Quantum Information HT24, No-signalling theorem

Sharing an entangled state is not enough for Alice and Bob to communicate.

Suppose:

• Alice wants to communicate value of a bit $x$ to Bob, using an entangled state of system $AB$ as the only resource
  • To communicate $x = 0$, Alice measures $A$ in ONB $A _ 0 = \{\alpha^0 _ m, m = 0, \cdots, d _ A - 1\}$
  • To communicate $x = 1$, Alice measures $A$ in ONB $A _ 1 = \{\alpha^1 _ m, m = 0, \cdots, d _ A - 1\}$
• Bob measures the system in the ONB $B = \{\beta _ m, m = 0, \cdots, d _ B - 1\}$

Then:

• The probability distribution of his outcomes does not depend on $x$.

Let $ \vert \Psi\rangle$ be the entangled state of the composite system.

\[\begin{aligned} p _ {AB}(m,n) &= \bigg\vert \langle \alpha _ m \vert \otimes \langle\beta _ n \vert \vert \Psi \rangle \bigg\vert^2 \\\\ &= \bigg[ \langle \Psi \vert \vert \alpha _ m \rangle\otimes \vert \beta _ n\rangle \bigg]\bigg[\langle \alpha _ m \vert \otimes \langle \beta _ n \vert \vert \Psi\rangle\bigg] \\\\ &= \langle\Psi \vert \bigg[ \vert \alpha _ m \rangle\otimes \vert \beta _ n\rangle \bigg]\bigg[\langle \alpha _ m \vert \otimes \langle \beta _ n \vert \bigg] \vert \Psi\rangle \\\\ &= \langle\Psi \vert \bigg( \vert \alpha _ m \rangle \langle \alpha _ m \vert \otimes \vert \beta _ n \rangle \langle \beta _ n \vert \bigg) \vert \Psi\rangle \end{aligned}\]

Then

\[\begin{aligned} p _ B(n) &= \sum^{d _ A - 1} _ {m = 0} p _ {AB}(m, n) \\\\ &= \sum^{d _ A - 1} _ {m = 0} \langle\Psi \vert \bigg( \vert \alpha _ m \rangle \langle \alpha _ m \vert \otimes \vert \beta _ n \rangle \langle \beta _ n \vert \bigg) \vert \Psi\rangle \\\\ &= \langle\Psi \vert \Bigg(\sum^{d _ A-1} _ {m = 0} \vert \alpha _ m \rangle \langle \alpha _ m \vert \otimes \vert \beta _ n \rangle \langle \beta _ n \vert \Bigg) \vert \Psi\rangle \\\\ &=\langle\Psi \vert \Bigg(\bigg(\sum^{d _ A-1} _ {m = 0} \vert \alpha _ m \rangle \langle \alpha _ m \vert \bigg) \otimes \vert \beta _ n \rangle \langle \beta _ n \vert \Bigg) \vert \Psi\rangle \\\\ &= \langle \Psi\bigg \vert I \otimes \vert \beta _ n \rangle \langle \beta _ n \vert \bigg \vert \Psi\rangle \end{aligned}\]