18 research outputs found
Polynomial identities for matrices over the Grassmann algebra
We determine minimal Cayley--Hamilton and Capelli identities for matrices
over a Grassmann algebra of finite rank. For minimal standard identities, we
give lower and upper bounds on the degree. These results improve on upper
bounds given by L.\ M\'arki, J.\ Meyer, J.\ Szigeti, and L.\ van Wyk in a
recent paper.Comment: 9 page
Remarks on the --permanent
We recall Vere-Jones's definition of the --permanent and describe the
connection between the (1/2)--permanent and the hafnian. We establish expansion
formulae for the --permanent in terms of partitions of the index set,
and we use these to prove Lieb-type inequalities for the --permanent
of a positive semi-definite Hermitian matrix and the
--permanent of a positive semi-definite real symmetric
matrix if is a nonnegative integer or . We are unable
to settle Shirai's nonnegativity conjecture for --permanents when
, but we verify it up to the case, in addition to
recovering and refining some of Shirai's partial results by purely
combinatorial proofs.Comment: 9 page
On vector configurations that can be realized in the cone of positive matrices
Let ,..., be vectors in an inner product space. Can we find a
natural number and positive (semidefinite) complex matrices ,...,
of size such that for all
? For such matrices to exist, one must have
for all . We prove that if then this trivial necessary
condition is also a sufficient one and find an appropriate example showing that
from this is not so --- even if we allowed realizations by positive
operators in a von Neumann algebra with a faithful normal tracial state.
The fact that the first such example occurs at is similar to what one
has in the well-investigated problem of positive factorization of positive
(semidefinite) matrices. If the matrix has a positive
factorization, then matrices ,..., as above exist. However, as we
show by a large class of examples constructed with the help of the Clifford
algebra, the converse implication is false.Comment: 8 page
Integral formula for quantum relative entropy implies data processing inequality
Integral representations of quantum relative entropy, and of the directional
second and higher order derivatives of von Neumann entropy, are established,
and used to give simple proofs of fundamental, known data processing
inequalities: the Holevo bound on the quantity of information transmitted by a
quantum communication channel, and, much more generally, the monotonicity of
quantum relative entropy under trace-preserving positive linear maps --
complete positivity of the map need not be assumed. The latter result was first
proved by M\"uller-Hermes and Reeb, based on work of Beigi. For a simple
application of such monotonicities, we consider any `divergence' that is
non-increasing under quantum measurements, such as the concavity of von Neumann
entropy, or various known quantum divergences. An elegant argument due to Hiai,
Ohya, and Tsukada is used to show that the infimum of such a `divergence' on
pairs of quantum states with prescribed trace distance is the same as the
corresponding infimum on pairs of binary classical states. Applications of the
new integral formulae to the general probabilistic model of information theory,
and a related integral formula for the classical R\'enyi divergence, are also
discussed.Comment: 16 pages. Accepted to Quantu
Classical simulations of communication channels
We investigate whether certain non-classical communication channels can be
simulated by a classical channel with a given number of states and a given
amount of noise. It is proved that any noisy quantum channel can be simulated
by the corresponding noisy classical channel. General probabilistic channels
are also studied.Comment: 12 page
Classical information storage in an -level quantum system
A game is played by a team of two --- say Alice and Bob --- in which the
value of a random variable is revealed to Alice only, who cannot freely
communicate with Bob. Instead, she is given a quantum -level system,
respectively a classical -state system, which she can put in possession of
Bob in any state she wishes. We evaluate how successfully they managed to store
and recover the value of in the used system by requiring Bob to specify a
value and giving a reward of value to the team.
We show that whatever the probability distribution of and the reward
function are, when using a quantum -level system, the maximum expected
reward obtainable with the best possible team strategy is equal to that
obtainable with the use of a classical -state system.
The proof relies on mixed discriminants of positive matrices and --- perhaps
surprisingly --- an application of the Supply--Demand Theorem for bipartite
graphs. As a corollary, we get an infinite set of new, dimension dependent
inequalities regarding positive operator valued measures and density operators
on complex -space.
As a further corollary, we see that the greatest value, with respect to a
given distribution of , of the mutual information that is
obtainable using an -level quantum system equals the analogous maximum for a
classical -state system.Comment: 13 page
Integral formula for quantum relative entropy implies data processing inequality
Integral representations of quantum relative entropy, and of the directional second and higher order derivatives of von Neumann entropy, are established, and used to give simple proofs of fundamental, known data processing inequalities: the Holevo bound on the quantity of information transmitted by a quantum communication channel, and, much more generally, the monotonicity of quantum relative entropy under trace-preserving positive linear maps â complete positivity of the map need not be assumed. The latter result was first proved by MĂŒller-Hermes and Reeb, based on work of Beigi. For a simple application of such monotonicities, we consider any `divergence' that is non-increasing under quantum measurements, such as the concavity of von Neumann entropy, or various known quantum divergences. An elegant argument due to Hiai, Ohya, and Tsukada is used to show that the infimum of such a `divergence' on pairs of quantum states with prescribed trace distance is the same as the corresponding infimum on pairs of binary classical states. Applications of the new integral formulae to the general probabilistic model of information theory, and a related integral formula for the classical RĂ©nyi divergence, are also discussed