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.

Filter by
Sorted by
Tagged with
6
votes
1answer
169 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
2answers
157 views

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\...
2
votes
1answer
153 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 ...
0
votes
1answer
60 views

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 ...
2
votes
3answers
138 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 $\...
3
votes
2answers
138 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 ...
5
votes
1answer
119 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 ...
5
votes
2answers
289 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\...
2
votes
2answers
155 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, ...
3
votes
1answer
61 views

How to find the POVM the optimally distinguishes 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 ...
3
votes
0answers
45 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 ...
2
votes
2answers
96 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 $...
1
vote
1answer
55 views

What are the matrices in the POVM for measuring the first $m$ qubits?

Suppose you have a quantum state $|w\rangle$ consisting of $m + n$ qubits, and you set up a measurement that measures the first $m$ qubits in the standard basis. What are the matrices in the ...
2
votes
0answers
41 views

How to define POVM measurement operators for a composite quantum state

I have an evolved quantum composite state $\hat{\rho}^{\otimes{N}}$ that I retrieved from a quantum channel $\mathcal{E}$, Now I do know how to define a POVM for the evolved states $\hat{\rho}$ that ...
1
vote
0answers
51 views

What does it mean “the N uses of classical-quantum channel”?

I was reading a paper Quantum Polar codes by Mark M. Wilde, where he discusses the N uses of the channel in the classical-quantum channel setting. What does he mean by "multiple channel uses"...
2
votes
1answer
443 views

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 ...
12
votes
1answer
1k views

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 ...
3
votes
1answer
160 views

How to construct a non-trivial (non-projective) POVM measurement example?

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 $...
3
votes
1answer
202 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 ...
4
votes
2answers
173 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 &...
3
votes
1answer
95 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. ...
4
votes
2answers
161 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 ...