arXiv:quant-ph/0008031

Algebraic measures of entanglement

Jean-Luc Brylinski

Published Aug 6, 2000• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.quant-ph/0008031

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arXiv:quant-ph/0008031

Algebraic measures of entanglement

Jean-Luc Brylinski

Published Aug 6, 2000• 0 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.48550/arxiv.quant-ph/0008031

Related papers

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

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Wojciech Bruzda, Shmuel Friedland, Karol Życzkowski

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Sophia Denker, Ismaël Septembre, Robin Krebs +1 more

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIEApr 1, 20090 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.

The rank of a tensor is analyzed in context of quantum entanglement. A pure quantum state

\bf v

of a composite system consisting of

d

subsystems with

n

levels each is viewed as a vector in the

d

-fold tensor product of

n

-dimensional Hilbert space and can be identified with a tensor with

d

indices, each running from

1

n

. We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state

{\bf v}

is called {\sl entangled}, if it {\sl cannot} be written in the product form,

{\bf v} \ne {\bf v}_1 \otimes {\bf v}_2 \otimes \cdots \otimes {\bf v}_d

, what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed..