3 research outputs found
Why the Entanglement of Formation is not generally monogamic
Differently from correlation of classical systems, entanglement of quantum
systems cannot be distributed at will - if one system A is maximally entangled
with another system B, it cannot be entangled at all to a third system C. This
concept, known as the monogamy of entanglement, manifests when the entanglement
of A with a pair BC, can be divided as contributions of entanglement between A
and B and A and C, plus a term \tau_{ABC} involving genuine tripartite
entanglement and so expected to be always positive. A very important measure in
Quantum Information Theory, the Entanglement of Formation (EOF), fails to
satisfy this last requirement. Here we present the reasons for that and show a
set of conditions that an arbitrary pure tripartite state must satisfy for EOF
to become a monogamous measure, ie, for \tau_{ABC} \ge 0. The relation derived
is connected to the discrepancy between quantum and classical correlations,
being \tau_{ABC} negative whenever the quantum correlation prevails over the
classical one. This result is employed to elucidate features of the
distribution of entanglement during a dynamical evolution. It also helps to
relate all monogamous instances of EOF to the Squashed Entanglement, an always
monogamous entanglement measure.Comment: 7 pages, 3 figures. Extended versio
Locally Inaccessible Information as a Fundamental Ingredient to Quantum Information
Quantum discord (QD) measures the fraction of the pairwise mutual information
that is locally inaccessible, in a multipartite system. Fundamental aspects
related to two important measures in quantum information theory the
Entanglement of Formation (EOF) and the conditional entropy, can be understood
in terms of the distribution of this form of Local Inaccessible Information
(LII). As such, the EOF for an arbitrarily mixed bipartite system AB can be
related to the gain or loss of LII due to the extra knowledge that a purifying
ancillary system E has on the pair AB. Similarly, a clear meaning of the
negativity of the conditional entropy for AB is given. We exemplify by showing
that these relations elucidate important and yet not well understood quantum
features, such as the bipartite entanglement sudden death and the distinction
between EOF and QD for quantifying quantum correlation. For that we introduce
the concept of LII flow which quantifies the LII shared in multipartite system
when a sequential local measurements are performed.Comment: We relate the entanglement and the conditional entropy exclusively as
a function of quantum discord. Final Versio
Calculation of quantum discord for qubit-qudit or N qubits
Quantum discord, a kind of quantum correlation, is defined as the difference
between quantum mutual information and classical correlation in a bipartite
system. It has been discussed so far for small systems with only a few
independent parameters. We extend here to a much broader class of states when
the second party is of arbitrary dimension d, so long as the first, measured,
party is a qubit. We present two formulae to calculate quantum discord, the
first relating to the original entropic definition and the second to a recently
proposed geometric distance measure which leads to an analytical formulation.
The tracing over the qubit in the entropic calculation is reduced to a very
simple prescription. And, when the d-dimensional system is a so-called X state,
the density matrix having non-zero elements only along the diagonal and
anti-diagonal so as to appear visually like the letter X, the entropic
calculation can be carried out analytically. Such states of the full bipartite
qubit-qudit system may be named "extended X states", whose density matrix is
built of four block matrices, each visually appearing as an X. The optimization
involved in the entropic calculation is generally over two parameters, reducing
to one for many cases, and avoided altogether for an overwhelmingly large set
of density matrices as our numerical investigations demonstrate. Our results
also apply to states of a N-qubit system, where "extended X states" consist of
(2^(N+2) - 1) states, larger in number than the (2^(N+1) - 1) of X states of N
qubits. While these are still smaller than the total number (2^(2N) - 1) of
states of N qubits, the number of parameters involved is nevertheless large. In
the case of N = 2, they encompass the entire 15-dimensional parameter space,
that is, the extended X states for N = 2 represent the full qubit-qubit system.Comment: 6 pages, 1 figur