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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 ...
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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 ...
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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 ...
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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 ...
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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 $\mathcal{H}^d$ ...
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### Give an explicit example of a $d = 4$ SIC-POVM

For $q=e^{2 \pi i/3}$, the set of $d^2$ vectors ($d=3$) \left( \begin{array}{ccc} 0 & 1 & -1 \\ 0 & 1 & -q \\ 0 & 1 & -q^2 \\ -1 & 0 & 1 \\ -q &...
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### 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 ...
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### 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 ...
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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 ...
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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 \sum_{i=1}^n \vert \langle \langle E_i \vert X \rangle \rangle \vert^2 = C \Vert V \Vert_2^2, \...
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### 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 ...
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### 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 ...
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Consider the (asymmetric) quantum discord, defined as (borrowing notation from Eq. 4.13c of Zurek's review): $$\mathcal D(\mathcal S:\mathcal A) = I(\mathcal S:\mathcal A) - \chi(\rho_{\mathcal A}),$$ ...