For $\ket{\uparrow_A}\ket{\downarrow_B}$ ,

\begin{align*} \text{Z}_A\text{Z}_B|↑_A⟩|\downarrow_B⟩ &= (+1)(-1)|↑_A⟩|\downarrow_B⟩\\ \text{Z}_A\text{Z}_B|\downarrow_A⟩|\uparrow_B⟩ &= (-1)(+1)|\downarrow_A⟩|\uparrow_B⟩\\ \Rightarrow \braket{\text{Z}_A\text{Z}_B} &= -1,\\ \text{Z}_A|↑_A⟩|\downarrow_B⟩ &= \left(+\frac{1}{2}\right)|↑_A⟩ + \left(-\frac{1}{2}\right)|\downarrow_B⟩\\ \text{Z}_A|\downarrow_A⟩|\uparrow_B⟩ &= \left(-\frac{1}{2}\right)|\downarrow_A⟩ + \left(+\frac{1}{2}\right)|\uparrow_B⟩\\ \Rightarrow \braket{\text{Z}_A} &= \frac{1}{2}(-1) +\frac{1}{2}(+1) = 0,\\ \text{Z}_B|↑_A⟩|\downarrow_B⟩ &= \left(+\frac{1}{2}\right)|↑_A⟩ + \left(-\frac{1}{2}\right)|\downarrow_B⟩\\ \text{Z}_B|\downarrow_A⟩|\uparrow_B⟩ &= \left(-\frac{1}{2}\right)|\downarrow_A⟩ + \left(+\frac{1}{2}\right)|\uparrow_B⟩\\ \Rightarrow \braket{\text{Z}_B} &= \frac{1}{2}(-1) +\frac{1}{2}(+1) = 0. \end{align*}

Hence,

\text{Covariance}(\text{Z}_A, \text{Z}_B) = \braket{\text{Z}_A\text{Z}_B} - \braket{\text{Z}_A}\braket{\text{Z}_B} = -1 - (0)(0) = -1.

Note: We say that for this state, the spin measurements are perfectly “anti-correlated.” Whenever you get spin-up for qubit $A$ , you get spin-down for qubit $B$ and vice versa.

Introduction

Traditional Computing

Traditional Bits in New Clothing

Qubits and Quantum States

Quantum Measurements

Quantum Gates and Quantum Circuits

Putting a Spin on Spin

Your Basis, My Basis

Multi-Qubit Systems, Entanglement, and Quantum Weirdness

Quantum Circuits and Multi-Qubit Applications

Quantum Computing Algorithms

Implement the Deutsch Algorithm with Qiskit

More Quantum Algorithms

Factoring Algorithm

Implement Shor’s Factoring Algorithm Using Qiskit

Fundamental Quantum Issues

Complexifying Quantum States

Present and Future QIS and QC

Quiz: Fundamental Quantum Issues