We study the rank of a general tensor $u$ in a tensor product $H_1\ot...\ot H_k$. The rank of $u$ is the minimal number $p$ of pure states $v_1,...,v_p$ such that $u$ is a linear combination of the $v_j$ '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 $(\C^2)^{\ot 3}$ 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.We study the rank of a general tensor $u$ in a tensor product $H_1\ot...\ot H_k$. The rank of $u$ is the minimal number $p$ of pure states $v_1,...,v_p$ such that $u$ is a linear combination of the $v_j$'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 $(\C^2)^{\ot 3}$ 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.

arXiv:quant-ph/0008031Aug 6, 20000 citations

The geometric measure of entanglement of multipartite states and the Z-eigenvalue of tensors

Liang Xiong, Jianzhou Liu, Qi Qin

Quantum Information ProcessingFeb 22, 202210 citations

Entanglement in C $^*$ -algebras: tensor products of state spacesEntanglement in C$^*$-algebras: tensor products of state spaces

Magdalena Musat, Mikael Rørdam

We analyze the Namioka-Phelps minimal and maximal tensor products of compact convex sets which arise as the state spaces of unital C $^*$ -algebras. Relatedly, we study entanglement in (infinite dimensional) C $^*$ -algebras. While the minimal Namioka-Phelps tensor product of the state spaces of two C $^*$ -algebras is the well-known set of separable (= un-entangled) states on the (minimal) tensor product of the C $^*$ -algebras, we also describe the more elusive maximal Namioka-Phelps tensor product of state spaces of C $^*$ -algebras. We show that the minimal and maximal tensor products of state spaces of C $^*$ -algebras agree precisely when one of the two C $^*$ -algebras is commutative, which confirms Barker's conjecture in the case where the compact convex sets are state paces of C $^*$ -algebras. Further, the Namioka-Phelps tensor product of the trace simplexes of two or more unital C $^*$ -algebras is shown to be the trace simplex of the (minimal or maximal) tensor product of the C $^*$ -algebras. This enables a systematic way of determining the trace simplex of a tensor product of C $^*$ -algebras.We analyze the Namioka-Phelps minimal and maximal tensor products of compact convex sets which arise as the state spaces of unital C$^*$-algebras. Relatedly, we study entanglement in (infinite dimensional) C$^*$-algebras. While the minimal Namioka-Phelps tensor product of the state spaces of two C$^*$-algebras is the well-known set of separable (= un-entangled) states on the (minimal) tensor product of the C$^*$-algebras, we also describe the more elusive maximal Namioka-Phelps tensor product of state spaces of C$^*$-algebras. We show that the minimal and maximal tensor products of state spaces of C$^*$-algebras agree precisely when one of the two C$^*$-algebras is commutative, which confirms Barker's conjecture in the case where the compact convex sets are state paces of C$^*$-algebras. Further, the Namioka-Phelps tensor product of the trace simplexes of two or more unital C$^*$-algebras is shown to be the trace simplex of the (minimal or maximal) tensor product of the C$^*$-algebras. This enables a systematic way of determining the trace simplex of a tensor product of C$^*$-algebras.

math.OACross-listedarXiv:2512.10410Dec 12, 2025

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.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.

Chapman & Hall/CRC eBooksFeb 14, 200220 citations

Entanglement Measures and Their Properties in Quantum Field Theory

Stefan Hollands, Ko Sanders

arXiv:1702.04924SpringerBriefs in mathematical physicsJan 1, 201885 citations

We study the rank of a general tensor

u

in a tensor product $H_1\ot...\ot H_k$. The rank of

u

is the minimal number

p

of pure states

v_1,...,v_p

such that

u

is a linear combination of the

v_j

'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 $(\C^2)^{\ot 3}$ 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.

Entanglement in C

^*

-algebras: tensor products of state spaces

We analyze the Namioka-Phelps minimal and maximal tensor products of compact convex sets which arise as the state spaces of unital C

^*

-algebras. Relatedly, we study entanglement in (infinite dimensional) C

^*

-algebras. While the minimal Namioka-Phelps tensor product of the state spaces of two C

^*

-algebras is the well-known set of separable (= un-entangled) states on the (minimal) tensor product of the C

^*

-algebras, we also describe the more elusive maximal Namioka-Phelps tensor product of state spaces of C

^*

-algebras. We show that the minimal and maximal tensor products of state spaces of C

^*

-algebras agree precisely when one of the two C

^*

-algebras is commutative, which confirms Barker's conjecture in the case where the compact convex sets are state paces of C

^*

-algebras. Further, the Namioka-Phelps tensor product of the trace simplexes of two or more unital C

^*

-algebras is shown to be the trace simplex of the (minimal or maximal) tensor product of the C

^*

-algebras. This enables a systematic way of determining the trace simplex of a tensor product of C

^*

-algebras.