Quantum Computing Stack Exchange
Quantum Computing Stack Exchange

Questions tagged [state-discrimination]

the distinguishability of quantum systems in different states, and the general process of extracting classical information from quantum systems

53 questions
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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 $\rho_x$. How does this quantity relate to…
john_smith
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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 distinguish between these two states --- the…
BlackHat18
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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 of figuring out which of the two states the…
glS
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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 distinguish between these two states --- the…
BlackHat18
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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, but one of the two. I need to guess which one of…
nishkr
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How to measure superposition coefficients to determine state?

There was a problem at the Winter 2019 Q# codeforces contest, which I cannot find a mathematical solution for. It goes like this: You are given 3 qubits that can be in one of the following states: $$|\psi_0\rangle=\frac{1}{\sqrt…
Tudor
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How to measure the multiple-use distinguishability of two quantum channels?

The diamond norm categorises the single-use distinguishability of two quantum channels. The operational setting is the following: you have access to an unknown unitary, which is promissed to be either $U_1$ or $U_2$ and the goal is to distinguish…
shashvat
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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 quantity $\sum_a p_a \langle\mu_a,\rho_a\rangle$…
glS
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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}\rangle = \frac{1}{\sqrt{2}}(|01\rangle +…
user16106
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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 $$ \sum_{x\in X} p_x\operatorname{Tr} M_x\rho_x \geq 1-\varepsilon. $$ Does…
user114158
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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 if the states are orthogonal. The resulting…
Jason Pereira
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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 $|w\rangle$ with probability $1-p$. I will choose some…
Pedro
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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 unknown $x$. Is it possible to distinguish the two?
Henry
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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 successfully identifying a state between two…
Dimitri
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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 the metric to be the average or the worst-case…
Stan
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