Questions tagged [povm]

For questions related to positive-operator valued measures (POVMs), that is, sets of positive semi-definite operators summing to the identity matrix.

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Finding the "dual" basis of an overcomplete basis for Quantum State Tomography

This question is related to this stack exchange post: What does the POVM corresponding to single-qubit state tomography look like? From what I understand, when we are interested in reconstructing a ...
junoswrld's user avatar
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When does a Hamiltonian result in the construction of an IC-POVM?

Consider a generic $d_A$-dimensional quantum system $A$, for example made of $n$ qubits. Now consider a second higher-dimensional system $B$, with $d_B=d_A^2$, and the quantum map $$ \rho_A \...
Tailor's user avatar
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What does the outcome $i$ mean when we measuring a quantum system?

The POVM element $E_{i}$ is associated with the measurement outcome $i$, such that the probability of obtaining it when making a measurement on the quantum state $\rho$ is given by: $p(i)=tr(\rho E_i)...
Karry's user avatar
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Are POVM elements invertible?

A POVM is a set $\mathcal{M} = \{A_i : A_i \geq 0, \sum{A_i }= \mathbf{I}\}_{i=1}^m$ on a Hilbert space $\mathcal{H}^d$ of dimension $d$, I want to know whether $A_i$ can be invertible linear map?
Karry's user avatar
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Motivation behind POVM and projective measurement

This is in reference to Quantum Computation and Quantum Information by Michael A. Nielsen and Isaac L. Chung [page 90, 92]. Any POVM elements $E_{m}$ are defined as $E_{m} = M_{m}^{\dagger}M_{m}$. A ...
Physkid's user avatar
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Is there a tight operator frame that is also a POVM?

We define the tight operator frame as a set of operators $\{E_i\}_{i=1}^{n}$ satisfying \begin{equation} \sum_{i=1}^n \vert \langle \langle E_i \vert X \rangle \rangle \vert^2 = C \Vert V \Vert_2^2, \...
Michael.Andy's user avatar
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Show that any measurement where the measurement operators and the POVM elements coincide is a projective measurement

The following question is exercise 2.62 from Nielsen and Chuang's "Quantum Computation and Quantum Information" Show that any measurement where the measurement operators and the POVM ...
QC123_367's user avatar
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How do we show that a measurement is a projective measurement

In order to show that a measurement is a projective measurement, is it sufficient to prove that the measurement operators $\{M_{m}\}$ satisfy the properties: Hermitian: $M_{m}^{T*} = M_{m}$ ...
QC123_367's user avatar
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What is the classical Fisher information of a parametrized coherent state $|\alpha_\theta\rangle$?

Suppose $|\alpha_\theta\rangle$ is a coherent state depending on the real parameter $\theta$. What is the classical Fisher information it carries? There are explicit formulae for the quantum Fisher ...
Bentanglement's user avatar
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Recover the noisy POVMs of Bell basis measurement

Considering Bell basis measurement, we have that the ideal POVMs are four Bell states, which can be obtained by reversing the following quantum circuits. Now, we add depolarizing errors to CX gate and ...
Michael.Andy's user avatar
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How can we derive the form of POVMs on a subspace from a projective measurement on a larger space?

Suppose we have the Hilbert space $\mathcal{H}_{0}$ describing the states of the system $s$, and the Hilbert space $\mathcal{H}_{e}$ describing the states of the environment. I have seen that [1][2], ...
Adrien Amour's user avatar
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67 views

Neumark dilation for qubit tetrahedron SIC-POVM

I would like to know if an analytic solution is known for the Neumark dilation of the qubit tetrahedron SIC-POVM defined by $$ M_0= \frac{1}{4\sqrt{3}} \Big( \sqrt{3}I + X +Y +Z \Big), \qquad M_1= \...
quantum_theo's user avatar
2 votes
1 answer
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Is $Tr[E_i E_j] \geq 0$ for $i\neq j$ and $\{E_k\}$ a POVM?

Suppose that $\{E_i\}$ form a POVM (i.e. a set of positive operators satisfying $\sum_{i} E_i = I$, where $I$ denotes identity). Is it the case that $Tr[E_i E_j] \geq 0$ for all $i \neq j$?
Michael.Andy's user avatar
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Can two measurements be represented as a single measurement when they are acted upon sequentially?

Let two different POVM measurements represent as $\mathcal{M}_1=\{\Pi_i\}_{i=1}^k$ where $\Pi_i$ is element of the $\mathcal{M}_1$ measurement and $\mathcal{M}_2=\{E_j\}_{j=1}^n$ where $E_j$ is the ...
PAB's user avatar
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What are examples where the quantum discord is achieved by a non-projective POVM?

Consider the (asymmetric) quantum discord, defined as (borrowing notation from Eq. 4.13c of Zurek's review): $$\mathcal D(\mathcal S:\mathcal A) = I(\mathcal S:\mathcal A) - \chi(\rho_{\mathcal A}),$$ ...
glS's user avatar
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Prove that the square root measurement $\Lambda_y=\frac14(\rho_{B^3})^{-\frac12}|\psi_y\rangle\langle\psi_y|(\rho_{B^{3}})^{-\frac{1}{2}}$ is a POVM

Consider $\textit{X}\sim \mathrm{Unif}([0,1,2,3]), |\mathcal{Y}|=|\mathcal{X}|=4$. Also for every random variable realization {\it x} we use three parallel quantum channels like the one employed ...
Noether's user avatar
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POVM construction with little input information

Let $E$ be part of a POVM $M = \{E,I-E\}$. Suppose that I know that $E = f(\rho_1, \rho_2)$. Suppose also that those two states are provided but we only know their type (dimension) and we also know $f$...
R.W's user avatar
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Given a POVM, what's the channel that optimally preserves coherence in the post-measurement outcomes?

It is well-known that a POVM $\boldsymbol\mu\equiv (\mu_a)_{a\in\Sigma}$ describes outcome probabilities, but not post-measurement outcomes, which in many scenarios exist and are of interest. To ...
glS's user avatar
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Do SIC-POVM elements for $d=2$ sum up to the identity?

I am studying SIC-POVM in dimension two and I want to check that the elements sum up to identity. $$\begin{aligned} & \left|\psi_1\right\rangle=|0\rangle \\ & \left|\psi_2\right\rangle=\frac{1}...
윤성원's user avatar
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What is a POVM?

I am having a hard time understanding what exactly a Measurement is by its definition? What I read is that a POVM $M$ is defined by its set of elements $M_i$. So is $M$ itself an operator? In circuit ...
TTa's user avatar
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1 answer
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How many measurements are needed to distinguish two fixed density matrices?

Suppose there are two fixed density matrices $\rho_1$ and $\rho_2$ are prepared for equal probability. Can we say something about the minimum number of measurements required to distinguish the two ...
Jon Megan's user avatar
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1 answer
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Does closeness in trace distance imply close measurement outcomes?

Suppose we have two density matrices $\rho$ and $\rho'$ such that $\|\rho - \rho'\|_1 \leq \varepsilon$. Let $\{\Lambda, I - \Lambda\}$ be elements of some POVM. If it holds that $$Tr(\Lambda\rho) \...
JRT's user avatar
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1 answer
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What is the most general way to describe post-measurement states?

Background Generally speaking, the description of post-measurement states associated with a POVM seems to always pass through, in some form or another, the formalism of Kraus operators. For example: ...
glS's user avatar
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1 answer
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Given a state $\rho$ and operator $0\le \Lambda\le I$, what does $\sqrt\Lambda \rho \sqrt\Lambda$ represent?

An expression that is found in a good number of results is $\sqrt\Lambda\rho\sqrt\Lambda$, for some pair of positive semidefinite operators $\rho,\Lambda\ge0$. For example, in the gentle operator ...
glS's user avatar
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2 votes
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Distinguishing $n$ pure states in an $n$ dimensional Hilbert space

Suppose we have $n$ pure states in an $n$ dimensional Hilbert space, and we would like to distinguish them using POVM or PVM. We get any one of the pure states with equal probability, and we may set ...
Stan's user avatar
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Why is $\| M|\psi\rangle \| \leq 1$ for POVM $M$?

In this question‘s answer it is mentioned that $\| M|\psi\rangle \| \leq 1$ for POVM Element $M$. I don‘t get why this is. My thoughts so far: for the set of POVM elements $\{M_a\}$ we know that all $...
Aemmel's user avatar
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2 votes
2 answers
65 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 ...
Eulerian's user avatar
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2 answers
348 views

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 ...
PAB's user avatar
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1 vote
1 answer
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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 ...
glS's user avatar
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4 votes
2 answers
836 views

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 $\mathcal{H}^d$ ...
Abhishek Banerjee's user avatar
1 vote
1 answer
51 views

Is there a construction and/or term for the following 'sandwich' measurement?

Suppose we have two projective measurements with elements $E_i$, $i=1...m$, and $F_j$, $j=1...n$. So we know $F_j^2=F_j$ and $E_i^2=E_i$ and $\sum_i E_i = \sum_j F_j = \mathbb{I}$. Then it is easy to ...
M. Stern's user avatar
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3 votes
1 answer
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Helstrom Measurement when two quantum states are close

I've been reading a paper about Entangled-quantum GAN (see this PDF) and wondering why descriptions below Eq.(3) in the paper are in fact true. To summarize the description, suppose we have two ...
user19468's user avatar
3 votes
1 answer
222 views

How to distinguish 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 ...
omerna's user avatar
  • 179
4 votes
1 answer
196 views

Are measurement probabilities on the two qubits of a maximally entangled state equal? [closed]

Suppose we get the Bell state $$ |\Phi ^{+}\rangle ={\frac {1}{{\sqrt {2}}}}(|0\rangle _{A}\otimes |0\rangle _{B}+|1\rangle _{A}\otimes |1\rangle _{B}). $$ If we now apply a unitary operator $U$ ...
Johny Dow's user avatar
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2 votes
2 answers
158 views

Show that there are unitaries $U_m$ such that $M_m=U_m \sqrt{E_m}$, for any measurement $M_m$ and associated POVM $E_m$

Nielsen and Chuang's QCQI, section 2.2.6, page 92, asks Suppose a measurement is described by measurement operators $M_m$. Show that there exist unitary operators $U_m$ such that $M_m=U_m\sqrt{E_m}$, ...
Anna Naden's user avatar
7 votes
1 answer
395 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 ...
glS's user avatar
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7 votes
1 answer
341 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 ...
BlackHat18's user avatar
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6 votes
0 answers
124 views

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-...
glS's user avatar
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5 votes
1 answer
270 views

Does a basis of maximally entangled states exist for two-qubit or two-qutrit system so that the density matrices of the basis states don't commute?

I want to find a basis of maximally entangled states $|\Psi_i\rangle$, for $\mathcal{H}^{2} \otimes \mathcal{H}^{2}$ and, $\mathcal{H}^{3} \otimes \mathcal{H}^{3}$ such that the density matrices of ...
Abhishek Banerjee's user avatar
2 votes
1 answer
228 views

Unambiguous discrimination using POVM with highest discriminate probability

I was studying Nielsen&Chuang's textbook (about page 92), and come up with a question that I cannot solve it. Given one of the two state $|\psi_1\rangle=|0\rangle$ and $|\psi_2\rangle=\frac{1}{\...
Shara's user avatar
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1 answer
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Moving between $\sum_{I}E_{i}=I$ and $\sum_{i}M^{\dagger}M=I$ for non-hermitian $M$

If $\sum_{i}E_{i}=I$ is a set of POVM's and $\sum_{i}M^{\dagger}M=I$ is a set of general measurement operators, I have always been confused on how to move from one to the other, in regards to the ...
GaussStrife's user avatar
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8 votes
1 answer
562 views

What are examples of extremal non-projective POVMs?

Fix some finite-dimensional space $\mathcal X$. Define a POVM as a collection of positive operators summing to the identity: $\mu\equiv \{\mu(a):a\in\Sigma\}\subset{\rm Pos}(\mathcal X)$ such that $\...
glS's user avatar
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2 votes
3 answers
356 views

Why are entanglement breaking channels, defined as $\Phi(\rho)=\sum_a \operatorname{Tr}(\mu(a)\rho)\sigma_a$, entanglement breaking?

Define an entanglement breaking channel $\Phi$ as a channel (CPTP map) of the form $$\Phi(\rho) = \sum_a \operatorname{Tr}(\mu(a)\rho) \sigma_a\tag A$$ for some POVM $\{\mu(a)\}_a$ and states $\...
glS's user avatar
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6 votes
2 answers
1k views

What does the POVM corresponding to single-qubit state tomography look like?

Let $\rho$ be a single-qubit state. A standard way to characterise $\rho$ is to measure the expectation values of the Pauli matrices, that is, to perform projective measurements in the three mutually ...
glS's user avatar
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11 votes
2 answers
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What is the relation between POVMs and observables (as Hermitian operators)?

Let $\renewcommand{\calH}{{\mathcal{H}}}\calH$ be a finite-dimensional Hilbert space. An observable $A$ is here a Hermitian operator, $A\in\mathrm{Herm}(\calH)$. A POVM is here a collection of ...
glS's user avatar
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4 votes
1 answer
666 views

How to find the POVM that optimally distinguishes between two given states?

A quantum state preparation machine emits a state $\rho_0$ with probability $2/3$ and emits the state $\rho_1$ with probability $1/3$. We aim to make the best guess which one is it using a set of two ...
Siddhant Singh's user avatar
4 votes
0 answers
56 views

Is there some notion of work associated with performing a measurement?

Let a measurement be described by POVM elements $M_i$ such that probability $p(i) = Tr[\rho M_i]$ for some state $\rho$. I want to know whether there is some notion of work associated with such ...
Mike's user avatar
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1 answer
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How to express a probability distribution $P(x,y,z)= \sum_\lambda P(x|y,\lambda)P(y|\lambda,z)P(z)P(\lambda)$ in terms of a trace of a density matrix?

I have been given and expression for a probability distribution $$P(x,y,z)= \sum_\lambda P(x|y,\lambda)P(y|\lambda,z)P(z)P(\lambda)$$ and I have been asked to show that the above expression can be ...
Shashaank's user avatar
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1 answer
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What additional conditions are there to make POVM measurement same as projective measurement?

We know that POVMs are applied in the more general cases where the system is not necessarily closed. So mathematically, how does going from open to closed system change the scenario in case of POVM so ...
user27286's user avatar
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3 votes
2 answers
323 views

What is the relation between observables (as defined in the measure-theoretic framework) and POVMs?

A POVM is typically defined as a collection of operators $\{\mu(a)\}_{a\in\Sigma}$ with $\mu(a)\in\mathrm{Pos}(\mathcal X)$ positive operators such that $\sum_{a\in\Sigma}\mu(a)=I$, where I take here $...
glS's user avatar
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