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7 votes

How does the conditional min-entropy $H_{\rm min}(A|B)_\rho$ relate to the conditional entropy $H(X|Y)_\rho$?

The conditional min-entropy $\text{H}_{\text{min}}(A | B)_{\rho}$ can be defined for an arbitrary state $\rho$ of a pair of registers $(A,B)$ as $$ - \inf_{\sigma} \,\text{D}_{\text{max}}(\rho \| \...
John Watrous's user avatar
  • 6,097
5 votes
Accepted

Is it possible to distinguish a pure state from a "partially uniform" state?

As mentioned by @DaftWullie, if you know $f$ then you can uncompute it and end up with $\frac{1}{2^n}\sum_x|x,0\rangle$ vs. $|x, 0\rangle$. You can then apply an $H$ gate on the first register, which ...
Tristan Nemoz's user avatar
  • 6,462
4 votes

How to distinguish between two very similar pure quantum states?

The claim does not specify what protocols for distinguishing quantum states are acceptable. In particular, it does not state whether we are allowed to err or reserve judgment. Below, we note success ...
Adam Zalcman's user avatar
  • 22.9k
4 votes
Accepted

What is the conditional min-entropy for diagonal ("classical") matrices?

Long story short: taking $\sigma_B = \rho_B$ is equivalent to taking the worst case min-entropy $$ \hat{H}_{\min}(A|B) = - \log \max_{a,b} P(A=a|B=b)\,, $$ and optimizing over $\sigma_B$ is equivalent ...
Rammus's user avatar
  • 5,853
4 votes
Accepted

Are projective measurements the only optimal measurements to discriminate between two states?

Suppose you are given either $\rho_1$ or $\rho_2$, and you also know that the probabilities you got one or the other are $p_1$ and $p_2$, respectively. If you have no prior knowledge of the ...
glS's user avatar
  • 25.4k
4 votes

How do you test a pair of unknown qubits for orthogonality with certainty?

It is not possible for a measurement to deterministically give one outcome or the other depending on whether two states are equal or orthogonal. Such a measurement would be some two-outcome POVM $\mu$ ...
glS's user avatar
  • 25.4k
4 votes

Distinguishing $\frac{| 0 \rangle + e^{i\theta} |1 \rangle}{\sqrt{2}} $ from $| 0 \rangle/|1 \rangle$ with probability $1/2 + \epsilon$

No, there is no way to do this with probability better than 1/2. Basically, you are saying that, ideally, you'd like a protocol where $$ |0\rangle\mapsto|0\rangle,\quad |1\rangle\mapsto|0\rangle,\quad ...
DaftWullie's user avatar
  • 58.8k
4 votes
Accepted

Unambiguous discrimination using POVM with highest discriminate probability

I'll write $\rho_1 = |\psi_1\rangle \langle \psi_1|$ and $\rho_2 = |\psi_2\rangle \langle \psi_2|$. We want the discrimination to be unambiguous so we want, $$ \mathrm{tr}[\rho_1 \Pi_2] = 0 = \mathrm{...
Rammus's user avatar
  • 5,853
4 votes
Accepted

Can we test whether $|\psi\rangle$ is orthogonal to $|\phi\rangle$ without creating a coherent superposition therebetween?

No, you can't do it (except for trivial things). Think about what you're asking for: a map that performs $$ |0\rangle|\psi\rangle|\psi^\perp\rangle\longrightarrow |1\rangle|\psi\rangle|\psi^\perp\...
DaftWullie's user avatar
  • 58.8k
3 votes
Accepted

How to find the POVM that optimally distinguishes between two given states?

The optimal probability of guessing correctly is $$ \frac12 + \frac12 \Big\|\frac23 \rho_0 - \frac13 \rho_1 \Big\|_1 $$ where $\| X \|_1 = \mathrm{Tr}[\sqrt{X^* X}]$ is the Schatten 1-norm. This ...
Rammus's user avatar
  • 5,853
3 votes
Accepted

Is the quantum state discrimination success probability always $\lambda_0\langle\mu(0),\rho_0\rangle+\lambda_1\langle\mu(1),\rho_1\rangle$?

Yes. Expression (X) is in principle more general, but it boils down to doing classical post-processing of the outcomes, and this can always be included in the POVM. So as long as you do optimize over ...
Mateus Araújo's user avatar
2 votes

In classical state discrimination, why does the trace distance quantify the probability of success?

Upon some additional reflection, I found a way to see that the expressions are equivalent. I'm not sure whether this is the way that was intended in the text though, there might be better/more direct ...
glS's user avatar
  • 25.4k
2 votes
Accepted

Do entangled measurements across multiple copies help in state distinguishability?

Very roughly speaking, yes, "entangled measures" (that is, global measures on multiple copies) make it easier to distinguish states. The intuitive reason is that, if $\langle\rho,\sigma\...
glS's user avatar
  • 25.4k
2 votes
Accepted

What projective measurement discriminates between a set of pairwise orthogonal states?

If the states $|\psi_j\rangle$ are orthogonal, then the set of projections $\{|\psi_j\rangle\!\langle \psi_j|\}_j$ form a projective measurement. Performing this measurement, if the input state is $|\...
glS's user avatar
  • 25.4k
2 votes

How to discriminate between $N$ states drawn from one of two ensembles?

Unless I'm mistaken, your special case is also the solution to the generalization, up to the fact that you have to redefine $\rho$ and $\sigma$. Forget about $Q$ for now. If Alice gives Bob one copy ...
Tristan Nemoz's user avatar
  • 6,462
2 votes
Accepted

Unambiguous State Discrimination

In general, I think not. You need to be more restrictive on the properties of your sets. To see this, let's define 3 measurement operators: $M_A$, $M_B$ and $M_U$, corresponding to the answers A (...
DaftWullie's user avatar
  • 58.8k
2 votes
Accepted

proof of Theorem 3.10 (Barnum-Knill) on pretty-good measurements in John Watrous' book

As also mentioned in this other answer of mine, that identity relies on the following general statement: given any pair of Hermitian matrices, $A,B$, and an Hermitian operator $C$ such that $\...
glS's user avatar
  • 25.4k
2 votes
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Ensemble state identification from ensemble state distinction

Your idea is essentially correct. I will just expand it a bit: Let $1_{A}$ denote the indicator random variable that takes value $1$ on event $A$ and $0$ otherwise. Let $\mathcal{M}$ be the POVM that ...
Qaunt_mickey's user avatar
2 votes
Accepted

Is unambiguous discrimination between $|+\rangle,|0\rangle,|1\rangle$ possible?

It is not possible to unambiguously distinguish between these states. To perform unambiguous state discrimination you need states orthogonal to all but one of the states to discriminate. So for ...
glS's user avatar
  • 25.4k
2 votes
Accepted

Niemark's theorem - simulating POVMs with PVMs

The tl;dr of how to go from a given POVM to a PVM is the following: Take your state on which you do the measurement to a larger Hilbert space using a linear isometry $A$. Do a projective measurement ...
rnva's user avatar
  • 842
2 votes

Quantum Information Retrieval from Bipartite Mixed States under LOCC: Maximizing Individual State Knowledge

None. What is meaningful is the state $\rho$, not the decomposition $\sum \lvert\psi_i\rangle\langle\psi_i$. There is many different such definitions, and there is no way to distinguish them by any ...
Norbert Schuch's user avatar
2 votes

What is a measurement that always identifies $|0\rangle$ but can misidentify $|1\rangle$?

If I understand the question, you're looking for a state strategy to discriminate $|0\rangle$ and $|1\rangle$, with zero error when the input is $|0\rangle$, but some error when it's $|1\rangle$. You ...
glS's user avatar
  • 25.4k
1 vote

Why is state discrimination possible to infidelity $\delta$ using $n=\Theta(1/\delta)$ states?

The statement state discrimination is possible to infidelity 𝛿 with 𝑛 copies is probably related to minimum error state discrimination. The minimum probability of error (while discriminating ...
Ghost-of-PPPF's user avatar
1 vote

Making an ambiguous and unambiguous state determinations together

For two states this doesn't work, as @glS pointed out: the probability of getting an inconclusive outcome is equal to $|\langle \psi_0 | \psi_1 \rangle| $ for both $|\psi_0\rangle$ and $|\psi_1\rangle$...
Mateus Araújo's user avatar
1 vote

What's the best entangling circuit to measure the Peres-Wootters double-trine state?

Following on from my previous answer about unambiguous discrimination, there is also an optimal solution for the minimum error probability, which I found here. The trick is to introduce a state $$|\...
DaftWullie's user avatar
  • 58.8k
1 vote
Accepted

What's the best entangling circuit to measure the Peres-Wootters double-trine state?

There's no simple answer to what is the "best". You need to define what you mean a bit more carefully. There are two common scenarios under which one optimises: minimum error: we might get ...
DaftWullie's user avatar
  • 58.8k
1 vote

How to understand the result of Scenario 3.1 in John Watrous' book?

This isn't directly the derivation that you need, but hopefully shows you all the elements that you need! Imagine that for one sample of $X$, with value $b$, you have a probability of getting the two ...
DaftWullie's user avatar
  • 58.8k
1 vote

What projective measurement discriminates between a set of pairwise orthogonal states?

Just measure in [K] basis. Since only one $|\psi_i\rangle$ among $i=1\ldots K$ can be measurement outcome, its probability $P_i=1$. I wonder what answer is expected. Maybe you should characterize ...
kludg's user avatar
  • 3,214
1 vote

Are mixtures of pairs of Bell states perfectly distinguishable by local operations?

The states $\sigma_0$ and $\sigma_1$ can be seen to have the form $$\sigma_0 = N[\mathbb P(e_1+e_4) + \mathbb P(e_2+e_3)], \\ \sigma_1 = N[\mathbb P(e_1-e_4) + \mathbb P(e_2-e_3)],$$ where $N$ are ...
glS's user avatar
  • 25.4k
1 vote

What's the idea behind "pretty good measurements"?

I'll reproduce here a standard argument used to prove the fundamental bound for pretty good measurements (PGMs), the the most part taken from Watrous' book, with some minor changes in notation, ...
glS's user avatar
  • 25.4k

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