7,897 research outputs found
Separability of qubit-qudit quantum states with strong positive partial transposes
We show that all states with strong positive partial transposes
(SPPT) are separable. We also construct a family of entangled SPPT
states, so the conjecture on the separability of SPPT states are completely
settled. In addition, we clarify the relation between the set of all separable states and the set of all SPPT states for the case of
.Comment: 3 page
Notes on extremality of the Choi map
It is widely believed that the Choi map generates an extremal ray in the cone
of all positive linear maps between -algebra of
all matrices over the complex field. But the only proven fact is
that the Choi map generates the extremal ray in the cone of all positive linear
map preserving all real symmetric matrices. In this note, we show
that the Choi map is indeed extremal in the cone . We also
clarify some misclaims about the correspondence between positive semi-definite
biquadratic real forms and postive linear maps, and discuss possible positive
linear maps which coincide with the Choi map on symmetric matrices.Comment: 12 page
Global geometric difference between separable and Positive partial transpose states
In the convex set of all 3\ot 3 states with positive partial transposes, we
show that one can take two extreme points whose convex combinations belong to
the interior of the convex set. Their convex combinations may be even in the
interior of the convex set of all separable states. In general, we need at
least extreme points to get an interior point by their convex combination,
for the case of the convex set of all m\ot n separable states. This shows a
sharp distinction between PPT states and separable states. We also consider the
same questions for positive maps and decomposable maps.Comment: 14 pages, 2 figure
Construction of exposed indecomposable positive linear maps between matrix algebras
We construct a large class of indecomposable positive linear maps from the
matrix algebra into the matrix algebra, which generate
exposed extreme rays of the convex cone of all positive maps. We show that
extreme points of the dual faces for separable states arising from these maps
are parametrized by the Riemann sphere, and the convex hulls of the extreme
points arising from a circle parallel to the equator have the exactly same
properties with the convex hull of the trigonometric moment curve studied from
combinatorial topology. Any interior points of the dual faces are
boundary separable states with full ranks. We exhibit concrete examples of such
states.Comment: 12 page
Geometry for separable states and construction of entangled states with positive partial transposes
We construct faces of the convex set of all bipartite separable
states, which are affinely isomorphic to the simplex with ten
extreme points. Every interior point of these faces is a separable state which
has a unique decomposition into 10 product states, even though ranks of the
state and its partial transpose are 5 and 7, respectively. We also note that
the number 10 is greater than , to disprove a conjecture on the
lengths of qubit-qudit separable states. This face is inscribed in the
corresponding face of the convex set of all PPT states so that sub-simplices
of share the boundary if and only if . This
enables us to find a large class of PPT entangled edge states with
rank five.Comment: 8 pages, 2 figure
Separable states with unique decompositions
We search for faces of the convex set consisting of all separable states,
which are affinely isomorphic to simplices, to get separable states with unique
decompositions. In the two-qutrit case, we found that six product vectors
spanning a five dimensional space give rise to a face isomorphic to the
5-dimensional simplex with six vertices, under suitable linear independence
assumption.
If the partial conjugates of six product vectors also span a 5-dimensional
space, then this face is inscribed in the face for PPT states whose boundary
shares the fifteen 3-simplices on the boundary of the 5-simplex. The remaining
boundary points consist of PPT entangled edge states of rank four. We also show
that every edge state of rank four arises in this way. If the partial
conjugates of the above six product vectors span a 6-dimensional space then we
have a face isomorphic to 5-simplex, whose interior consists of separable
states with unique decompositions, but with non-symmetric ranks. We also
construct a face isomorphic to the 9-simplex. As applications, we give answers
to questions in the literature \cite{chen_dj_semialg,chen_dj_ext_PPT}, and
construct 3\ot 3 PPT states of type (9,5). For the qubit-qudit cases with
, we also show that -dimensional subspaces give rise to faces
isomorphic to the -simplices, in most cases.Comment: 26 pages, 1 figure. A part on SPA conjecture is withdraw
Multi-partite separable states with unique decompositions and construction of three qubit entanglement with positive partial transpose
We investigate conditions on a finite set of multi-partite product vectors
for which separable states with corresponding product states have unique
decomposition, and show that this is true in most cases if the number of
product vectors is sufficiently small. In the three qubit case, generic five
dimensional spaces give rise to faces of the convex set consisting of all
separable states, which are affinely isomorphic to the five dimensional simplex
with six vertices. As a byproduct, we construct three qubit entangled PPT edge
states of rank four with explicit formulae. This covers those entanglement
which cannot be constructed from unextendible product basis.Comment: 19 pages, 1 figur
Construction of three qubit genuine entanglement with bi-partite positive partial transposes
We construct tri-qubit genuinely entangled states which have positive partial
transposes with respect to bi-partition of systems. These examples disprove a
conjecture [L. Novo, T. Moroder and O. G\" uhne, Phys.Rev.A {88}, 012305
(2013)] which claims that PPT mixtures are necessary and sufficient for
biseparability of three qubits.Comment: 6 page
Geometry of the faces for separable states arising from generalized Choi maps
We exhibit examples of separable states which are on the boundary of the
convex cone generated by all separable states but in the interior of the convex
cone generated by all PPT states. We also analyze the geometric structures of
the smallest face generated by those examples. As a byproduct, we obtain a
large class of entangled states with positive partial transposes.Comment: 15 pages, 1 figur
A Formal Approach to Power Optimization in CPSs with Delay-Workload Dependence Awareness
The design of cyber-physical systems (CPSs) faces various new challenges that
are unheard of in the design of classical real-time systems. Power optimization
is one of the major design goals that is witnessing such new challenges. The
presence of interaction between the cyber and physical components of a CPS
leads to dependence between the time delay of a computational task and the
amount of workload in the next iteration. We demonstrate that it is essential
to take this delay-workload dependence into consideration in order to achieve
low power consumption.
In this paper, we identify this new challenge, and present the first formal
and comprehensive model to enable rigorous investigations on this topic. We
propose a simple power management policy, and show that this policy achieves a
best possible notion of optimality. In fact, we show that the optimal power
consumption is attained in a "steady-state" operation and a simple policy of
finding and entering this steady state suffices, which can be quite surprising
considering the added complexity of this problem. Finally, we validated the
efficiency of our policy with experiments.Comment: 27 pages, 8 figures, 3 table
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