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Consider a measurement of $\sigma_{z}$ that will correctly identify $|0\rangle$ every time however sometimes incorrectly identifies $|1\rangle$ as $|0\rangle$.

I would like to model how such a faulty detector will skew my quantum state tomography and how I can account for it in error bars.

Measurements of $\sigma_{x}$ and $\sigma_{y}$ are performed by qubit rotations and the use of this device.

Has anyone seen this kind of thing investigated and if so could you send me any material you have on it?

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2 Answers 2

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If I understand the question, you're looking for a state strategy to discriminate $|0\rangle$ and $|1\rangle$, with zero error when the input is $|0\rangle$, but some error when it's $|1\rangle$.

You could then consider the POVM with elements $$\left\{\frac{2|0\rangle\!\langle0|+|1\rangle\!\langle1|}{2}, \frac{|1\rangle\!\langle1|}{2}\right\},$$ assigning the first outcome to $|0\rangle$ and the other one to $|1\rangle$. With this strategy, if you enter with $|0\rangle$ you have zero probability of getting the second outcome, but if you enter with $|1\rangle$, you have a 50/50 probability of getting either outcome.

If you're looking for a more "concrete" implementation of such a strategy, you can think of it in terms of Neumark's dilation. Namely, Consider the isometry $$V = \begin{pmatrix}1 & 0 \\ 0&\frac1{\sqrt2} \\ 0&0\\0&\frac{1}{\sqrt2}\end{pmatrix}.$$ Then, measuring in the computational basis after evolution through this $V$ amounts to measuring the input in the above POVM. More explicitly, you can implement this as a two-qubit circuit with the first qubit being the input, the second qubit starting with $|0\rangle$, where you apply a single controlled-Hadamarad gate and then measure only the second qubit at the and in the computational basis. Finding $|0\rangle$ tells you the input was $|0\rangle$, but finding $|1\rangle$ does not tell you anything.

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Consider the measurement as the only component affecting a state $\sigma$. You can model your problem as a quantum channel $\mathcal{N}\circ \mathcal{I}$, with $\mathcal{I}$ being the identity channel, followed by $\mathcal{N}$, which represents the error caused by the measurement.

At this point, we focus on characterising channel $\mathcal{N}$. Given your specifics

$$\mathcal{N}(\sigma) = p |0\rangle\langle 1| \sigma|1\rangle\langle 0| + (1-p)\sigma.$$

Use quantum process tomography to assert your noise model. Namely, estimate the Kraus operators of $\mathcal{N}'$ and verify that $\mathcal{N}' \sim \mathcal{N}$.

EDIT: Notice that the Spontaneous Emission channel differs from $\mathcal{N}$, as reported here.

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