Questions tagged [state-discrimination]
the distinguishability of quantum systems in different states, and the general process of extracting classical information from quantum systems
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The lower bound on the probability of error in quantum hypothesis testing
Prove the following lower bound on the probability of error $P_e$ in a quantum hypothesis test to distinguish $\rho$ from $\sigma$:
\begin{align}
P_e \geq \frac{1}{2} \left(1-\sqrt{1-F(\rho, \...
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geometric intepretation of Helstrom formula
Let's suppose Alice transmit to Bob either one of the two states:
$$|\psi_{\pm}\rangle = \cos(\theta)|H\rangle \pm \sin(\theta)|V\rangle, \quad \theta \in [-\frac{\pi}{4}, +\frac{\pi}{4}]$$
The ...
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Can we test whether $|\psi\rangle$ is orthogonal to $|\phi\rangle$ without creating a coherent superposition therebetween?
Let a first register store $|\psi\rangle$ and a second register store $|\phi\rangle$, and let us be promised that either $\vert\langle\psi|\phi\rangle\vert^2=0$ or $\vert\langle\psi|\phi\rangle\vert^2=...
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Improving Quantum State Distinguishability through Embedding
Consider the following map, $\mathcal{E}:\mathcal{L}(H_A) \rightarrow \mathcal{L}(H_{AB})$,
$$
\mathcal{E}(\rho_A|U_{AB}, U_{AC}) = {\rm Tr_C} \left[ U_{AC}U_{AB}(\rho_A\otimes |0_B\rangle\langle 0_B|\...
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Is unambiguous discrimination between $|+\rangle,|0\rangle,|1\rangle$ possible?
I have a quantum state that is either $|{+}\rangle$ or it is $|{0}\rangle$, $|{1}\rangle$. Is there a way to determine this with a single measurement?
I am assuming not since $|{0}\rangle$, $|{1}\...
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How to discriminate between $N$ states drawn from one of two ensembles?
Consider the following quantum discrimination problem:
Suppose, there are two sets of states, $P = \{ \rho_i \}$ and $Q = \{\sigma_i\}$. Both Alice and Bob know which states are in each set. We can ...
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Probability of making an error in identifying quantum state
In my notes I have the following:
In a measurement described by POVM $\{\pî_j\}$, we associate $\pî_j$
with state $\rhô_j$, that is if outcome $j$ is obtained, we take this
to indicate that the ...
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Unambiguous State Discrimination
I have a collection of possible states that are not necessarily orthogonal to each other -- suppose $A_1, A_2, ... A_N, B_1, B_2... B_N$. I get a new state $C$, and I want to determine whether its in ...
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Is it possible to distinguish a pure state from a "partially uniform" state?
Let $f$ be a random function mapping $n$ bits to $m$ bits. Let $|\phi\rangle$ be a state that is whether (1) $\sum_x2^{-n/2}|x,f(x)\rangle$ or (2) a pure state $|x,f(x)\rangle$ for some random and ...
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Making an ambiguous and unambiguous state determinations together
Measurement operators for those that one creates for unambiguous state discrimination, $\Pi_0,~\Pi_1,~\Pi_?$, are such that
$\Pi_0 + \Pi_1 + \Pi_? = \mathbb{1}$ and the probabilities for a measurement ...
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What is the value of average success probability in unambiguous state discrimination?
Unambiguous state discrimination, as outlined here starting at p. 421, introduces an average success probability $P = 1 - Q^{POVM}$ where $Q^{POVM}$ is the average failure probability. However, I fail ...
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What's the best entangling circuit to measure the Peres-Wootters double-trine state?
The early-90's, pre-teleportation work of Peres and Wootters studied the now-called "double-trine" or Mercedes-Benz states of two qubits in one of three product states:
$$|A\rangle\otimes|B\...
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How to understand the result of Scenario 3.1 in John Watrous' book?
In scenario 3.1, Bob's goal is to correctly determine the value stored in $\textbf{Y} $ using only the information from the observation of $\textbf{X}$. How to understand the claim that "The ...
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proof of Theorem 3.10 (Barnum-Knill) on pretty-good measurements in John Watrous' book
The near-optimality result for pretty-good measurements as given in Barnum & Knill's original paper holds only for a commuting ensemble of states, Theorem 2 in
https://arxiv.org/pdf/quant-ph/...
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Optimality of discriminating POVM among all quantum channels
It is proven in the very helpful answer here that using the optimal POVM for unambiguous discrimination of the equally-likely non-orthogonal states $\lvert 0\rangle$ and $\lvert +\rangle$, we can ...
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Show pretty good measurements give the error probability bound $P_e^{\rm PGM}\le\sum_{i\neq j}\sqrt{p_i p_j} F(\rho_i\rho_j)$
While reading some online lecture notes (Link to pdf) about state discrimination, I stumbled upon the following bound for the success probability of discriminating between a set of states $\rho_i$, ...
glS♦
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Holevo bound and indistinguishability of non-orthogonal quantum states
I was trying to understand the fact that non-orthogonal quantum states cannot be reliably distinguished and I came across this link.
The explanation finishes with the result that the probability of ...
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Ensemble state identification from ensemble state distinction
I am trying to derive Fact 5. in paper 1:
Let $\mathscr{E}=\{\sigma_1,.., \sigma_m\}$ be an ensemble of quantum states in $\mathbb{C}^n$. If there is a POVM $\mathscr{M}$ for the state distinction ...
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Why is state discrimination possible to infidelity $\delta$ using $n=\Theta(1/\delta)$ states?
In (Haah et al. 2015), in the first section, the authors study the asymptotic behaviours of fidelity and trace distance between $\rho^{\otimes n}$ and $\sigma^{\otimes n}$ for some given pair of ...
glS♦
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What projective measurement discriminates between a set of pairwise orthogonal states?
If we know a set of $K$ states $\{\lvert\psi_j\rangle:j\in[K]\}$ such that they are pairwise orthogonal, and we are given an unknown state $\lvert\psi_i\rangle$, what sort of projective measurement ...
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What is the conditional min-entropy for diagonal ("classical") matrices?
The conditional min-entropy, discussed e.g. in these notes by Watrous, as well as in this other post, can be defined as
$$\mathsf{H}_{\rm min }(\mathsf{X} \mid \mathsf{Y})_{\rho}\equiv -\inf _{\sigma \...
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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What's the idea behind "pretty good measurements"?
In the context of quantum state discrimination, the task of finding the POVM $\mu$ that optimally discriminates between the elements of an ensemble $a\mapsto (p_a,\rho_a)$, amounts to maximising the ...
glS♦
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What's the structure of the measurement $\mu$ that optimally discriminates an ensemble $\{(p_i,\rho_i)\}_i$?
As discussed e.g. in this post, given two states $\rho$ and $\sigma$, the measurement that allows to optimally discriminate between them (i.e. the measurement providing the highest average probability ...
glS♦
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Is there a way to know if a probability amplitude is negative or positive?
I have a quantum system that outputs a state similar to:
$|\psi\rangle = \frac{1}{2}(|00\rangle + |01\rangle + |10\rangle \pm |11\rangle)$.
So my question is: is there a way (by measurement or by ...
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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 mixtures of pairs of Bell states perfectly distinguishable by local operations?
Consider the four Bell states
$$ |\psi^{00}\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle), \hspace{2mm}
|\psi^{01}\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle),\hspace{2mm}
|\psi^{10}\...
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Optimality of the SWAP test versus weak Schur sampling for testing unitarily invariant properties
Consider the following setting.
I am either given the density matrix $|\psi\rangle \langle \psi|^{\otimes k}$ or the density matrix $\frac{\mathbb{I}^{\otimes k}}{2^{nk}}$, where $\mathbb{I}$ is the $...
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Do entangled measurements across multiple copies help in state distinguishability?
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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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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Are almost perfectly distinguishable ensembles almost orthogonal?
Let $\varepsilon>0$ and consider an ensemble of states $\{p_x\rho_x\}_{x\in X}$ and suppose there exists a measurement with POVM representation $\{M_x\}_{x\in X}$ such that
$$ \sum_{x\in X} p_x\...
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How do you test a pair of unknown qubits for orthogonality with certainty?
If you want to check if a pair of unknown qubits are the same, a standard test is the controlled SWAP test. This gives a result of 0 with certainty if the states are the same and 1 with a 50% chance ...
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Distinguishing $\frac{| 0 \rangle + e^{i\theta} |1 \rangle}{\sqrt{2}} $ from $| 0 \rangle/|1 \rangle$ with probability $1/2 + \epsilon$
I am given one copy of one of two quantum states -
$\frac{| 0 \rangle + e^{i\theta} | 1 \rangle}{\sqrt{2}} $, for some unknown fixed $\theta$.
One of $| 0 \rangle/|1 \rangle$ - don't know which one, ...
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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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Conditional Statements on a Quantum Computer
I've been struggling with this for a while now, and whilst I suspect the answer is unknown or very complicated, I thought I'd ask regardless.
Suppose I have some states $|\psi_i\rangle = \sum_{j=1}^n ...
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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 the quantum state discrimination success probability always $\lambda_0\langle\mu(0),\rho_0\rangle+\lambda_1\langle\mu(1),\rho_1\rangle$?
Consider the standard quantum state discrimination setup: Alice sends Bob either $\rho_0$ or $\rho_1$. She picks $\rho_0$ and $\rho_1$ with probabilities $\lambda_0$ and $\lambda_1$, respectively. Bob ...
glS♦
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In classical state discrimination, why does the trace distance quantify the probability of success?
Consider the following task: we are given a probability distribution $p_y:x\mapsto p_y(x)$ with $y\in\{0,1\}$ (e.g. we are given some black box that we can use to draw samples from either $p_0$ or $...
glS♦
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Quantum state discrimination and lower bound for conditional von Neumann entropy
Consider two quantum states $\rho_A$ and $\sigma_A$, and define the classical-quantum state over a classical binary system $B$ and $A$,
$$\omega_{AB}^\epsilon :=\epsilon \vert 0 \rangle \langle 0 \...
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Finding the optimal projective measurement to distinguish between two pure states
I would like some help on what should be a simple computation that I'm failing to see through to the end. Suppose I have a qubit which can be in the state $|v\rangle$ with probability $p$, or $|w\...
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How does the conditional min-entropy $H_{\rm min}(A|B)_\rho$ relate to the conditional entropy $H(X|Y)_\rho$?
Suppose we have a classical quantum state $\sum_x |x\rangle \langle x|\otimes \rho_x$, one can define the smooth-min entropy $H_\min(A|B)_\rho$ as the best probability of guessing outcome $x$ given $\...
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