This work is a technical examination of concepts in Arveson’s paper Probability of Entanglement [3]. For finite dimensional Hilbert spaces H with dimension n and K with dimension m such that 2 ≤ n ≤ m, every extension of a faithful state of B(H) to a state of B(K ⊗ H) is parameterized by a complex nm2 × n matrix v with the property that v∗v is the identity on B(H). The set of all such nm2 × n matrices is a Steifel manifold V mn(H,K) which corresponds to the set of isometries from H to K⊗mn. We show that the set of extensions of a faithful state E(!) is isomorphic to V mn(H,K)/U(Cmn) where U(Cmn) is the set of unitary linear operators from Cmn → Cmn. We can construct a filtration of extensions: E1(!) ⊆ · · · ⊆ Er(!) ⊆ · · · ⊆ Emn(!) where Er(!) is the set of extensions with rank at most r. Each Er(!) is isomorphic to V r(H,K)/U(Cr). We show that there is a unique measure μ on V r(H,K) defined by the inner product on B(H, r · K) where hu, vi = Tr(v∗u) for u, v ∈ B(H, r · K). We show that entangled extensions of every rank exist and for r < √dimH every extension in Er(!) is entangled. With respect to μ, for a fixed r with 1 ≤ r ≤ mn the set of extensions with rank strictly less than r in Er(!) has measure zero. For r < dimH almost every extension is entangled, in other words the probability that an extension of a faithful state with rank less than dimH is not entangled is zero.
This work is a technical examination of concepts in Arveson’s paper Probability of Entanglement [3]. For finite dimensional Hilbert spaces H with dimension n and K with dimension m such that 2 ≤ n ≤ m, every extension of a faithful state of B(H) to a state of B(K ⊗ H) is parameterized by a complex nm2 × n matrix v with the property that v∗v is the identity on B(H). The set of all such nm2 × n matrices is a Steifel manifold V mn(H,K) which corresponds to the set of isometries from H to K⊗mn. We show that the set of extensions of a faithful state E(!) is isomorphic to V mn(H,K)/U(Cmn) where U(Cmn) is the set of unitary linear operators from Cmn → Cmn. We can construct a filtration of extensions: E1(!) ⊆ · · · ⊆ Er(!) ⊆ · · · ⊆ Emn(!) where Er(!) is the set of extensions with rank at most r. Each Er(!) is isomorphic to V r(H,K)/U(Cr). We show that there is a unique measure μ on V r(H,K) defined by the inner product on B(H, r · K) where hu, vi = Tr(v∗u) for u, v ∈ B(H, r · K). We show that entangled extensions of every rank exist and for r < √dimH every extension in Er(!) is entangled. With respect to μ, for a fixed r with 1 ≤ r ≤ mn the set of extensions with rank strictly less than r in Er(!) has measure zero. For r < dimH almost every extension is entangled, in other words the probability that an extension of a faithful state with rank less than dimH is not entangled is zero.
Abstract. We show that states on tensor products of matrix algebras whose ranks are relatively small are almost surely entangled, but that states of maximum rank are not. More precisely, let M = Mm(C) and N = Mn(C) be full matrix algebras with m ≥ n, fix an arbitrary state ω of N, and let E(ω) be the set of all states of M ⊗N that extend ω. The space E(ω) contains states of rank r for every r = 1, 2,..., m · rank ω, and it has a filtration into compact subspaces E 1 (ω) ⊆ E 2 (ω) ⊆ · · · ⊆ E m·rank ω = E(ω), where E r (ω) is the set of all states of E(ω) having rank ≤ r. We show first that for every r, there is a real-analytic manifold V r, homogeneous under a transitive action of a compact group G r, which parameterizes E r (ω). The unique G r-invariant probability measure on V r promotes to a probability measure P r,ω on E r (ω), and P r,ω assigns probability 1 to states of rank r. The resulting probability space (E r (ω),P r,ω) represents “choosing a rank r extension of ω at random”. Main result: For every r = 1, 2,..., [rank ω/2], states of (E r (ω), P r,ω) are almost surely entangled. 1.
G.A. RAGGIO Abstract. We give a brief and incomplete survey of the problem of entan- glement of of composite quantum systems. 1960's - 1980's there were serious attempts (all of them anticipated by J. von Neu- mann) to do away with the underlying Hilbert space and work directly with the abstract algebraic structure modeled by C � -algebras or W � -algebras (abstract von Neumann algebras)(1, 2, 3). This was particularly fruitful when dealing with sys- tems of infinitely many degrees of freedom (quantum fields, thermodynamic limits, etc.). The problem of entanglement which we want to address here can be formu- lated quite straightforwardly in this algebraic framework of quantum theory. But we will stick to the quantum mechanics of the 1930's and keep the Hilbert space. Mainly because the results which are available concern the finite dimensional case. The states of the quantum system specified by H are associated with the lin- ear functionals f : B(H) → C which are positive (i.e., f(a) ≥ 0 for all positive a ∈ B(H)), normalized (i.e., f(1) = 1) and normal (i.e., f(supn{an}) = supnf(an) for any increasing family of selfadjoint operators {an : n = 1,2, � � � } which is uni- formly bounded). It is this normality condition which guarantees that the are in one-to-one correspondence with positive trace class operators d of unit trace (density operators) via the formula B(H) ∋ a 7→ tr(da). This gives an extremely convenient representation of and one often confuses the state f as a linear functional with the associated density operator d for which f(a) = tr(da). States are automatically continuous, and satisfy f(a � ) = f(a). The complex number f(a) is interpreted probabilistically as the expected value of the observable associated with the operator a when the system is in the state f. The normalization condi- tion is thus necessary for the consistency of this interpretation and the normality condition is seen as a non-commutative version of the �-additivity of probability measures. Clearly states, which we will denote by �(H), form a convex set which is closed
A density operator (state) on a tensor product H ⊗ K of Hilbert spaces is separable if it is in the convex closure of the subset of all tensor product states. Non-separable states are called entangled. These concepts are of great importance in quantum information theory, but they have been studied in depth only in the finite-dimensional context [1]. In this note we give a general integral representation for separable states and provide the first example of separable states that are not countably decomposable. We also prove a structure theorem for quantum communication channels that are entanglement-breaking, generalizing the finite-dimensional result of [2]. In the finite-dimensional case such channels can be characterized as having a Stinespring–Kraus representation (3) with operators Vj of rank 1. The above example implies the existence of infinite-dimensional entanglement-breaking channels having no such representation. In what follows, H, K ,... are separable Hilbert spaces, T(H) is the Banach space of trace-class operators and S(H) is the convex subset of all density operators on H. For brevity we shall also call them states, having in mind that a density operator ρ uniquely determines a normal state on the algebra B(H) of all bounded operators on H. Equipped with the trace-norm topology, S(H) is a complete separable metric space. If π is a Borel probability measure on S(H), then the Bochner integral
The restricted numerical range W R ( A ) of an operator A acting on a D -dimensional Hilbert space is defined as a set of all possible expectation values of this operator among pure states which belong to a certain subset R of the set of pure quantum states of dimension D . One considers for instance the set of real states, or in the case of composite spaces, the set of product states and the set of maximally entangled states. Combining the operator theory with a probabilistic approach we introduce the restricted numerical shadow of A —a normalized probability distribution on the complex plane supported in W R ( A ). Its value at point is equal to the probability that the expectation value 〈ψ| A |ψ〉 is equal to z , where |ψ〉 represents a random quantum state in subset R distributed according to the natural measure on this set, induced by the unitarily invariant Fubini–Study measure. Studying restricted shadows of operators of a fixed size D = N A N B we analyse the geometry of sets of separable and maximally entangled states of the N A × N B composite quantum system. Investigating trajectories formed by evolving quantum states projected into the plane of the shadow, we study the dynamics of quantum entanglement. A similar analysis extended for operators on D = 2 3 -dimensional Hilbert space allows us to investigate the structure of the orbits of GHZ and W quantum states of a three-qubit system.
This paper, a continuation of a previous one ( Iwai 2007 J. Phys. A: Math. Theor. 40 1361 ), studies the geometry of multi-qubit entanglement with respect to bipartite partitions. An n -qubit system ( C 2 ) ⊗ n is isomorphic with , the linear space of 2 ℓ × 2 m complex matrices, where ℓ + m = n . According to the isomorphism, the local transformation group U (2 ℓ ) × U (2 m ) acts on the space of normalized states in . Let M and G denote the space of normalized states and the local transformation group acting on M , respectively. According to the orbit types, M is stratified into strata, among which a principal stratum and the sets of separable states and maximally entangled states will be identified. For , a function F ( C ) = det( I − CC *) proves to serve as a measure of entanglement, where I denotes the 2 ℓ × 2 ℓ identity matrix. The F ( C ) attains the minimal and the maximal values, respectively, on the sets of separable states and maximally entangled states, and further takes no extremal values on the principal stratum. The F ( C ) projects to a function on the factor space G \ M . A naturally defined metric on M also projects to that on G \ M , which serves to measure the distance between the separable states, F −1 (0), and the states, F −1 ( k ), of prescribed value k of measure. To be precise, one has to restrict M to the principal stratum, when projecting the metric. Three- and four-qubit systems will be studied intensively, and then multi-qubit systems discussed.
Let H and K be (finite or infinite dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from B(H) into B(K) is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is give which shows that a state ρ on H⊗K is separable if and only if (Φ⊗I)ρ≥0 for all positive finite rank elementary operators Φ. Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.