4,263 research outputs found
On the volume of the set of mixed entangled states
A natural measure in the space of density matrices describing N-dimensional
quantum systems is proposed. We study the probability P that a quantum state
chosen randomly with respect to the natural measure is not entangled (is
separable). We find analytical lower and upper bounds for this quantity.
Numerical calculations give P = 0.632 for N=4 and P=0.384 for N=6, and indicate
that P decreases exponentially with N. Analysis of a conditional measure of
separability under the condition of fixed purity shows a clear dualism between
purity and separability: entanglement is typical for pure states, while
separability is connected with quantum mixtures. In particular, states of
sufficiently low purity are necessarily separable.Comment: 10 pages in LaTex - RevTex + 4 figures in eps. submitted to Phys.
Rev.
Efficient algorithm for multi-qudit twirling for ensemble quantum computation
We present an efficient algorithm for twirling a multi-qudit quantum state.
The algorithm can be used for approximating the twirling operation in an
ensemble of physical systems in which the systems cannot be individually
accessed. It can also be used for computing the twirled density matrix on a
classical computer. The method is based on a simple non-unitary operation
involving a random unitary. When applying this basic building block
iteratively, the mean squared error of the approximation decays exponentially.
In contrast, when averaging over random unitary matrices the error decreases
only algebraically. We present evidence that the unitaries in our algorithm can
come from a very imperfect random source or can even be chosen
deterministically from a set of cyclically alternating matrices. Based on these
ideas we present a quantum circuit realizing twirling efficiently.Comment: 11 pages including 6 figures, revtex4; v2: presentation improved,
sections VI and VII added; v3: small changes before publicatio
How often is a random quantum state k-entangled?
The set of trace preserving, positive maps acting on density matrices of size
d forms a convex body. We investigate its nested subsets consisting of
k-positive maps, where k=2,...,d. Working with the measure induced by the
Hilbert-Schmidt distance we derive asymptotically tight bounds for the volumes
of these sets. Our results strongly suggest that the inner set of
(k+1)-positive maps forms a small fraction of the outer set of k-positive maps.
These results are related to analogous bounds for the relative volume of the
sets of k-entangled states describing a bipartite d X d system.Comment: 19 pages in latex, 1 figure include
Almost all quantum channels are equidistant
In this work we analyze properties of generic quantum channels in the case of
large system size. We use random matrix theory and free probability to show
that the distance between two independent random channels converges to a
constant value as the dimension of the system grows larger. As a measure of the
distance we use the diamond norm. In the case of a flat Hilbert-Schmidt
distribution on quantum channels, we obtain that the distance converges to
, giving also an estimate for the maximum success
probability for distinguishing the channels. We also consider the problem of
distinguishing two random unitary rotations.Comment: 30 pages, commets are welcom
Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive
We investigate the set a) of positive, trace preserving maps acting on
density matrices of size N, and a sequence of its nested subsets: the sets of
maps which are b) decomposable, c) completely positive, d) extended by identity
impose positive partial transpose and e) are superpositive. Working with the
Hilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided bounds
for the volumes of all five sets. A sample consequence is the fact that, as N
increases, a generic positive map becomes not decomposable and, a fortiori, not
completely positive.
Due to the Jamiolkowski isomorphism, the results obtained for quantum maps
are closely connected to similar relations between the volume of the set of
quantum states and the volumes of its subsets (such as states with positive
partial transpose or separable states) or supersets. Our approach depends on
systematic use of duality to derive quantitative estimates, and on various
tools of classical convexity, high-dimensional probability and geometry of
Banach spaces, some of which are not standard.Comment: 34 pages in Latex including 3 figures in eps, ver 2: minor revision
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