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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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What is the Helstrom measurement?

I have been reading the paper Belief propagation decoding of quantum channels by passing quantum messages by Joseph Renes for decoding Classical-Quantum channels and I have crossed with the concept of ...
Josu Etxezarreta Martinez's user avatar
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 ...
glS's user avatar
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8 votes
2 answers
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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 ...
glS's user avatar
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8 votes
1 answer
689 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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7 votes
1 answer
504 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
430 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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7 votes
0 answers
146 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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6 votes
3 answers
1k views

Why are POVMs useful? Are they just an axiomatic way to define measurement?

I know the definition of projective measurement, generalized measurement, POVM. I understand the usage of generalized measurement for the reason that it can model experiments "easier" (for example ...
Marco Fellous-Asiani's user avatar
6 votes
2 answers
632 views

Are three POVM measurements on a single qubit physically realizable?

In Nielsen and Chuang Quantum Computation and Quantum Information book section 2.2.6, a POVM of three elements are used to measure a single qubit in order to know for sure whether the state is $|0\...
czwang's user avatar
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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 ...
TTa's user avatar
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6 votes
2 answers
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Does Neumark's/Naimark's extension theorem only apply to rank-1 POVMs?

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 $\mathcal{H}^d$ ...
junfan02's user avatar
  • 305
5 votes
2 answers
573 views

Give an explicit example of a $d = 4$ SIC-POVM

For $q=e^{2 \pi i/3}$, the set of $d^2$ vectors ($d=3$) \begin{equation} \left( \begin{array}{ccc} 0 & 1 & -1 \\ 0 & 1 & -q \\ 0 & 1 & -q^2 \\ -1 & 0 & 1 \\ -q &...
Paul B. Slater's user avatar
5 votes
1 answer
610 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 ...
MrRobot's user avatar
  • 253
5 votes
1 answer
306 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 ...
junfan02's user avatar
  • 305
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 $...
kludg's user avatar
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5 votes
1 answer
216 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 ...
Jon Megan's user avatar
  • 497
4 votes
2 answers
231 views

POVM three-qubit circuit for symmetric quantum states

I have been reading this paper but don't yet understand how to implement a circuit to determine in which state the qubit is not for a cyclic POVM. More specifically, I want to implement a cyclic POVM ...
xbk365's user avatar
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4 votes
2 answers
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What is the relation between POVMs and projective measurements?

I'm a little confused about the terminology of measurement. So say that we have the single qubit state $|\phi \rangle=c_0|0\rangle+c_1|1\rangle$. If we perform the projective measurement $P_0=|0\...
bhapi's user avatar
  • 869
4 votes
1 answer
171 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) \...
JRT's user avatar
  • 520
4 votes
1 answer
103 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 ...
Shadumu's user avatar
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4 votes
1 answer
873 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
2 answers
360 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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4 votes
1 answer
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How do I efficiently implement a POVM using a fixed universal gate set and the ability to measure in the standard basis?

Let's say I am given a Hamiltonian \begin{equation} H = \sum_{i = 1}^{m} H_{i}, \end{equation} where $H$ acts on $n$-qubits, and each $H_{i}$ acts non-trivially on at most $k$ qubits. The eigenvalues ...
BlackHat18's user avatar
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4 votes
1 answer
139 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 ...
glS's user avatar
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4 votes
1 answer
224 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 ...
glS's user avatar
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4 votes
1 answer
216 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
  • 157
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
  • 191
3 votes
2 answers
446 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 ...
Pratapaditya Bej's user avatar
3 votes
1 answer
245 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
  • 189
3 votes
1 answer
375 views

How do you embed a POVM matrix in a Unitary?

In QuantumKatas Measurement Task 2.3 - Peres-Wooter's Game, we are given 3 states A,B and C. We construct a POVM of these states. But how do we convert that POVM into a Unitary that we can apply. ...
vasjain's user avatar
  • 802
3 votes
1 answer
741 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: ...
glS's user avatar
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3 votes
1 answer
52 views

How large does the isometry in Naimark's theorem need to be for a 3-outcome POVM?

I am interested in the POVM example Nielsen and Chuang give in the discussion about indistinguishability. They define the POVM $E_1 = \frac{\sqrt{2}}{1+\sqrt{2}} |1\rangle \langle 1|$, $E_2 = \frac{\...
BenPhys's user avatar
  • 31
3 votes
1 answer
143 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 ...
user19468's user avatar
2 votes
2 answers
126 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 $...
Aemmel's user avatar
  • 23
2 votes
3 answers
441 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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2 votes
1 answer
327 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
  • 165
2 votes
1 answer
192 views

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
  • 129
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 $i \neq j$?
Michael.Andy's user avatar
2 votes
2 answers
184 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
2 votes
2 answers
371 views

Does the dilation in Naimark's theorem produce a state?

A POVM, as defined for example in (Peres and Wooters 1991), is defined by a set of positive operators $\mu(a)$ satisfying $\sum_a \mu(a)=\mathbb 1$. We do not require the $\mu(a)$ to be projectors, ...
glS's user avatar
  • 25.4k
2 votes
1 answer
101 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 ...
yosh's user avatar
  • 127
2 votes
1 answer
124 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: $$...
shashvat's user avatar
  • 807
2 votes
1 answer
176 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 ...
junoswrld's user avatar
2 votes
1 answer
100 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, \...
Michael.Andy's user avatar
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 ...
Eulerian's user avatar
  • 183
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)...
karry's user avatar
  • 629
2 votes
1 answer
80 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?
karry's user avatar
  • 629
2 votes
1 answer
76 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 ...
Pratapaditya Bej's user avatar
2 votes
0 answers
73 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 ...
yosh's user avatar
  • 127
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): $$\mathcal D(\mathcal S:\mathcal A) = I(\mathcal S:\mathcal A) - \chi(\rho_{\mathcal A}),$$ ...
glS's user avatar
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