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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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2 votes
3 answers
431 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 $\...
8 votes
2 answers
1k views

What's the POVM corresponding to single-qubit state tomography?

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 ...
5 votes
1 answer
604 views

What is an example of a separable measurement that is not LOCC?

Could you give me an example of a measurement which is separable but not LOCC (Local Operations Classical Communication)? Given an ensable of states $\rho^{N}$, a separable measurement on it is a POVM ...
4 votes
1 answer
102 views

Can post-measurement states have entropy larger than the original state?

Given a set of measurement operators $\{M_i\}$ that sum to unity, consider the post-measurement states on some $\rho$ as $\rho_i:=(\sqrt{M_i}\rho\sqrt{M_i})/p_i$ and $p_i:=\mathrm{Tr}(M_i\rho)$. It's ...
1 vote
0 answers
22 views

The POVM representation for 1 qubit using multiqubit Clifford group

I am studying the multiqubit Clifford group with the aim of utilizing it to generate a POVM for a single qubit. Similar to how the SIC-POVM is generated using four matrices, I aim to accomplish a ...
2 votes
0 answers
71 views

What does it mean to take the maximum over all POVMs?

What does it mean to take the maximum over all POVMs (in the definition of accessible information for example) ? The set of POVMs is infinite, how can we be sure that the maximum is attained by one of ...
2 votes
1 answer
91 views

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, \...
1 vote
1 answer
86 views

Was Deutsch contemplating a positive-operator valued measurement to distinguish balancedness from constancy?

This is a follow up to a couple of questions on Deutsch's foundational paper on quantum Turing machines. In it, he determines $f(0)\oplus f(1)$ with a single query by measuring a state prepared as $\...
2 votes
1 answer
100 views

Niemark's theorem - simulating POVMs with PVMs

I am having trouble understanding Niemark's theorem from books (e.g watrous). The wikipedia page is clearer but most calculations are not justified. I want to contruct the PVM associated with the ...
2 votes
1 answer
119 views

How can one derive the POVM to unambiguously discriminate between $|0\rangle$ and $|+\rangle$?

Page 92 of Nielsen and Chuang describes a POVM that can determine if a given state is either $|0\rangle$ or $|+\rangle$ with no error, but with some chance of an inconclusive result. The POVM is: $$...
1 vote
2 answers
71 views

Are the outcomes of a quantum measurement a random variable?

What does it mean when we say that the outcome of a quantum measurement is a random variable (or a quantum ensemble) such as in renes notes (page 51) or in this paper (page 4 B-1) ? Does it mean that ...
2 votes
1 answer
157 views

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 ...
1 vote
0 answers
39 views

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 \...
2 votes
1 answer
79 views

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?
2 votes
1 answer
45 views

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)...
1 vote
0 answers
50 views

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 ...
0 votes
1 answer
95 views

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: $M_{m}^{T*} = M_{m}$ ...
0 votes
1 answer
142 views

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 ...
1 vote
1 answer
94 views

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 ...
2 votes
1 answer
301 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}{\...
7 votes
1 answer
414 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 ...
1 vote
0 answers
41 views

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], ...
4 votes
1 answer
127 views

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 ...
1 vote
0 answers
75 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= \...
2 votes
0 answers
65 views

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): $$\mathcal D(\mathcal S:\mathcal A) = I(\mathcal S:\mathcal A) - \chi(\rho_{\mathcal A}),$$ ...
2 votes
1 answer
33 views

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 $i \neq j$?
2 votes
1 answer
74 views

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 ...
0 votes
1 answer
116 views

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 ...
2 votes
0 answers
45 views

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 $f$...
-1 votes
1 answer
84 views

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}...
6 votes
1 answer
2k views

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 ...
5 votes
1 answer
205 views

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 ...
8 votes
1 answer
673 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 $\...
4 votes
1 answer
167 views

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) \...
3 votes
1 answer
685 views

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: ...
4 votes
1 answer
220 views

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 ...
2 votes
0 answers
105 views

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 ...
2 votes
2 answers
124 views

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 $...
3 votes
1 answer
241 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 ...
2 votes
2 answers
68 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 ...
11 votes
2 answers
1k views

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 ...
3 votes
2 answers
433 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 ...
6 votes
2 answers
1k 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 $\mathcal{H}^d$ ...
1 vote
1 answer
194 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 ...
5 votes
1 answer
1k views

What are examples of non-trivial POVM measurements?

We know that generalized (POVM) measurement is defined by matrices $M_i$ which are Positive semidefinite Add up to a unit matrix, $\sum_i M_i = \mathbb{I}$ and the probability of obtaining outcome $...
1 vote
1 answer
52 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 ...
3 votes
1 answer
139 views

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 ...
4 votes
1 answer
214 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 $U$ ...
2 votes
2 answers
176 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 $M_m=U_m\sqrt{E_m}$, ...
7 votes
1 answer
485 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 ...