(The questions How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems? and Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces? seem to be asking something different from what follows, but perhaps I have not read them closely enough.) $\newcommand{\tp}{\otimes}$ $\newcommand{\ptp}{\hat{\otimes}}$ $\newcommand{\itp}{\check{\otimes}}$

In what follows: $\tp$ denotes the algebraic tensor product of complex vector spaces; $\ptp$ denotes the projective tensor product of Banach spaces; $\itp$ denotes the injective tensor product of Banach spaces; $\tp_2$ denotes the Hilbertian tensor product of two Hilbert spaces.

Let $H_1$, $H_2$ and $H_3$ be Hilbert spaces (you can assume finite--dimensional of arbitrary dimension, although ultimately I am after the separable infinite-dimensional case).

A consequence of the singular value decomposition (or Schmidt decomposition) is that each $w\in H_1\tp H_2$ can be written as $$ w = \sum_{i=1}^N \lambda_i e_i \tp f_i $$ where $$ \Vert w \Vert_{H_1\ptp H_2} = \sum_{i=1}^N |\lambda_i| \tag{1}$$ $$ \Vert w \Vert_{H_1\tp_2 H_2} = \left(\sum_{i=1}^N |\lambda_i|^2\right)^{1/2} \tag{2}$$ $$ \Vert w \Vert_{H_1\itp H_2} = \max_{1\leq i\leq N} |\lambda_i| \tag{$\infty$}$$

Question. Can we do the same for $H_1\tp H_2\tp H_3$?

That is, for given $w\in H_1\tp H_2\tp H_3$ we want vectors $(e_i)$, $(f_i)$, $(g_i)$ and scalars $(\lambda_i)$ such that $w=\sum_i \lambda_i e_i\tp f_i \tp g_i$ and the 3-variable analogues of $(1)$, $(2)$ and $(\infty)$ hold. I must admit this seems overly optimistic to me, so I wondered if there were standard counterexamples known, perhaps recorded in the quantum computing literature, or perhaps just folklore for specialists in Banach space theory.

(Remark: the SVD decomposition actually tells us that $(e_1,\dots, e_N)$ and $(f_1,\dots,f_N)$ are orthonormal. It isn't immediately clear to me if this is already forced by requiring $(1)$, $(2)$ and $(\infty)$, although I haven't given it any thought. In any case it doesn't seem to be directly needed in my intended application.)

(The questions How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems? and Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces? seem to be asking something different from what follows, but perhaps I have not read them closely enough.) $\newcommand{\tp}{\otimes}$ $\newcommand{\ptp}{\hat{\otimes}}$ $\newcommand{\itp}{\check{\otimes}}$

In what follows: $\tp$ denotes the algebraic tensor product of complex vector spaces; $\ptp$ denotes the projective tensor product of Banach spaces; $\itp$ denotes the injective tensor product of Banach spaces; $\tp_2$ denotes the Hilbertian tensor product of two Hilbert spaces.

Let $H_1$, $H_2$ and $H_3$ be Hilbert spaces (you can assume finite--dimensional of arbitrary dimension, although ultimately I am after the separable infinite-dimensional case).

A consequence of the singular value decomposition (or Schmidt decomposition) is that each $w\in H_1\tp H_2$ can be written as $$ w = \sum_{i=1}^N \lambda_i e_i \tp f_i $$ where $$ \Vert w \Vert_{H_1\ptp H_2} = \sum_{i=1}^N |\lambda_i| \tag{1}$$ $$ \Vert w \Vert_{H_1\tp_2 H_2} = \left(\sum_{i=1}^N |\lambda_i|^2\right)^{1/2} \tag{2}$$ $$ \Vert w \Vert_{H_1\itp H_2} = \max_{1\leq i\leq N} |\lambda_i| \tag{$\infty$}$$

Question. Can we do the same for $H_1\tp H_2\tp H_3$?

That is, for given $w\in H_1\tp H_2\tp H_3$ we want vectors $(e_i)$, $(f_i)$, $(g_i)$ and scalars $(\lambda_i)$ such that $w=\sum_i \lambda_i e_i\tp f_i \tp g_i$ and the 3-variable analogues of $(1)$, $(2)$ and $(\infty)$ hold. I must admit this seems overly optimistic to me, so I wondered if there were standard counterexamples known, perhaps recorded in the quantum computing literature, or perhaps just folklore for specialists in Banach space theory.

(Remark: the SVD decomposition actually tells us that $(e_1,\dots, e_N)$ and $(f_1,\dots,f_N)$ are orthonormal. It isn't immediately clear to me if this is already forced by requiring $(1)$, $(2)$ and $(\infty)$, although I haven't given it any thought. In any case it doesn't seem to be directly needed in my intended application.)

(The questions How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems? and Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces? seem to be asking something different from what follows, but perhaps I have not read them closely enough.) $\newcommand{\tp}{\otimes}$ $\newcommand{\ptp}{\hat{\otimes}}$ $\newcommand{\itp}{\check{\otimes}}$

In what follows: $\tp$ denotes the algebraic tensor product of complex vector spaces; $\ptp$ denotes the projective tensor product of Banach spaces; $\itp$ denotes the injective tensor product. $\tp_2$ denotes the Hilbertian tensor product of two Hilbert spaces.

Let $H_1$, $H_2$ and $H_3$ be Hilbert spaces (you can assume finite--dimensional of arbitrary dimension, although ultimately I am after the separable infinite-dimensional case).

A consequence of the singular value decomposition (or Schmidt decomposition) is that each $w\in H_1\tp H_2$ can be written as $$ w = \sum_{i=1}^N \lambda e_i \tp f_i $$ where $$ \Vert w \Vert_{H_1\ptp H_2} = \sum_{i=1}^N |\lambda_i| \tag{1}$$ $$ \Vert w \Vert_{H_1\tp_2 H_2} = \left(\sum_{i=1}^N |\lambda_i|^2\right)^{1/2} \tag{2}$$ $$ \Vert w \Vert_{H_1\itp H_2} = \max_{1\leq i\leq N} |\lambda_i| \tag{$\infty$}$$

Question. Can we do the same for $H_1\tp H_2\tp H_3$?

That is, for given $w\in H_1\tp H_2\tp H_3$ we want vectors $(e_i)$, $(f_i)$, $(g_i)$ and scalars $(\lambda_i)$ such that $w=\sum_i \lambda_i e_i\tp f_i \tp g_i$ and the 3-variable analogues of $(1)$, $(2)$ and $(\infty)$ hold. I must admit this seems overly optimistic to me, so I wondered if there were standard counterexamples known, perhaps recorded in the quantum computing literature, or perhaps just folklore for specialists in Banach space theory.

(Remark: the SVD decomposition actually tells us that $(e_1,\dots, e_N)$ and $(f_1,\dots,f_N)$ are orthonormal. It isn't immediately clear to me if this is already forced by requiring $(1)$, $(2)$ and $(\infty)$, although I haven't given it any thought. In any case it doesn't seem to be directly needed in my intended application.)

(The questions How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems? and Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces? seem to be asking something different from what follows, but perhaps I have not read them closely enough.) $\newcommand{\tp}{\otimes}$ $\newcommand{\ptp}{\hat{\otimes}}$ $\newcommand{\itp}{\check{\otimes}}$

In what follows: $\tp$ denotes the algebraic tensor product of complex vector spaces; $\ptp$ denotes the projective tensor product of Banach spaces; $\itp$ denotes the injective tensor product of Banach spaces; $\tp_2$ denotes the Hilbertian tensor product of two Hilbert spaces.

Let $H_1$, $H_2$ and $H_3$ be Hilbert spaces (you can assume finite--dimensional of arbitrary dimension, although ultimately I am after the separable infinite-dimensional case).

A consequence of the singular value decomposition (or Schmidt decomposition) is that each $w\in H_1\tp H_2$ can be written as $$ w = \sum_{i=1}^N \lambda_i e_i \tp f_i $$ where $$ \Vert w \Vert_{H_1\ptp H_2} = \sum_{i=1}^N |\lambda_i| \tag{1}$$ $$ \Vert w \Vert_{H_1\tp_2 H_2} = \left(\sum_{i=1}^N |\lambda_i|^2\right)^{1/2} \tag{2}$$ $$ \Vert w \Vert_{H_1\itp H_2} = \max_{1\leq i\leq N} |\lambda_i| \tag{$\infty$}$$

Question. Can we do the same for $H_1\tp H_2\tp H_3$?

That is, for given $w\in H_1\tp H_2\tp H_3$ we want vectors $(e_i)$, $(f_i)$, $(g_i)$ and scalars $(\lambda_i)$ such that $w=\sum_i \lambda_i e_i\tp f_i \tp g_i$ and the 3-variable analogues of $(1)$, $(2)$ and $(\infty)$ hold. I must admit this seems overly optimistic to me, so I wondered if there were standard counterexamples known, perhaps recorded in the quantum computing literature, or perhaps just folklore for specialists in Banach space theory.

(Remark: the SVD decomposition actually tells us that $(e_1,\dots, e_N)$ and $(f_1,\dots,f_N)$ are orthonormal. It isn't immediately clear to me if this is already forced by requiring $(1)$, $(2)$ and $(\infty)$, although I haven't given it any thought. In any case it doesn't seem to be directly needed in my intended application.)