An extremal element of the convex set of composite quantum states in $M_2\otimes M_3$, whose marginals are all normalised identities has been constructed. It is found to be a mixed state and is entangled as well.
An extremal element of the convex set of composite quantum states in M2⊗M3, whose marginals are all normalised identities has been constructed. It is found to be a mixed state and is entangled as well.
An extremal element of the convex set of composite quantum states in M2⊗M3, whose marginals are all normalised identities has been constructed. It is found to be a mixed state and is entangled as well.
We study the convex set C(ρ1,ρ2) of all bipartite quantum states with fixed marginal states ρ1 and ρ2. The extremal states in this set have recently been characterized by Parthasarathy [Ann. Henri Poincaré (to appear), quant-ph/0307182]. Here we present an alternative necessary and sufficient condition for a state in C(ρ1,ρ2) to be extremal. Our approach is based on a canonical duality between bipartite states and a certain class of completely positive maps and has the advantage that it is easier to check and to construct explicit examples of extremal states. In dimension 2×2 we give a simple new proof for the fact that all extremal states in C(121,121) are precisely the projectors onto maximally entangled wave functions. We also prove that in higher dimension this does not hold and construct an explicit example of an extremal state in C(131,131) that is not maximally entangled. Generalizations of this result to higher dimensions are also discussed.
It is well - known that the set of states of a given quantum mechanical system is to be closed from the point of view of the operational approach if we want to make mixed states or convex combinations. That is, s1 and s2 are states, then the same is to be true for λs1+(1−λ)s2, where 0<λ<1. We can define a convex combination of elements in a linear space, but unfortunately, in the general case the linear space is artificial for the set of states and has no physical meaning, but the procedure of forming the mixtures of states has a natural meaning. This is why we provide an abstract definition of the mixtures, which is independent of the linearity notion. We call this space a convex structure. In the work we consider state spaces, generalized state spaces, in which we select pure states, define operations and effects associated with the operations. We also consider ultraproducts of the sequences of these structures, operations and effects.
Absolutely maximally entangled states require every reduction of at most half of the parties to be maximally mixed, a condition that is both rigid and often impossible for qubit systems. Previous work introduced the quantum extremal number, which maximizes the number of exactly maximally mixed half-body marginals, and determined the exact value Qex(8,4)=56. The present work develops a robust extension of this extremal problem. For a subsystem A, the marginal maximal-mixing defect is defined by DA=2∣A∣Tr(ρA2)−1=2∣A∣ρA−2∣A∣IA22, and Qex,εD(n,k) is defined as the maximum number of k-body marginals satisfying DA≤ε in an n-qubit pure state. This counting problem differs from approximate k-uniformity, which requires all k-body marginals to obey a common error bound.
For pure states on 4m qubits, the following local stability inequality is established: ∑i∈TDT∖{i}≥1(∣T∣=2m+1). It follows that, whenever ε<1/(2m+1), the hypergraph of ε-good 2m-subsets is K2m+1(2m)-free. Combined with the known exact eight-qubit construction, this yields the stability plateau Qex,εD(8,4)=56,0≤ε<51. For odd systems of 2k+1 qubits, the exact forbidden hypergraph Hk is used to derive explicit finite-error stability radii. In particular, Qex,εD(9,4)≤120 for 0≤ε<1/17. These results turn exact quantum Turán obstructions into quantitative robustness statements and identify intervals on which quantum extremal numbers are stable under imperfect marginal mixedness.
Let be the convex set consisting of all states on the tensor product B B of the algebra B = M n (C) of all n n matrices over the complex numbers C with the property that the restrictions BI and IB are the unique tracial states on B I and I B . We find necessary and sufficient conditions for such a state, called a marginal tracial state, to be extremal in . We also give a characterization of those extreme points in which are pure states. We conjecture that all extremal marginal tracial states are pure states.