in the limit
of heavy damping of the ancilla. We discuss the relation of our work to
previous work on feedback in compound quantum systems, and also to previous
work on adiabatic elimination in general.Comment: 18 pages, 12 figures including two subplots as jpeg attachment
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