Pure Product States Research Articles

The Rate of Optimal Purification Procedures

Purification is a process in which decoherence is partially reversed by using several input systems which have been subject to the same noise. The purity of the outputs generally increases with the number of input systems, and decreases with the number of required output systems. We construct the optimal quantum operations for this task, and discuss their asymptotic behaviour as the number of inputs goes to infinity. The rate at which output systems may be generated depends crucially on the type of purity requirement. If one tests the purity of the output systems one at a time, the rate is infinite: this fidelity may be made to approach 1, while at the same time the number of outputs goes to infinity arbitrarily fast. On the other hand, if one also requires the correlations between outputs to decrease, the rate is zero: if fidelity with the pure product state is to go to 1, the number of outputs per input goes to zero. However, if only a fidelity close to 1 is required, the optimal purifier achieves a positive rate, which we compute.

Operational criterion and constructive checks for the separability of low-rank density matrices

We consider low rank density operators $\varrho$ supported on a $M\times N$ Hilbert space for arbitrary $M$ and $N$ ($M\leq N$) and with a positive partial transpose (PPT) $\varrho^{T_A}\ge 0$. For rank $r(\varrho) \leq N$ we prove that having a PPT is necessary and sufficient for $\varrho$ to be separable; in this case we also provide its minimal decomposition in terms of pure product states. It follows from this result that there is no rank 3 bound entangled states having a PPT. We also present a necessary and sufficient condition for the separability of generic density matrices for which the sum of the ranks of $\varrho$ and $\varrho^{T_A}$ satisfies $r(\varrho)+r(\varrho^{T_A}) \le 2MN-M-N+2$. This separability condition has the form of a constructive check, providing thus also a pure product state decomposition for separable states, and it works in those cases where a system of couple polynomial equations has a finite number of solutions, as expected in most cases.

Entangling power of quantum evolutions

We analyze the entangling capabilities of unitary transformations $U$ acting on a bipartite $d_1\times d_2$-dimensional quantum system. To this aim we introduce an entangling power measure $e(U)$ given by the mean linear entropy produced acting with $U$ on a given distribution of pure product states. This measure admits a natural interpretation in terms of quantum operations. For a uniform distribution explicit analytical results are obtained using group-theoretic arguments. The behaviour of the features of $e(U)$ as the subsystem dimensions $d_1$ and $d_2$ are varied is studied both analytically and numerically. The two-qubit case $d_1=d_2=2$ is argued to be peculiar.

Local description of quantum inseparability

We show how to decompose any density matrix of the simplest binary composite systems, whether separable or not, in terms of only product vectors. We determine for all cases the minimal number of product vectors needed for such a decomposition. Separable states correspond to mixing from one to four pure product states. Inseparable states can be described as pseudomixtures of four or five pure product states, and can be made separable by mixing them with one or two pure product states.

Separability criterion and inseparable mixed states with positive partial transposition

It is shown that any separable state on the Hilbert space H = H 1 ⊗ H 2 can be written as a convex combination of N pure product states with N ≤ (dim H ) 2. Then a new separability criterion for mixed states in terms of the range of the density matrix is obtained. It is used in the construction of inseparable mixed states with positive partial transposition in the case of 3 × 3 and 2 × 4 systems. The states represent an entanglement which is hidden in a more subtle way than known so far.

Pure States on Some Group-Invariant C ∗ -Algebras

Let W be a UHF algebra of Glimm type n?, i.e., 9t 0 Nk, where N = N, = N2 = ... are n X n matrix algebras. We define an AF-subalgebra W G of 9t, consisting of those elements of 9. invariant under a group of automorphisms {ag: g E G = SU(n)} of product type. StG is shown to be generated by an embedding of S(oo), the discrete group of finite permutations on countably many symbols. Let co be a pure product state on 9t, xc its restriction to c G Let e E N be a one-dimensional projection with corresponding projections e k E N,. Then if both (i) kvlw(ek) = oo, and (ii) 0 < EklIw(ek)[w(ek)] < oo hold, WG is not pure. G is shown to be pure if there exist orthogonal one-dimensional projections {p,: I < i s n} of N with corresponding projections pk E Nk such that W(pk) = 0 or 1, I < i s n, k ? 1, and 0 < Ek2Iw(Pk) < 00 for at most one i.

Product states and C∗-dynamical system of product type

A study is made of a special type of C ∗-dynamical system, consisting of a class n ∞, uniformly hyperfinite C ∗-algebra, a compact Lie group, and a natural product action of G on ▪ by ∗-automorphisms. In particular, a simple class of representations of the fixed point algebra ▪ G is examined, those induced by restrictions of pure product states of ▪. Sufficient conditions are obtained for (1) irreducibility and (2) unitary equivalence of such (irreducible) representations, in the case of an arbitrary compact Lie group G with continuous unitary representation in C n . These are used to produce necessary and sufficient conditions when G = U(1) or G = SU(2) acts by a product-type conjugation on the class 2 ∞ algebra.