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"simple tensors" --> "rank-one tensors"
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Jess Riedel
Jess Riedel
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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 simplerank-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?

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 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?

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 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?

"product state" --> "simple tensor"
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Jess Riedel
Jess Riedel
  • 846
  • 5
  • 19

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 "product states"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 statessimple 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?

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 "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 states, 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 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?

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 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?

One more question
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Jess Riedel
Jess Riedel
  • 846
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Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more Hilbertvector spaces?

(I use I've rewritten the question in math notation, and I've left the old version in physics bra-ket notation since I'm a physicists and this question is inspired by quantum-infohere. 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"?)

Background

A simple consequence of the singular value decomposition is that any vector ("state") $\vert v \rangle$$v$ in a Hilbertvector space $\mathcal{H}$$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} = \mathcal{H}_1 \otimes \mathcal{H}_2$$v \in V = U \otimes W$,

has a special decomposition in terms of product states"product states"

$\vert v \rangle = \sum_{i=1}^d \lambda_i \vert i \rangle_1 \otimes \vert i \rangle_2$$v = \sum_{i=1}^d \lambda_i u_i \otimes w_i $, $\qquad d = \mathrm{min}(d_1,d_2)$$\qquad u_i \in U$, $\qquad w_i \in W$, $\qquad d = \mathrm{min}(d_U,d_W)$

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} = \bigotimes_{n=1}^N \mathcal{H}_n$$V = \bigotimes_{n=1}^N V_n$ cannot be expressed as a sum of fewer than $\tilde{d}$ product states, with $\tilde{d} \gg d_n = \mathrm{dim}\mathcal{H}_n$$\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 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:

$\vert GHZ \rangle = \vert 0 \rangle_1 \otimes \cdots \otimes \vert 0 \rangle_N + \vert 1 \rangle_1 \otimes \cdots \otimes \vert 1 \rangle_N$$v_{GHZ} = a_1 \otimes \cdots \otimes a_N + b_1 \otimes \cdots \otimes b_N$, $\qquad \langle 0 \vert 1 \rangle_n = 0$$\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 (Shannon) 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

$\vert v \rangle = \sum_{i=1}^{\tilde{d}} \lambda_i \vert i \rangle$$v = \sum_{i=1}^{\tilde{d}} \lambda_i v_i$, $\qquad \vert i \rangle = \bigotimes_{n=1}^N \vert \psi_i^n \rangle_n$$\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 $\{ \vert i \rangle \}$$\{ v_i \}$ are orthonormal. Note that we allow $\langle \psi_i^n \vert \psi_j^n \rangle_n \neq 0$$\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 Hilbertvector 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?

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

(I use bra-ket notation since I'm a physicists and this question is inspired by quantum-info. I'm happy to switch over to standard math notation if it would convince a mathematician to take a look.)

Background

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

$\vert v \rangle \in \mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2$,

has a special decomposition in terms of product states

$\vert v \rangle = \sum_{i=1}^d \lambda_i \vert i \rangle_1 \otimes \vert i \rangle_2$, $\qquad d = \mathrm{min}(d_1,d_2)$

built from fixed the orthonormal bases $\{ \vert i \rangle_1 \}$ and $\{ \vert i \rangle_2 \}$.

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 $\mathcal{H} = \bigotimes_{n=1}^N \mathcal{H}_n$ cannot be expressed as a sum of fewer than $\tilde{d}$ product states, with $\tilde{d} \gg d_n = \mathrm{dim}\mathcal{H}_n$ 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 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:

$\vert GHZ \rangle = \vert 0 \rangle_1 \otimes \cdots \otimes \vert 0 \rangle_N + \vert 1 \rangle_1 \otimes \cdots \otimes \vert 1 \rangle_N$, $\qquad \langle 0 \vert 1 \rangle_n = 0$ for all $n$.

Specific Question

If we guess that the (Shannon) 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

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

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

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?

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

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 "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 states, 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 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?

Source Link
Jess Riedel
Jess Riedel
  • 846
  • 5
  • 19