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 ...
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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 \...
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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)...
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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?
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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 ...
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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, \...
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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 ...
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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}$
...
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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 ...
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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 ...
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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], ...
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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= \...
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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 $i \neq j$?
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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 ...
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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♦
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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 ...
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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 $f$...
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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♦
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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.
$$\begin{aligned}
& \left|\psi_1\right\rangle=|0\rangle \\
& \left|\psi_2\right\rangle=\frac{1}...
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What is a POVM?
I am having a hard time understanding what exactly a Measurement is by its definition? What I read is that a POVM $M$ is defined by its set of elements $M_i$. So is $M$ itself an operator? In circuit ...
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How many measurements are needed to distinguish two fixed density matrices?
Suppose there are two fixed density matrices $\rho_1$ and $\rho_2$ are prepared for equal probability. Can we say something about the minimum number of measurements required to distinguish the two ...
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Does closeness in trace distance imply close measurement outcomes?
Suppose we have two density matrices $\rho$ and $\rho'$ such that $\|\rho - \rho'\|_1 \leq \varepsilon$. Let $\{\Lambda, I - \Lambda\}$ be elements of some POVM. If it holds that
$$Tr(\Lambda\rho) \...
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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♦
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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♦
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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 ...
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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 $...
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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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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 ...
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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♦
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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 $\mathcal{H}^d$ ...
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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 ...
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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 ...
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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 ...
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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 $U$ ...
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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 $M_m=U_m\sqrt{E_m}$, ...
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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♦
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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 ...
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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♦
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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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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}{\...
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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 ...
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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♦
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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♦
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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♦
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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♦
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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 ...
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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 ...
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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 ...
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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 ...
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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♦
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