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THE PROBABILITY OF ENTANGLEMENT

William Arveson

Published Nov 27, 2012• 13 citations
OpenAlex

Abstract

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.

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The space of extensions that restrict to faithful states

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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.

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Bipartite states of low rank are almost surely entangled

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We show that a bipartite state on a tensor product of two matrix algebras is almost surely entangled if its rank is not greater than that of one of its reduced density matrices.

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Algebraic measures of entanglement

Jean-Luc Brylinski

Abstract We study the rank of a general tensor u in a tensor product space H1 ⊗ · · · ⊗Hk. The rank of u is the minimal number p of decomposable states v1, · · · , vp such that u is a linear combination of the vj ’s. This rank is an algebraic measure of the degree of entanglement of u. Motivated by quantum computation, we completely describe the rank of an arbitrary tensor in (C 2 )⊗3 and give normal forms for tensor states up to local unitary transformations. We also obtain partial results for (C 2 )⊗4; in particular, we show that the maximal rank of a tensor in (C 2 )⊗4 is equal to 4.

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Unitary-orbit classification and a refinement theorem for context-independent projective probabilities

Michael P. Rubin

Let $\mathcal H$ be a finite-dimensional complex Hilbert space, and let $w(P,\mathsf M)$ be a normalized probability weight assigned to an outcome projection $P$ as it occurs in a projective measurement $\mathsf M$. For fixed $P$, let $G_P\cong\mathcal U(P^\perp)$ be the group of unitaries acting identically on $\operatorname{ran}P$ and arbitrarily on $P^\perp$. We classify the $G_P$-orbits of projective measurements containing $P$: two measurements lie in the same orbit exactly when the multisets of ranks of their complementary outcomes agree. The orbit with profile $λ=(r_1,\ldots,r_k)$ is a compact homogeneous space of real dimension $(d-\operatorname{rank}P)^2-\sum_j r_j^2$. On maximal rank-one measurements there is one orbit, so context independence is equivalent to $G_P$-invariance, with equality of the corresponding uniform defects; invariance under two-level complementary unitaries already suffices. For arbitrary projective measurements, every context containing an outcome $P\neq I$ coarsens to the unique binary context $\{P,I-P\}$. Consequently, refinement consistency alone is equivalent to context independence. The finite orbit space carries a natural rank-profile refinement graph, whose diameter is $d-\operatorname{rank}P-1$ when $P\neq I$. We prove a stability theorem that compares the binary-coarsening path with a shortest path in this graph followed by one complementary unitary. In dimension at least three, Gleason's theorem converts the maximal-context invariance condition and the all-context refinement condition, on their respective domains, into the Born form $\operatorname{Tr}(ρP)$. The results are structural characterizations, not independent physical derivations of context independence.

quant-pharXiv:2608.16936•Aug 11, 2026•0 citations
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.
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 study the rank of a general tensor u in a tensor product space H1 ⊗ · · · ⊗Hk. The rank of u is the minimal number p of decomposable states v1, · · · , vp such that u is a linear combination of the vj ’s. This rank is an algebraic measure of the degree of entanglement of u. Motivated by quantum computation, we completely describe the rank of an arbitrary tensor in (C 2 )⊗3 and give normal forms for tensor states up to local unitary transformations. We also obtain partial results for (C 2 )⊗4; in particular, we show that the maximal rank of a tensor in (C 2 )⊗4 is equal to 4.
We study the rank of a general tensor uuu in a tensor product $H_1\ot...\ot H_k$. The rank of uuu is the minimal number ppp of pure states v1,...,vpv_1,...,v_pv1​,...,vp​ such that uuu is a linear combination of the vjv_jvj​'s. This rank is an algebraic measure of the degree of entanglement of uuu. Motivated by quantum computation, we completely describe the rank of an arbitrary tensor in $(\C^2)^{\ot 3}$ and give normal forms for tensor states up to local unitary transformations. We also obtain partial results for $(\C^2)^{\ot 4}$; in particular, we show that the maximal rank of a tensor in $(\C^2)^{\ot 4}$ is equal to 4.
The main goal of this paper is to study the geometric structures associated with the representation of tensors in subspace based formats. To do this we use a property of the so-called minimal subspaces which allows us to describe the tensor representation by means of a rooted tree. By using the tree structure and the dimensions of the associated minimal subspaces, we introduce, in the underlying algebraic tensor space, the set of tensors in a tree-based format with either bounded or fixed tree-based rank. This class contains the Tucker format and the Hierarchical Tucker format (including the Tensor Train format). In particular, we show that the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank of an algebraic tensor product of normed vector spaces is an analytic Banach manifold. Indeed, the manifold geometry for the set of tensors with fixed tree-based rank is induced by a fibre bundle structure and the manifold geometry for the set of tensors with bounded tree-based rank is given by a finite union of connected components. In order to describe the relationship between these manifolds and the natural ambient space, we introduce the definition of topological tensor spaces in the tree-based format. We prove under natural conditions that any tensor of the topological tensor space under consideration admits best approximations in the manifold of tensors in the tree-based format with bounded tree-based rank. In this framework, we also show that the tangent (Banach) space at a given tensor is a complemented subspace in the natural ambient tensor Banach space and hence the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank is an immersed submanifold. This fact allows us to extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces.
Let H\mathcal HH be a finite-dimensional complex Hilbert space, and let w(P,M)w(P,\mathsf M)w(P,M) be a normalized probability weight assigned to an outcome projection PPP as it occurs in a projective measurement M\mathsf MM. For fixed PPP, let GP≅U(P⊥)G_P\cong\mathcal U(P^\perp)GP​≅U(P⊥) be the group of unitaries acting identically on ran⁡P\operatorname{ran}PranP and arbitrarily on P⊥P^\perpP⊥. We classify the GPG_PGP​-orbits of projective measurements containing PPP: two measurements lie in the same orbit exactly when the multisets of ranks of their complementary outcomes agree. The orbit with profile λ=(r1,…,rk)λ=(r_1,\ldots,r_k)λ=(r1​,…,rk​) is a compact homogeneous space of real dimension (d−rank⁡P)2−∑jrj2(d-\operatorname{rank}P)^2-\sum_j r_j^2(d−rankP)2−∑j​rj2​. On maximal rank-one measurements there is one orbit, so context independence is equivalent to GPG_PGP​-invariance, with equality of the corresponding uniform defects; invariance under two-level complementary unitaries already suffices. For arbitrary projective measurements, every context containing an outcome P≠IP\neq IP=I coarsens to the unique binary context {P,I−P}\{P,I-P\}{P,I−P}. Consequently, refinement consistency alone is equivalent to context independence. The finite orbit space carries a natural rank-profile refinement graph, whose diameter is d−rank⁡P−1d-\operatorname{rank}P-1d−rankP−1 when P≠IP\neq IP=I. We prove a stability theorem that compares the binary-coarsening path with a shortest path in this graph followed by one complementary unitary. In dimension at least three, Gleason's theorem converts the maximal-context invariance condition and the all-context refinement condition, on their respective domains, into the Born form Tr⁡(ρP)\operatorname{Tr}(ρP)Tr(ρP). The results are structural characterizations, not independent physical derivations of context independence.