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
  • 6,162
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
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
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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
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2 votes
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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
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
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2 votes
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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
  • 24.8k
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
1 vote

Expected trace distance between two types of random ensembles

Assume each of these states to be independently distributed. The question is too challenging as stated, so let us look at the expected fidelity between the two states, where trace distance is the ...
Quantum Mechanic's user avatar
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
  • 24.8k
1 vote

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

Though gIS' answer concerns deterministic measurements, its conclusion doesn't change when considering probabilistic measurements. Though it's not a formal proof, we can use ...
Tristan Nemoz's user avatar
  • 6,162
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
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
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

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