# Tensor product matrices order relation

• A
• Jufa
In summary, the conversation discusses the need to prove that a certain quantity involving bra and ket states is greater than or equal to zero for all product states. While it is easy to demonstrate for a product state, the challenge lies in proving it for a general and potentially entangled state. The conversation also raises the question of whether the inequality holds for a sum of product states, specifically orthogonal ones.
Jufa
TL;DR Summary
Having that ##A \geq \pm B## how can one prove that indeed ##A^{\otimes n} \geq \pm B^{\otimes n}##
We mainly have to prove that this quantity## \bra{\varphi} A^{\otimes n } \ket{\varphi} \pm \bra{\varphi} B^{\otimes n } \ket{\varphi} ##

is greater or equal than zero for all ##\ket{\varphi}##.

Being ##\ket{\varphi}## a product state it is straightforward to demonstrate such inequality. I am struggling though to demonstrate it for a general, perhaps entangled ##\ket{\varphi}##, because of the cross products that show up.

Can you show that if the inequality is true for a product state that its true for a sum of product states? Specifically for a sum of orthogonal product states?

## 1. What is a tensor product matrix?

A tensor product matrix is a mathematical operation that combines two or more matrices to create a new matrix. It is denoted by the symbol ⊗ and is commonly used in linear algebra and multilinear algebra.

## 2. What is the order relation for tensor product matrices?

The order relation for tensor product matrices is a way to compare the sizes of two matrices. It is defined as follows: given two matrices A and B, A is said to be less than or equal to B if the dimensions of A are less than or equal to the dimensions of B.

## 3. How is the order relation for tensor product matrices useful?

The order relation for tensor product matrices is useful for determining the compatibility of matrices in various mathematical operations. It allows for the comparison of matrix sizes and ensures that the matrices being combined have the appropriate dimensions.

## 4. Can the order relation for tensor product matrices be extended to more than two matrices?

Yes, the order relation for tensor product matrices can be extended to any number of matrices. In this case, the order relation is defined as follows: given n matrices A1, A2, ..., An, A1 is said to be less than or equal to A2 if the dimensions of A1 are less than or equal to the dimensions of A2, and so on for all n matrices.

## 5. Are there any special properties of the order relation for tensor product matrices?

Yes, the order relation for tensor product matrices has several important properties. It is reflexive, meaning that a matrix is always less than or equal to itself. It is also transitive, meaning that if A is less than or equal to B and B is less than or equal to C, then A is also less than or equal to C. Additionally, the order relation is antisymmetric, meaning that if A is less than or equal to B and B is less than or equal to A, then A and B must be equal.

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