given A and B share EPR pairs $ (|00⟩+|11⟩)/√2$

both are free to measure their own qubit with the following measurement settings

A measures with $[ |0⟩, |1⟩ ]$

B measures with $[ sin(3π/8)|0⟩ + cos(3π/8)|1⟩, -sin(π/8)|0⟩ + cos(π/8)|1⟩]$

  • $\begingroup$ Hi winner! Welcome to QCSE! Can you clarify what you mean by "how do I calculate measurement values?" Do you want to know the probability of measuring $0$ or $1$ in each basis? Are you familiar with the Born rule? $\endgroup$ – Mark S Jul 13 at 2:56
  • $\begingroup$ Hi @MarkS, I am trying to know what will be the value when A measures in [|0⟩,|1⟩] basis or bell basis $\endgroup$ – Win Jul 13 at 3:17
  • $\begingroup$ Thanks for the edit. Can you explain what you know, and what you've reviewed? Where are you getting stuck? $\endgroup$ – Mark S Jul 13 at 11:48
  • $\begingroup$ I am new to quantum computing, please correct me if I am not clear enough $\endgroup$ – Win Jul 13 at 16:18
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    $\begingroup$ Are you asking how to calculate the 4 probabilities in each case, or are you asking how to calculate an expectation value (which is what your question actually says)? If the latter, what expectation value are you wanting to calculate? That would be some function of the measurement outcomes that you want the average of. $\endgroup$ – DaftWullie Jul 15 at 7:40

The key to figuring out the probability of any measurement result is Born's rule, which says that if you have a state $\left|\psi\right\rangle$ the probability of observing measurement outcome $\left|\phi\right\rangle$ is given by $$ \begin{align} \Pr(\phi | \psi) = \left|\left\langle \phi | \psi \right\rangle \right|^2. \end{align} $$ In the example you described, let's consider the probability with which Alice observes $\left|0\right\rangle$ and Bob observes $\sin(3\pi / 8) \left| 0\right\rangle + \cos(3\pi / 8) \left| 1\right\rangle$. This outcome occurs with probability $$ \begin{align} & \Pr(A=0,B=0|\text{EPR}) \\ & \quad = \frac12\left| \sin(3\pi / 8) \left\langle 00 | 00 \right\rangle + \cos(3\pi / 8) \left\langle 01 | 00 \right\rangle + \sin(3\pi / 8) \left\langle 00 | 11 \right\rangle + \cos(3\pi / 8) \left\langle 01 | 11 \right\rangle \right|^2 \\ & \quad = \frac12\left| \sin(3\pi / 8) \right|^2 \approx 0.85. \end{align} $$ Here, we used that the four computational basis states $\left\{\left|00\right\rangle, \left|01\right\rangle, \left|10\right\rangle, \left|11\right\rangle \right\}$ are all orthogonal to each other, so that the $\left\langle 00 | 00\right\rangle$ is the only one that survives.

The probability for each of the other three outcomes can be worked out in a similar fashion using Born's rule.

  • $\begingroup$ thank you Chris, can you also explain to me how to calculate the expectation value when Alice observes $|0⟩$ and Bob observes $sin(3π/8)|0⟩+cos(3π/8)|1⟩$ $\endgroup$ – Win Jul 13 at 17:10

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