6 votes

Relating min-entropy with conditional entropy

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 \| \...
  • 4,598
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 ...
  • 48.1k
3 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 ...
  • 14.7k
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 ...
  • 4,252
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 ...
3 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 ...
  • 19.5k
2 votes
Accepted

Umambiguous 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{...
  • 4,252
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\...
  • 19.5k
2 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 ...
  • 4,252
1 vote

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 $|\...
  • 19.5k
1 vote
Accepted

Helstrom Measurement when two quantum states are close

Heuristically, when $\sigma$ and $\rho$ are "close," we can write $$\sigma=(1-\epsilon)\rho+\epsilon\varrho$$ for some small positive number $\epsilon$ and some other normalized state $\...
1 vote

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 $\mu$ ...
  • 19.5k
1 vote

Conditional Statements on a Quantum Computer

Disclaimer: Just few ideas, this is not full answer. XOR function is implemented by CNOT gate since: $|00\rangle \rightarrow |00\rangle$ $|01\rangle \rightarrow |01\rangle$ $|10\rangle \rightarrow |...
1 vote

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 ...
  • 19.5k
1 vote
Accepted

Finding the optimal projective measurement to distinguish between two pure states

$\newcommand{\ket}[1]{\lvert #1\rangle}\newcommand{\PP}{\mathbb{P}}$Given an arbitrary state $\ket a$, let us write the corresponding density matrix/projector as $\PP_a$. Any such density matrix can ...
  • 19.5k

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