User:Scythe33/Thalipyal

A demonstration of Thalipyal's inequality. I figured this out as I was walking home from class.

We have:

$$ |\psi \rangle = \alpha |\phi \rangle + \beta |\varphi \rangle $$

Without loss of generality:

$$ Sch(|\psi \rangle ) = x, Sch(|\phi \rangle ) = n, Sch(|\varphi \rangle ) = m, m \leq n $$

We have:

$$ \sum_{i=0}^{x-1} \lambda_i^{\psi} |\psi^A_i \rangle | \psi^B_i \rangle = \alpha \sum_{i=0}^{n-1} \lambda_i^{\phi} |\phi^A_i \rangle | \phi^B_i \rangle + \beta \sum_{i=0}^{m-1} \lambda_i^{\varphi} |\varphi^A_i \rangle | \varphi^B_i \rangle$$

Subtracting $$ \beta |\varphi \rangle $$ from both sides, we obtain:

$$\sum_{i=0}^{x-1} \lambda_i^{\psi} |\psi^A_i \rangle | \psi^B_i \rangle - \beta \sum_{i=0}^{m-1} \lambda_i^{\varphi} |\varphi^A_i \rangle | \varphi^B_i \rangle = \alpha \sum_{i=0}^{n-1} \lambda_i^{\phi} |\phi^A_i \rangle | \phi^B_i \rangle$$

Now left-multiply both sides by $$ \langle \phi^A_k |, 0 \leq k < n $$. We obtain:

$$\sum_{i=0}^{x-1} \lambda_i^{\psi} \langle \phi^A_i |\psi^A_i \rangle | \psi^B_i \rangle - \beta \sum_{i=0}^{m-1} \lambda_i^{\varphi} \langle \phi^A_i |\varphi^A_i \rangle | \varphi^B_i \rangle = \alpha \lambda_k^{\phi} | \phi^B_k \rangle $$

There are n such equations, since there are n terms in the Schmidt decomposition of $$| \phi \rangle $$. Furthermore, the RHSs of each of the n equations are orthogonal vectors in system B, whereas the LHSs are linear combinations of m + x vectors in system B. Since the LHS must span the RHS, and consists of linear combinations of m + x vectors, we require:

$$m + x \geq n \vdash x \geq n - m \vdash Sch(| \psi \rangle ) \geq | Sch(| \phi \rangle ) - Sch(| \varphi \rangle ) | $$

and the theorem is proved.