Questions tagged [measurement]

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

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24 views

How to choose $\beta$ in Gaussian derivative component of DRAG pulse?

From definition of the DRAG pulse it is: $$f(x)=Gaussian+1j*\beta*(-(x-duration/2)/\sigma^2)Gaussian,$$ where $Gaussian(x, amp, \sigma)=amp*e^{-(1/2)*(x-duration/2)^2/\sigma^2}$. If I try it in ...
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Schur transform and the outcome probabilities for a particular type of state

I was reading about the Schur transform and its applications in knowing about an unknown quantum state. Consider $\rho^{\otimes k}$ --- $k$ copies of an unknown $n$ qubit quantum density matrix $\rho$...
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48 views

Do we need ancillary qubits to implement orthogonal measurements?

Consider an $n$ qubit state $|\psi\rangle$. Let's say I want to implement an $m$ outcome orthogonal measurement on $|\psi\rangle$, where $m \neq n$. Denote the set of $m$ orthogonal measurement ...
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45 views

How is transformation for measurement in an arbitrary basis derived?

I started with Qiskit today and find it very exciting. As a first question I want to understand how to measure an arbitrary state $|\Psi\rangle$ not in the basis of ...
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1answer
299 views

Why does quantum distinguishability ensure no faster-than-light communication?

On page 56-57 in Nielsen and Chuang, for a proposed scenario, it's said that: if Bob had access to a device that could distinguish the four states $|0\rangle$, $|1\rangle$, $|+\rangle$, $|−\rangle$ ...
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61 views

How to compute the measurement probabilities of $|\phi\rangle=\sum_k c_k |k\rangle$ in a rotated basis $V|k\rangle$?

I came across the following question and have some conceptual questions. Consider a general quantum state $|\phi\rangle$ of dimension $N$ spanned by some standard basis $\{|k\rangle,k=0,1,...N-1\}$. ...
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40 views

What does the $I$ mean when measuring ${\rm Tr}(\rho (I\otimes\sigma\otimes\cdots))$ in quantum tomography?

In Nielsen and Chuang's QCQI, I learned that the quantum tomography for n qubit can be described easily in math as we need to measure $Tr(\rho W_k),\forall k$ where $W_k\in\{I,\sigma_x,\sigma_y,\...
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As regards measurement how would a quantum-full-adder perform multiple additions simultaneously?

Here in this video from 15:14 Arvin Ash demonstrates a quantum-full-adder circuit, he goes on further to illustrate how it can perform multiple operations ...
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392 views

Is there any simple mathematical proof that measurement destroys entanglement?

Is there a simple mathematical way to prove that measurement destroys entanglement? I can see that this is indeed true if I just take a specific measurement on an entangled state. What I am looking ...
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60 views

Measurement of single qubit operator $U$ which is both Hermitian and unitary with eigenvalues $±1$

Suppose we have a single qubit operator $U$ with eigenvalues $±1$, so that $U$ is both Hermitian and unitary, so it can be regarded both as an observable and a quantum gate. Suppose we wish to measure ...
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90 views

Are SIC-POVMs optimal for quantum state reconstruction?

Mutually unbiased bases (MUBs) are pairs of orthonormal bases $\{u_j\}_j,\{v_j\}_j\in\mathbb C^N$ such that $$|\langle u_j,v_k\rangle|= \frac{1}{\sqrt N},$$ for all $j,k=1,...,N$. These are useful for ...
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214 views

How can we measure a quantum system when the sum of amplitudes-squared does not equal one?

How can we measure a quantum system if the sum of amplitudes-squared does not equal one? For example, if we want to measure $|a\rangle = 0.25|0\rangle + 0.25|1\rangle$, how can we measure it?
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How are Pauli operators measured experimentally?

We know the theory of Random Pauli measurements: Consider a system of $n$ qubits, and let $d=2^n$, the Pauli matrix set is $P=\otimes_{i=1}^n \sigma_i$, where $\sigma_i\in \{ I,\sigma_x,\sigma_y,\...
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35 views

Can someone please explain how the syndrome bit still ends up being 0 in this quantum error correction circuit using repetition code?

I'm not too great at dealing with superpositions and applying the CNOT gate when superpositions are involved. Can you go through it in detail each gate using math/matrices etc. It's based on the ...
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110 views

Do entangled measurements across multiple copies help in state distinguishability?

Consider two density matrices $\rho$ and $\sigma$. The task is to distinguish between these two states, given one of them --- you do not know beforehand which one. There is an optimal measurement to ...
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195 views

How to analyze the following quantum circuit?

I'm trying to analyze the following quantum circuit The goal here is to analyze the final outputs at q3 & q4. For inputs, at q0 & q1, one of the Bell state $$|\psi\rangle = \frac{|01\rangle + ...
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89 views

Are projective measurements the only optimal measurements to discriminate between two states?

Consider two density matrices $\rho$ and $\sigma$. The task is to distinguish between these two states, given one of them --- you do not know beforehand which one. There is an optimal measurement to ...
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35 views

Find expectation value of the observable $X_1\otimes Z_2$ for a maximally entangled two-qubit system

In this exercise I need to find the expectation value of the observable $M=X_1 \otimes Z_2$ for two qubit system measured in the state $\dfrac{|00\rangle + |11\rangle}{\sqrt{2}}$. I know that $E[M]=\...
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82 views

What happens measuring the first qubit of a $GHZ$ state in the basis $\{|+\rangle, |-\rangle\}$?

This exercise ask me to explain what happens when the first qubit is measured in the diagonal base ($|+\rangle,|-\rangle$), considering this state: $|GHZ\rangle=\dfrac{1}{\sqrt{2}} (|000\rangle+|111\...
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99 views

Return only the measurements of a circuit

I have written the following program in jupyter: ...
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46 views

Pad get_counts() with zeros for unmeasured states? (Qiskit)

After executing a job on qiskit, the typically procedure to get the measurement data from the quantum computer is to call get_counts() like so... ...
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133 views

Intuition for why $\frac{|00\rangle+|11\rangle}{\sqrt{2}}$ can be written as $\frac{|++\rangle+|--\rangle}{\sqrt{2}}$

In analyzing measurement of $\frac{|00\rangle+|11\rangle}{\sqrt{2}}$ in the local $|+\rangle$, $|−\rangle$ basis, through algebra manipulation, the initial state is first written as $\frac{|++\rangle+|...
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322 views

What does "measurement destroys information" mean?

I am reading a paper on quantum cryptography. The author used two facts: quantum- information cannot be copied and Furthermore, measurements destroy information... For the first statement, I came ...
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Are almost perfectly distinguishable ensembles almost orthogonal?

Let $\varepsilon>0$ and consider an ensemble of states $\{p_x\rho_x\}_{x\in X}$ and suppose there exists a measurement with POVM representation $\{M_x\}_{x\in X}$ such that $$ \sum_{x\in X} p_x\...
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Why is the complexity of $n$-qubit state tomography not upper bounded as $O(3^n)$?

Consider the task of fully determining an $n$-qubit state $\rho$ which can be written as \begin{equation}\tag{1} \rho = \sum_{p \in \{I, X, Y, Z\}^n} \text{Tr}(\rho P_{p}) P_{p} \end{equation} and ...
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216 views

Do I need to use classical bits in measurement?

I wonder if I have to add classical bits while performing measurement for a single-qubit quantum circuit. The difference to initialize the circuit is ...
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81 views

Applying projectors with mid-circuit measurements

I am trying to apply a non-unitary projector (see image) to my two-qubit quantum circuit using mid-circuit measurements. $$ \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 &...
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1answer
94 views

Measuring tensor products of Pauli operators

Is there a neat way to derive and efficiently implement a measurement circuit for tensor products of arbitrary Pauli operators like $XZZXZ$ in Qiskit ? I tried using the ...
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1answer
76 views

Measure $\langle \hat{X}\rangle$ and $\langle \hat{Y}\rangle$ from counts

I'm confused about how I can measure $\langle \hat{X}\rangle$ and $\langle \hat{Y}\rangle$ using counts. Here's my code for X: x-basis: ...
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90 views

What are the constraints for the coefficents of the basis states in quantum computing?

$\newcommand{\ket}[1]{\left|#1\right>}$ It's known that the Kolmogorov axioms characterise a probability distribution: Probability of an event is a non-negative real number. The sum of all ...
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How does the extremality of a POVM reflect on its Naimark dilation isometry?

Let $\mu:\Sigma\to\mathrm{Pos}(\mathcal X)$ be some POVM, with $\Sigma$ the finite set of possible outcomes, and $\mathrm{Pos}(\mathcal X)$ the set of positive semidefinite operators on a finite-...
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93 views

How to normalise when the probability of measurement is zero?

In one of the answers to this question on measuring one qubit it is explained that given a general two-qubit state $$ |\psi\rangle = \begin{bmatrix} \alpha_{00} \\ \alpha_{01} \\ \alpha_{10} \\ \...
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62 views

Doing $|0\rangle$ then Hadamard gate then measurement

I am starting to use quantum experience and following exactly first example from lecture. After initializing the qubit to $|0\rangle$, then applying a Hadamard gate, the probability for measuring $|1\...
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65 views

What are the conditions under which an unknown quantum state is learnable with arbitrary precision?

Assume that we have an unknown quantum state and we need to learn that unknown state with arbitrary precision. Under what conditions can we learn the unknown state with arbitrary precision? One ...
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57 views

Finding the measurement basis for single qubit with given probability of outcome $0$

I have the general state of a single qubit $|\psi \rangle = \alpha|0\rangle + \beta|1\rangle $. Assume I am given a probability $p$ such that $0 < p <1$. Now I need to find the basis in which ...
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27 views

What is the correct notation to denote operations conditional on a measurement outcome?

What is the correct mathematical notation to describe the following setup? I have classical state in register $A$ which I can think of as $\sum_i p_i \vert i\rangle\langle i \vert_A$. I measure this ...
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48 views

How does a quantum circuit calculating the inverse of a non-injective function act?

Lets say I have a non-injective function $f()$, adding image for reference. Now lets say I build a quantum circuit to calculate $f^{-1}()$. If the input register has the value $i$, does the output ...
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50 views

Manual measurement error mitigation returning a negative number of counts

I have a simple 2 qubit circuit which I am trying to protect from errors using the measurement error mitigation technique laid out here: https://qiskit.org/textbook/ch-quantum-hardware/measurement-...
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64 views

Get the inverse of a Hermitian operator for measurement in qiskit

I am using qiksit to measure the Hamiltonian H, which isbasically sum of Pauli strings, say something like 2*X^X+0.5*Z^Y. For ...
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39 views

In what sense are Pauli matrices measurement operators?

Neilson and Chuang's textbook shows a nice example of measuring in the $Z$ basis on page 89 in section 2.2.5. The Hermitians for measuring in the $Z$ basis, $|0\rangle\langle 0|$ and $|1\rangle\langle ...
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56 views

Qiskit equivalent of get_expectation_value() in ProjectQ

Is there a corresponding function that has the same function as function get_expectation_value() in Project Q? Get the expectation value of qubit_operator w.r.t. the current wave function ...
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51 views

Calculating probability that two entangled qubits are the same when measured in different bases

Given the entangled state \begin{equation} |\Phi^+\rangle = \frac{1}{\sqrt 2} |00\rangle + \frac{1}{\sqrt 2} |11\rangle \end{equation} I am trying to calculate the probability that the two qubits end ...
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83 views

Do sequences of operations (including measurements) applied to different halves of an entangled pair always commute?

Let us say $A$ has one half of an entangled qubit pair, and $B$ has the other half. $A$ may be able to perform any type of operation on their half of the pair, such as unitary operations, entangling ...
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87 views

Why is $\langle \psi| \sigma_z |\psi \rangle=\cos(\phi_1)\cos(\phi_2)$ for $|\psi\rangle=R_y(\phi_2)R_x(\phi_1)|0\rangle$?

I'm trying some example with the rotation gates and stuck here: $$\langle \psi| \sigma_z |\psi \rangle = \langle 0 | R_x(\phi_1)^\dagger R_y(\phi_2)^\dagger \sigma_z R_y(\phi_2) R_x(\phi_1) | 0 \...
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103 views

Distinguishing $\frac{| 0 \rangle + e^{i\theta} |1 \rangle}{\sqrt{2}} $ from $| 0 \rangle/|1 \rangle$ with probability $1/2 + \epsilon$

I am given one copy of one of two quantum states - $\frac{| 0 \rangle + e^{i\theta} | 1 \rangle}{\sqrt{2}} $, for some unknown fixed $\theta$. One of $| 0 \rangle/|1 \rangle$ - don't know which one, ...
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1answer
139 views

How to get measurement result on Qiskit

I want to get measurement result from my circuit output, like '00', '01', '10', or '11' because want to process it further classically. Any suggestion? We can see the highest probability '01'. How I ...
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1answer
67 views

What are the possible initial states that can be prepared in a lab for use in a quantum computation?

So here's something that's been bothering me. Given the time evolution of the wavefunction can only be unitary or discontinuous as a process of the measurement. So let the observables for our ...
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1answer
55 views

Computing $H(Z|B)$ in a bipartite density matrix $\rho_{AB}$

Let's say Bob prepares a bipartite quantum state $\rho_{AB}$ to be shared between him and Alice. Bob sends Alice's part to her lab. Alice measures her subsystem $A$ in the computational basis $\...
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2answers
72 views

Inconsistent result trying to simulate measurement of a 2 qubit system

Consider a 2 qubit system in the initial state: $$ \begin{pmatrix} 0 \\ \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} \\ 0 \end{pmatrix} $$ the so-called Bell-pair. Now let's measure the spin of the first ...
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58 views

What is the observable to measure spin along some direction for a 2 qubit system?

Let $M$ be the $2 \times 2$ matrix corresponding to the observable to measure spin along some arbitrary axis $\vec{v}$. This matrix is given by following formula: \begin{equation} M = v_x X + v_y Y +...

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