# Three qubit identity

Problem:

I'm having trouble seeing how this paper claims the following identity to be true for a three qubit system (labelled by $$A$$, $$B$$, and $$C$$) under a pure state $$|\psi \rangle$$ with real coefficients

$$\langle X_{A} X_{B}\rangle ^2 + \langle Z_{A} Z_{B}\rangle ^2 + \langle Z_{A} X_{B}\rangle ^2 + \langle X_{A} Z_{B}\rangle ^2 = 1 + \langle Y_{A} Y_{B}\rangle ^2 - \langle Y_{A} Y_{C}\rangle ^2 - \langle Y_{B} Y_{C}\rangle ^2$$

where $$X$$, $$Y$$, and $$Z$$ are the three Pauli matrices and the subscript is the qubit label. The expectation values have the usual meaning $$\langle X_A X_B \rangle \equiv \text{tr} \rho X_A \otimes X_B \otimes 1$$ where $$\rho \equiv \text{tr}_C |\psi \rangle \langle \psi |$$.

My Guess:

I'm thinking that this identity arises out of something like $$X^2 + Y^2 + Z^2=3$$ but that approach leads me nowhere since the square is outside the expectation brackets. Another problem is that while on the LHS the $$C$$ space is being traced over, on the RHS it's either $$A$$ or $$B$$ that's being traced over, and the square on top of everything just makes it worse. I suspect this is a well known but tediously derived identity, in which case I'll be happy to be pointed to some source material which grinds this out.

• Is the last expectation value supposed to be $\left<Y_B Y_C\right>$? – Joseph Geipel Dec 20 '20 at 1:16