43 research outputs found
Generalized conditional entropy in bipartite quantum systems
We analyze, for a general concave entropic form, the associated conditional
entropy of a quantum system A+B, obtained as a result of a local measurement on
one of the systems (B). This quantity is a measure of the average mixedness of
A after such measurement, and its minimum over all local measurements is shown
to be the associated entanglement of formation between A and a purifying third
system C. In the case of the von Neumann entropy, this minimum determines also
the quantum discord. For classically correlated states and mixtures of a pure
state with the maximally mixed state, we show that the minimizing measurement
can be determined analytically and is universal, i.e., the same for all concave
forms. While these properties no longer hold for general states, we also show
that in the special case of the linear entropy, an explicit expression for the
associated conditional entropy can be obtained, whose minimum among projective
measurements in a general qudit-qubit state can be determined analytically, in
terms of the largest eigenvalue of a simple 3x3 correlation matrix. Such
minimum determines the maximum conditional purity of A, and the associated
minimizing measurement is shown to be also universal in the vicinity of maximal
mixedness. Results for X states, including typical reduced states of spin pairs
in XY chains at weak and strong transverse fields, are also provided and
indicate that the measurements minimizing the von Neumann and linear
conditional entropies are typically coincident in these states, being
determined essentially by the main correlation. They can differ, however,
substantially from that minimizing the geometric discord.Comment: 11 pages, 2 figures; References adde
Generalized conditional entropy optimization for qudit-qubit states
We derive a general approximate solution to the problem of minimizing the
conditional entropy of a qudit-qubit system resulting from a local projective
measurement on the qubit, which is valid for general entropic forms and becomes
exact in the limit of weak correlations. This entropy measures the average
conditional mixedness of the post-measurement state of the qudit, and its
minimum among all local measurements represents a generalized entanglement of
formation. In the case of the von Neumann entropy, it is directly related to
the quantum discord. It is shown that at the lowest non-trivial order, the
problem reduces to the minimization of a quadratic form determined by the
correlation tensor of the system, the Bloch vector of the qubit and the local
concavity of the entropy, requiring just the diagonalization of a
matrix. A simple geometrical picture in terms of an associated correlation
ellipsoid is also derived, which illustrates the link between entropy
optimization and correlation access and which is exact for a quadratic entropy.
The approach enables a simple estimation of the quantum discord. Illustrative
results for two-qubit states are discussed.Comment: 11 pages, 6 figures. Final published versio
Conditional purity and quantum correlation measures in two qubit mixed states
We analyze and show experimental results of the conditional purity, the
quantum discord and other related measures of quantum correlation in mixed
two-qubit states constructed from a pair of photons in identical polarization
states. The considered states are relevant for the description of spin pair
states in interacting spin chains in a transverse magnetic field. We derive
clean analytical expressions for the conditional local purity and other
correlation measures obtained as a result of a remote local projective
measurement, which are fully verified by the experimental results. A simple
exact expression for the quantum discord of these states in terms of the
maximum conditional purity is also derived.Comment: 16 pages, 5 figures, minor changes, to be published in J. Phys.
Biased Random Access Codes
A Random Access Code (RAC) is a communication task in which the sender
encodes a random message into a shorter one to be decoded by the receiver so
that a randomly chosen character of the original message is recovered with some
probability. Both the message and the character to be recovered are assumed to
be uniformly distributed. In this paper, we extend this protocol by allowing
more general distributions of these inputs, which alters the encoding and
decoding strategies optimizing the protocol performance, either with classical
or quantum resources. We approach the problem of optimizing the performance of
these biased RACs with both numerical and analytical tools. On the numerical
front, we present algorithms that allow a numerical evaluation of the optimal
performance over both classical and quantum strategies and provide a Python
package designed to implement them, called RAC-tools. We then use this
numerical tool to investigate single-parameter families of biased RACs in the
and scenarios. For RACs in the
scenario, we derive a general upper bound for the cases in which the inputs are
not correlated, which coincides with the quantum value for and, in some
cases for . Moreover, it is shown that attaining this upper bound
self-tests pairs or triples of rank-1 projective measurements, respectively. An
analogous upper bound is derived for the value of RACs in the
scenario which is shown to be always attainable using mutually unbiased
measurements if the distribution of input strings is unbiased
Bipartite representations and many-body entanglement of pure states of indistinguishable particles
We analyze a general bipartite-like representation of arbitrary pure states
of -indistinguishable particles, valid for both bosons and fermions, based
on - and -particle states. It leads to exact Schmidt-like
expansions of the state for any and is directly related to the
isospectral reduced - and -body density matrices and
. The formalism also allows for reduced yet still exact
Schmidt-like decompositions associated with blocks of these densities, in
systems having a fixed fraction of the particles in some single particle
subspace. Monotonicity of the ensuing -body entanglement under a certain set
of quantum operations is also discussed. Illustrative examples in fermionic and
bosonic systems with pairing correlations are provided, which show that in the
presence of dominant eigenvalues in , approximations based on a few
terms of the pertinent Schmidt expansion can provide a reliable description of
the state. The associated one- and two-body entanglement spectrum and entropies
are also analyzed.Comment: 17 pages, 5 figure
Quantum Discord and entropic measures of quantum correlations: Optimization and behavior in finite spin chains
We discuss a generalization of the conditional entropy and one-way
information deficit in quantum systems, based on general entropic forms. The
formalism allows to consider simple entropic forms for which a closed
evaluation of the associated optimization problem in qudit-qubit systems is
shown to become feasible, allowing to approximate that of the quantum discord.
As application, we examine quantum correlations of spin pairs in the exact
ground state of finite spin chains in a magnetic field through the quantum
discord and information deficit. While these quantities show a similar
behavior, their optimizing measurements exhibit significant differences, which
can be understood and predicted through the previous approximations. The
remarkable behavior of these quantities in the vicinity of transverse and
non-transverse factorizing fields is also discussed.Comment: 10 pages, 3 figure