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4 "simple tensors" --> "rank-one tensors"

I've rewritten the question in math notation, and I've left the old version in physics bra-ket notation here.

## Background

A simple consequence of the singular value decomposition is that any vector $v$ in a vector space $V$ formed by the tensor product of two smaller spaces ("subsystems") $U$ and $W$ of dimension $d_U$ and $d_W$,

$v \in V = U \otimes W$,

has a special decomposition in terms of simple rank-one tensors (aka product states)

$v = \sum_{i=1}^d \lambda_i u_i \otimes w_i$, $\qquad u_i \in U$, $\qquad w_i \in W$, $\qquad d = \mathrm{min}(d_U,d_W)$

built from the fixed orthonormal bases $\{ u_i \}$ and $\{ w_i \}$.

This decomposition has many nice properties, and it's simple to see that there's no way to generalize it to the case of 3 or more subsystems while keeping all of them. For instance, a generic state in $V = \bigotimes_{n=1}^N V_n$ cannot be expressed as a sum of fewer than $\tilde{d}$ simple tensors, with $\tilde{d} \gg d_n = \mathrm{dim}V_n$ for all $n=1,\ldots,N$.

## Vague Question

Is there a useful/natural/canonical decomposition of a vector into a "small" number of orthogonal simple tensors for the case of 3 or more subsystems? At the very least, the GHZ state should take it's canonical form in such a decompostion:

$v_{GHZ} = a_1 \otimes \cdots \otimes a_N + b_1 \otimes \cdots \otimes b_N$, $\qquad a_n, b_n \in V_n$, $\qquad \langle a_n ; b_n \rangle = 0$ for all $n$.

## Specific Question

If we guess that the entropy function

$H[\{ p_i \} ] = -\sum_i p_i \ln p_i$

is appropriate, we can define the "minimum-entropy product-state decomposition" to be the decomposition

$v = \sum_{i=1}^{\tilde{d}} \lambda_i v_i$, $\qquad v_i = \bigotimes_{n=1}^N \psi_i^n$, $\qquad \psi_i^n \in V_n$

with the minimum value for the entropy $H[\{ p_i = \lambda_i^2 \} ]$, under the condition that the $\{ v_i \}$ are orthonormal. Note that we allow $\langle \psi_i^n ; \psi_j^n \rangle \neq 0$ for $i \neq j$.

The natural questions to ask are: Is this decomposition generically unique (i.e. unique except for some set of measure zero in the global vector space)? Is it continuous? (What other properties should this decomposition have to satisfy to be useful?)

I am 99% sure that in the case of $N=2$, this reduces to the Schmidt decomposition and that the answers to both questions is "yes".

Is any of this sensitive to our choice of the entropy function $H$, as opposed to some other permutation-invariant and majoritization-preserving function of the spectrum?

3 "product state" --> "simple tensor"

I've rewritten the question in math notation, and I've left the old version in physics bra-ket notation here.(Is there standard math terminology for "product state" and "subsystem"?)

## Background

A simple consequence of the singular value decomposition is that any vector $v$ in a vector space $V$ formed by the tensor product of two smaller spaces ("subsystems") $U$ and $W$ of dimension $d_U$ and $d_W$,

$v \in V = U \otimes W$,

has a special decomposition in terms of "simple tensors (aka product states")

$v = \sum_{i=1}^d \lambda_i u_i \otimes w_i$, $\qquad u_i \in U$, $\qquad w_i \in W$, $\qquad d = \mathrm{min}(d_U,d_W)$

built from the fixed orthonormal bases $\{ u_i \}$ and $\{ w_i \}$.

This decomposition has many nice properties, and it's simple to see that there's no way to generalize it to the case of 3 or more subsystems while keeping all of them. For instance, a generic state in $V = \bigotimes_{n=1}^N V_n$ cannot be expressed as a sum of fewer than $\tilde{d}$ product statessimple tensors, with $\tilde{d} \gg d_n = \mathrm{dim}V_n$ for all $n=1,\ldots,N$.

## Vague Question

Is there a useful/natural/canonical decomposition of a vector into a "small" number of orthogonal product states simple tensors for the case of 3 or more subsystems? At the very least, the GHZ state should take it's canonical form in such a decompostion:

$v_{GHZ} = a_1 \otimes \cdots \otimes a_N + b_1 \otimes \cdots \otimes b_N$, $\qquad a_n, b_n \in V_n$, $\qquad \langle a_n ; b_n \rangle = 0$ for all $n$.

## Specific Question

If we guess that the entropy function

$H[\{ p_i \} ] = -\sum_i p_i \ln p_i$

is appropriate, we can define the "minimum-entropy product-state decomposition" to be the decomposition

$v = \sum_{i=1}^{\tilde{d}} \lambda_i v_i$, $\qquad v_i = \bigotimes_{n=1}^N \psi_i^n$, $\qquad \psi_i^n \in V_n$

with the minimum value for the entropy $H[\{ p_i = \lambda_i^2 \} ]$, under the condition that the $\{ v_i \}$ are orthonormal. Note that we allow $\langle \psi_i^n ; \psi_j^n \rangle \neq 0$ for $i \neq j$.

The natural questions to ask are: Is this decomposition generically unique (i.e. unique except for some set of measure zero in the global vector space)? Is it continuous? (What other properties should this decomposition have to satisfy to be useful?)

I am 99% sure that in the case of $N=2$, this reduces to the Schmidt decomposition and that the answers to both questions is "yes".

Is any of this sensitive to our choice of the entropy function $H$, as opposed to some other permutation-invariant and majoritization-preserving function of the spectrum?

2 One more question

# Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more Hilbertvector spaces?

(I use bra-ket

I've rewritten the question in math notationsince I'm a physicists , and this question is inspired by quantum-infoI've left the old version in physics bra-ket notation here. I'm happy to switch over to (Is there standard math notation if it would convince a mathematician to take a look.)terminology for "product state" and "subsystem"?)

A simple consequence of the singular value decomposition is that any vector ("state") $\vert v \rangle$ v$in a Hilbert vector space$\mathcal{H}$V$ formed by the tensor product of two smaller spaces ("subsystems") $\mathcal{H}_1$ U$and$\mathcal{H}_2$W$ of dimension $d_1$ d_U$and$d_2$,d_W$,

$\vert v \rangle \in \mathcal{H} V = \mathcal{H}_1 U \otimes \mathcal{H}_2$,W$, has a special decomposition in terms of "product states"$\vert v \rangle = \sum_{i=1}^d \lambda_i \vert i \rangle_1 u_i \otimes w_i $,$\qquad u_i \vert i in U$,$\qquad w_i \rangle_2$, in W$, $\qquad d = \mathrm{min}(d_1,d_2)$mathrm{min}(d_U,d_W)$built from fixed the fixed orthonormal bases$\{ \vert i \rangle_1 u_i \}$and$\{ \vert i \rangle_2 w_i \}$. This decomposition has many nice properties, and it's simple to see that there's no way to generalize it to the case of 3 or more subsystems while keeping all of them. For instance, a generic state in$\mathcal{H} V = \bigotimes_{n=1}^N \mathcal{H}_n$V_n$ cannot be expressed as a sum of fewer than $\tilde{d}$ product states, with $\tilde{d} \gg d_n = \mathrm{dim}\mathcal{H}_n$ mathrm{dim}V_n$for all$n=1,\ldots,N$.$\vert GHZ \rangle v_{GHZ} = \vert 0 \rangle_1 a_1 \otimes \cdots \otimes \vert 0 \rangle_N a_N + \vert 1 \rangle_1 b_1 \otimes \cdots \otimes \vert 1 b_N$,$\qquad a_n, b_n \rangle_N$, in V_n$, $\qquad \langle 0 \vert 1 a_n ; b_n \rangle_n rangle = 0$ for all $n$.

If we guess that the (Shannon) entropy function

$\vert v \rangle = \sum_{i=1}^{\tilde{d}} \lambda_i \vert i \rangle$, v_i$,$\qquad \vert i \rangle v_i = \bigotimes_{n=1}^N \vert psi_i^n$,$\qquad \psi_i^n \rangle_n$, in V_n$

with the minimum value for the entropy $H[\{ p_i = \lambda_i^2 \} ]$, under the condition that the $\{ \vert i \rangle v_i \}$ are orthonormal. Note that we allow $\langle \psi_i^n \vert ; \psi_j^n \rangle_n rangle \neq 0$ for $i \neq j$.

The natural questions to ask are: Is this decomposition generically unique (i.e. unique except for some set of measure zero in the global Hilbert vector space)? Is it continuous? (What other properties should this decomposition have to satisfy to be useful?)

1