The Exercise

Two quantum systems have state spaces $V$ and $W$, complex inner product spaces, and the composite system has state space $V \otimes W$. An operator $A$ on the first system and an operator $B$ on the second give the operator $A \otimes B$ on the composite system. The exercise asks to show that if $A$ and $B$ are positive, so is $A \otimes B$.

Positive Operators and Tensor Products

An operator $T$ on a complex inner product space $\mathcal{H}$ is positive if, for every vector $|\varphi\rangle$ of $\mathcal{H}$, $$\langle \varphi | T | \varphi \rangle \ge 0 .$$(1)

The space $V \otimes W$ is spanned by the product vectors $|v\rangle \otimes |w\rangle$, whose inner products are $\langle v \otimes w | v' \otimes w' \rangle = \langle v | v' \rangle \langle w | w' \rangle$.

The tensor product $A \otimes B$ of an operator $A$ on $V$ and an operator $B$ on $W$ is the operator on $V \otimes W$ that acts on product vectors as $$(A \otimes B)\bigl(|v\rangle \otimes |w\rangle\bigr) = A|v\rangle \otimes B|w\rangle$$(2) and on the rest of $V \otimes W$ by linearity.

The Claim

What Is Checked

The claim is stated over two complex inner product spaces $V$ and $W$, a positive operator on each in the sense of (1), and their tensor product (2), an operator on $V \otimes W$. What the library proves about these constructions is the inequality (3).