13 research outputs found
Finite size mean-field models
We characterize the two-site marginals of exchangeable states of a system of
quantum spins in terms of a simple positivity condition. This result is used in
two applications. We first show that the distance between two-site marginals of
permutation invariant states on N spins and exchangeable states is of order
1/N. The second application relates the mean ground state energy of a
mean-field model of composite spins interacting through a product pair
interaction with the mean ground state energies of the components.Comment: 20 page
Extending additivity from symmetric to asymmetric channels
We prove a lemma which allows one to extend results about the additivity of
the minimal output entropy from highly symmetric channels to a much larger
class. A similar result holds for the maximal output -norm. Examples are
given showing its use in a variety of situations. In particular, we prove the
additivity and the multiplicativity for the shifted depolarising channel.Comment: 8 pages. This is the latest version of the first half of the original
paper. The other half will appear in another pape
Pauli Diagonal Channels Constant on Axes
We define and study the properties of channels which are analogous to unital
qubit channels in several ways. A full treatment can be given only when the
dimension d is a prime power, in which case each of the (d+1) mutually unbiased
bases (MUB) defines an axis. Along each axis the channel looks like a
depolarizing channel, but the degree of depolarization depends on the axis.
When d is not a prime power, some of our results still hold, particularly in
the case of channels with one symmetry axis. We describe the convex structure
of this class of channels and the subclass of entanglement breaking channels.
We find new bound entangled states for d = 3.
For these channels, we show that the multiplicativity conjecture for maximal
output p-norm holds for p=2. We also find channels with behavior not exhibited
by unital qubit channels, including two pairs of orthogonal bases with equal
output entropy in the absence of symmetry. This provides new numerical evidence
for the additivity of minimal output entropy
Comments on Hastings' Additivity Counterexamples
Hastings recently provided a proof of the existence of channels which violate
the additivity conjecture for minimal output entropy. In this paper we present
an expanded version of Hastings' proof. In addition to a careful elucidation of
the details of the proof, we also present bounds for the minimal dimensions
needed to obtain a counterexample.Comment: 38 page
Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1
For all p > 1, we demonstrate the existence of quantum channels with
non-multiplicative maximal output p-norms. Equivalently, for all p >1, the
minimum output Renyi entropy of order p of a quantum channel is not additive.
The violations found are large; in all cases, the minimum output Renyi entropy
of order p for a product channel need not be significantly greater than the
minimum output entropy of its individual factors. Since p=1 corresponds to the
von Neumann entropy, these counterexamples demonstrate that if the additivity
conjecture of quantum information theory is true, it cannot be proved as a
consequence of any channel-independent guarantee of maximal p-norm
multiplicativity. We also show that a class of channels previously studied in
the context of approximate encryption lead to counterexamples for all p > 2.Comment: Merger of arXiv:0707.0402 and arXiv:0707.3291 containing new and
improved analysis of counterexamples. 17 page