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Questions tagged [measurement]

For questions related to measurement and its effects as relevant to quantum computation and quantum information.

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Expected Minimum Clique Coverage?

I'm curious if there is literature on this and, if there is, where to find it. Here I'm using ``clique" to mean a set of observables which all commute with each other. I also include the identity ...
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1 vote
0 answers
34 views

Conditional measurement using classical bit in qiskit

I am trying to implement Figure 6 in Implementing a distance-based classifier with a quantum interference circuit by Schuld et. al The specific part that I'm struggling with is the controlled (on |0&...
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3 votes
2 answers
80 views

Is a "Bell measurement" equivalent to a projective measurement in a different basis?

Let \begin{align} H = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1\end{bmatrix}\in M_2(\mathbb C), \quad \mathrm{CNOT} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 &...
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1 vote
2 answers
46 views

What does distinguishability mean in this case?

In a lecture, we were given the following example to explain the operational significance of the trace distance. Suppose that Alice prepares one of two (known) states $\rho_0$ or $\rho_1$ with equal ...
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How to interpret the measurement $\mu$ giving a fidelity equal to the Bhattacharyya coefficient: ${\rm F}(P,Q)={\rm B}(P,Q|\mu)$?

(Bhattacharyya coefficient vs state fidelity) Given two vectors $u,v$ with nonnegative entries, their Bhattacharyya coefficient is $$\mathrm B(u,v)\equiv \sum_a \sqrt{u_a v_a}.$$ Given two positive ...
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1 vote
0 answers
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Measurement Base of Fusion Gate

I am reading this article on "Fusion-based quantum computation" and they claim there about Fusion Gate type-2 that it collapses the state to 2 types of eigenstates, and returns 2 results - $...
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  • 801
1 vote
2 answers
86 views

What's a circuit to compute the expectation value of an observable?

Apologies if the question is naive. I want to compute the expectation value of some operator for a particular model. It seems I can do that with Hadamard test. I have the circuit for the ground state ...
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4 votes
0 answers
36 views

What's the structure of the measurement $\mu$ that optimally discriminates an ensemble $\{(p_i,\rho_i)\}_i$?

As discussed e.g. in this post, given two states $\rho$ and $\sigma$, the measurement that allows to optimally discriminate between them (i.e. the measurement providing the highest average probability ...
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1 vote
3 answers
92 views

Is there a way to know if a probability amplitude is negative or positive?

I have a quantum system that outputs a state similar to: $|\psi\rangle = \frac{1}{2}(|00\rangle + |01\rangle + |10\rangle \pm |11\rangle)$. So my question is: is there a way (by measurement or by ...
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0 votes
0 answers
20 views

Figuring out which experiment is being performed from the results of the experiment

Consider two different experiments involving qubits. In Experiment 1, a qubit is prepared in the mixed state $I/2$, where $I$ is the $2 × 2$ identity matrix. Alice then chooses an orthonormal basis $B$...
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1 vote
1 answer
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Sending messages via CNOT gates

Consider a situation in which each of Alice, Bob and Charlie holds a qubit, where the three qubits are acted on by a 3-qubit gate $U$. Assume that the gate $U$ can be implemented by first performing a ...
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3 votes
2 answers
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Can any rank-$n$ POVM be realized as a rank-one POVM?

Let, $\mathcal{M}$ is a POVM measurement whose elements are $M_i=\sum_{k=1}^np_{ki}|\phi_{ki}\rangle\langle\phi_{ki}|$ with $p_{ki}\geq 0$ and $\sum_{i=1}^sM_i=I$ where $|\phi_{ki}\rangle$ is a ...
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3 votes
0 answers
32 views

Why does running a pulse schedule for Bell circuit give incorrect counts?

As part of another project I need to be able to run a custom pulse schedule for a bell ciruit and measure the counts. However I am having a problem where running the scheduled circuit is giving counts ...
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1 vote
1 answer
90 views

Characterise, via Naimark's theorem, the POVM corresponding to a PVM in a dilated space

Let $F\equiv\{F^a\}_a$ be a POVM in some finite-dimensional Hilbert space $\mathcal X$. It is well-known that one can always understand $F$ as a projective measurement (PVM) in an isometrically ...
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4 votes
2 answers
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Confusion regarding Neumark's/Naimark's extension of POVM

Starting with the definitions used. A PVM is a set $\mathcal{P} = \{P_i: P_i^2 = P_i, P_iP_j = \delta_{ij}P_j, \sum{P_i} = \mathbf{I}\}_{i,j=1}^n$, where $n\leq d$ on a Hilbert space $\mathcal{H}^d$ ...
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0 votes
1 answer
20 views

Get relative phase between matrix elemets

I have a N qubits system and I would appreciate a lot some suggestions on how I can get the relative phase between $$ \langle 0 \mid U^\dagger Z_n U \mid 0 \rangle$$ and $$ \langle 0 \mid U^\dagger ...
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4 votes
2 answers
105 views

Tapering off qubits

Suppose you have a Hamiltonian of the form $$ H = ZXXX + YXXX + XXXX $$ where $Z,X,Y$ are the usual Pauli matrices with $ZXXX = Z \otimes X \otimes X \otimes X$ and similar for the other two terms. ...
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4 votes
1 answer
69 views

Are projections determined by their action on a full-rank density matrix?

Consider (self-adjoint) projections $P$ and $Q$ defined on a finite-dimensional Hilbert space. If $\rho$ is the maximally-mixed state, then we have that $P \rho P = Q \rho Q$ implies $P = Q$, since $\...
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1 answer
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Calculating the expectation value of an observable in Qiskit

Consider the following quantum circuit that consists of a three qubit quantum register and an ancilla qubit: Let $W$ and $V$ be unitary operators. $U(t)$ is an implementation of unitary time ...
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3 votes
2 answers
140 views

What is the state after a projective measurement?

Given an observable $M = \sum_m \lambda_m P_m$ and assuming that $P_m = |v_m\rangle \langle v_m|$, the state after measurement after getting result $\lambda_m$ is given as $$ \frac{P_m |\psi\rangle}{||...
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2 votes
1 answer
27 views

Qubit State (theta,phi) Dependence on the Readout Error

Do the readout errors on the publicly available IBM quantum computers have any dependency on the state being measured? That is, if we are measuring a qubit in the state $\cos({\theta/2}) |0> + e^{i\...
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3 votes
1 answer
121 views

What exactly makes VQE faster than classical optimization?

I have been trying to understand Variational Quantum Eigensolver (VQE), particularly from the non-linear binary programming perspective. But after reading a few resources I'm still confused about ...
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6 votes
1 answer
132 views

Is it possible to extract $x_1$ and $x_2$ from $|\phi\rangle=\frac1{\sqrt2}(|x_1,0^n\rangle+|0^n,x_2\rangle)$ with non-negligible probability?

Let $\left\vert \phi\right\rangle=\frac 1{\sqrt2}\left\vert x_1,0^n\right\rangle+\frac1{\sqrt2}\left\vert 0^n,x_2\right\rangle$ be a $2n$-bit quantum state for some unknown $x_1,x_2\in\{0,1\}^n$. My ...
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1 vote
2 answers
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How does a unitary transformation preserve a measurement?

Apologies if my question is worded poorly or unclear. I am still new to quantum mechanics and am having trouble understanding this concept. In my textbook, it says: Instead of measuring |ψ⟩ in a ...
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1 vote
1 answer
81 views

Measuring a state $\frac{1}{2}|0\rangle-\frac{\sqrt 3}{2}|1\rangle$ in the $X$ and $Z$-bases?

If a qubit is in the state $|\psi\rangle = \frac {1}{2}|0\rangle - \frac{\sqrt 3}{2} |1\rangle$, how do I measure it in the $Z$-basis, i.e. $\{|0\rangle,|1\rangle\}$, and the $X$ basis, i.e. $\{|+\...
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3 votes
3 answers
105 views

Measuring in rotated basis for mult-qubit circuits

In a previous question (and others), someone asked about measuring on a basis other than the computational one, but for one qubit circuit. Here it was asked for a specific basis. Suppose I have a two-...
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1 vote
0 answers
53 views

Is $\rho | \psi \rangle$ invariant in the Wigners friend thought experiment?

Background Let's say I have a gas of $N$ particles where I cannot distinguish between the particles at a temperature $T$. Its density matrix is given by $\rho$. Note, if my friend happens to measure ...
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1 vote
0 answers
50 views

Replace condition in Knill-Laflamme quantum error conditions with probability

The Knill-Laflamme conditions say that in order to have a quantum error correcting code for some error set $\mathcal{E}$, we must satisfy both of the following conditions for all $E_a,E_b \in \mathcal{...
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3 votes
0 answers
30 views

Trading gates for qubits in the Hadamard test for $\langle 0| V^\dagger G U | 0\rangle$... using the SWAP test?

In this paper, the measurement of the real part of the matrix element of a Hermitian&unitary operator $G$ between the states $U|0\rangle$ and $V|0\rangle$ (i.e. $\langle 0| V^\dagger G U | 0\...
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5 votes
1 answer
101 views

Implementing $a^\dagger|\psi\rangle$

One way or another, I would like to implement the action of the second-quantized creation operator on a quantum state: $|\psi\rangle\mapsto a^\dagger|\psi\rangle$. The motivation, of course, comes ...
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4 votes
2 answers
289 views

How to find the stabilizer generators for a post-measurement state?

My question is closely related to this one. A bit of vocabulary and a reminder of basic properties: I consider the total Hilbert space of the problem has dimension $2^n$. I call a "well defined ...
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2 votes
0 answers
33 views

Amplifying states using partial information

Assume we have an initial state $(H^{\otimes n}|0\rangle)|0\rangle = \frac{1}{\sqrt{N}}\sum_{i=0}^{N-1}|i\rangle|0\rangle$ in two $n$-qubit registers (let $N=2^n$). Then we feed this state through a ...
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0 votes
0 answers
24 views

Can we find which qubits are multiplexed for readout in IBMQ systems?

This question has a couple of parts: Do all IBM systems (except IBM Armonk) utilize frequency division multiplexing for qubit readout? If yes, is there a way of finding which qubits constitute a ...
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0 votes
1 answer
65 views

Parameterized swap test and perfect swap test

Suppose one has parameterized a swap test by using an ansatz $U(\theta) = \exp(-i\theta \text{ CSWAP})$, and one tries to find an angle $\theta$ such that one can distinguish given two quantum states ...
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0 votes
1 answer
62 views

The meaning of non-destructive measurements in QEC

I am starting to study quantum error correction and it's the first time that I see the operators (usually Pauli) as projective measurements which seem to just identify a syndrome but do not destroy ...
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3 votes
1 answer
198 views

Can joint measurement be achieved in two labs far apart?

Consider the following scenario: Alice and Bob are in two labs far apart, and they each have one qubit. Can joint measurement (for bipartite projective measurement, they are measurements that cannot ...
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0 votes
1 answer
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Retrieve image from Histogram data of Qiskit

I am trying to understand the quantum image processing. Please refer the link- https://qiskit.org/textbook/ch-applications/image-processing-frqi-neqr.html. I am not able to understand how to process ...
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0 votes
1 answer
42 views

How do we define qubit measurements in a plane?

When does $\vec{a} \cdot \vec{\sigma}$ define a measurement in x-y, y-z, and x-z planes?
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  • 476
1 vote
1 answer
72 views

Measurement of the second quantum register in the Shor's algorithm [duplicate]

I've read that the measurement of the ancilla qubits is not fundamental for Shor's algorithm, but I don't understand how the algorithm works if I remove it. Without those measurements, do I have $r$ ...
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  • 141
1 vote
2 answers
87 views

Error with IBM's Quantum Composer Qiskit Histogram when Resets are involved?

I am having trouble understanding what IBM's histogram is doing. I have independently verified, in agreement with the statevector given from IBM, that after all these steps the statevector should be ...
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  • 168
1 vote
1 answer
28 views

What is a histogram of a job a plot of, if not the squared amplitudes of the final state-vector?

The histogram of a circuit is the result of running the circuit (with measurement) many times, right? Does this correspond to the squared amplitudes of the final state-vector? If not, why?
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  • 168
1 vote
1 answer
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What happens to the elements of the simulated state vector when we set a qubit to $|0\rangle$?

In IBM's Qiskit online simulator, we have the (non-reversible) ability to set a specific qubit to $| 0\rangle$. This is convenient but I'm left confused as to what happens to the elements of the ...
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  • 168
2 votes
1 answer
92 views

Density matrix after measuring Bell state in CHSH game

In these notes, the author says the following about the CHSH game Does Alice and Bob’s ability to succeed more than 75% of the time mean that they are communicating? Well, we know it’s not possible ...
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3 votes
0 answers
65 views

Partial measurement can be replaced with constant overhead

While reading the chapter on Quantum Computation (starting on page 401) of the draft version of the Arora & Barak book I came across exercise §4 on page 431 that reads as: Suppose that $f$ is ...
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3 votes
1 answer
192 views

Does one measurement affect the following measurement errors in mid-circuit measurement?

Suppose I have a quantum circuit with a few measurements (say $N$) on a single qubit. Before all the measurements, I generated calibration circuits and the 2 by 2 calibration matrix for that qubit of ...
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1 vote
1 answer
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Mathematics of Measurement then Partial Trace

Say we have the following quantum state: $$ |\psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle +|10\rangle)$$ To measure the first qubit and then further trace out the first qubit, my notes have the ...
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3 votes
2 answers
29 views

How to find $\langle Z_1...Z_{N-1} \rangle$ knowing $\langle Z_1...Z_{N} \rangle$ and $\langle Z_i \rangle$

I have a quantum circuit with $N$ qubits represented by the unitary $U$. The initial state is $| 00...0\rangle$ and $\psi=U|00...0\rangle$. Given $\langle\psi| Z_1...Z_{N} |\psi\rangle$ and $\langle \...
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  • 141
2 votes
1 answer
123 views

How to distingush between two very similar pure quantum states?

I'm trying to prove the claim that Given two pure states: $|\psi_i\rangle$ and $|\phi_i\rangle$ such that $|\,|\psi_i\rangle - |\phi_i\rangle\,|\le \delta$ then no measurement can distinguish ...
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  • 119
2 votes
0 answers
48 views

Individual processing of quantum circuit measurment results

When superconducting transmon qubits are measured with a readout pulse, the raw readout signal is demodulated, and results appear as clouds on the IQ plane, with one point in the cloud representing ...
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  • 1,240
2 votes
1 answer
59 views

Expectation value of an observable containing a single projector vs Born rule for the projector

Suppose I have a state $|\psi\rangle$ and I want to estimate the probability of obtaining a computational basis state $|x\rangle$. Then by Born rule: $$ p(x) = |\langle x|\psi\rangle|^2 = Tr[|x\rangle ...
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