8 votes
Accepted

Can $2^n$ bits be sent with $n$ instances of quantum teleportation?

Quantum teleportation can send a single qubit from Alice to Bob, with two classical bits Correct, on the condition that Alice and Bob also have an entangled qubit pair shared between them. This ...
Chris E's user avatar
  • 1,235
4 votes
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How to derive the quantum Fisher information from the relative entropy?

Expressing the derivative $\partial_i\rho$ in terms of its eigenvalues and eigenvectors will show us that these two are not equal. I will assume a full-rank density matrix $\rho$ to streamline the ...
Quantum Mechanic's user avatar
4 votes
Accepted

How to show $T(\rho,\sigma)≥\sum_i|r_i − s_i|$ with $r_i,s_i$ eigenvalues of $\rho,\sigma$?

By the min-max theorem, we have$^1$ $$ \begin{align} t_k&=\max_{\quad U\\\dim U=k}\min_{|x\rangle\in U\\\langle x|x\rangle=1}\langle x|V|x\rangle\tag1\\ &=\max_{\quad U\\\dim U=k}\min_{|x\...
Adam Zalcman's user avatar
  • 19.1k
3 votes

Why does the bit flip channel produce a uniform contraction of $1-2p$?

The operation elements corresponding to the bit flip channel are, $E_0=\sqrt{p}I=\sqrt{p}\begin{bmatrix}1&0\\0&1\end{bmatrix}$ and $E_1=\sqrt{1-p}X=\sqrt{1-p}\begin{bmatrix}0&1\\1&0\...
Sooraj S's user avatar
  • 721
3 votes
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How to calculate the log of a density matrix?

By definition, a $\log$ of a matrix $M$ is a matrix $L$ such that $\exp(L)=M$. Note that I wrote a matrix because in general it may not be uniquely defined. Recall the definition of $\exp(L)$: $$\exp(...
Tristan Nemoz's user avatar
3 votes

What are explicit examples of smoothed conditional min(max) entropies?

I'll just give a classical example, which is a typical motivating example for these smooth quantities. Consider an $n$-bit distribution of the form $$ p(x) = \begin{cases} 1-\delta \qquad \text{ if }x=...
Rammus's user avatar
  • 4,906
3 votes

Is it possible to prevent quantum communication detection?

You don't need quantum in order to communicate while evading detection. There's nearly a century of research on what is known as LPI/LPD (Low-probability of intercept/low-probability of detection) ...
Punk_Physicist's user avatar
3 votes
Accepted

Is it possible to prevent quantum communication detection?

Regardless of whether Alice and Bob share some amount of entangled states, there is no way for them to communicate without having Alice physically sending something to Bob. "Something" means ...
glS's user avatar
  • 21.7k
3 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
  • 4,906
3 votes

References of group theory for quantum information theory

Short answer: there are no shortcuts. Things like Schur–Weyl duality are from representation theory. This generally assumes you understand group theory to begin with. Most physics-oriented texts I've ...
esabo's user avatar
  • 166
3 votes

General Proof of the Statement that You Need $1$ Ebit and $2$ Bits to Teleport $1$ Qubit?

To prove that teleportation of a single qubit cannot be accomplished with less than 2 classical bits, you can assume the opposite to show that No-Signalling was violated (as your comment suggests). ...
forky40's user avatar
  • 5,715
3 votes
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What is the conditional min-entropy of a pure bipartite state?

I'll use an equivalent definition of the min-entropy $$ \begin{aligned} H_{\min}(A|B) = - \log_2 \min& \quad \lambda \\ \mathrm{s.t.}& \quad \rho_{AB} \leq \lambda I_A \otimes \sigma_B \\ &...
Rammus's user avatar
  • 4,906
2 votes
Accepted

$2$ ebits $+$ $1$ bit $ = 2$ bits?

The protocol you describe is correct, but the resource estimation is wrong. Furthermore, something like superdense coding with purely classical bits is prohibited by the No-Signaling Principle. This ...
forky40's user avatar
  • 5,715
2 votes
Accepted

Additivity of degradable and anti-degradable quantum capacities

I have been able to find the answer to this question, so I will post it myself for anyone that would be interested. The result is proven in Useful States and Entanglement Distillation by Leditzky, ...
Josu Etxezarreta Martinez's user avatar
2 votes

Does proving $Q^{(1)}(\mathcal{N}\otimes\mathcal{N})=Q^{(1)}(\mathcal{N})+Q^{(1)}(\mathcal{N})$ imply additivity for arbitrary $n$?

I found out the paper Quantum Channel Capacities (by Graeme Smith) were the author states: "Some entropic function f(N ) is shown to be an achievable rate, and its regularization equal to the ...
Josu Etxezarreta Martinez's user avatar
2 votes

With $\vert\Psi^+\rangle$ the Bell state, can $\sqrt{\rho}\vert\Psi^+\rangle\langle\Psi^+\vert\sqrt{\rho}$ be simplified?

TL;DR: Yes, $\Omega_{AB}(\rho)$ can be simplified. It turns out that it's one $n$th of the projector onto the ray spanned by the vectorization of $\rho^{1/2}$, see $(3)$ below. First, observe that $\...
Adam Zalcman's user avatar
  • 19.1k
2 votes
Accepted

How to prove the strong convexity of the trace distance?

Let $A$ be the set of indices $i$ such that $p_i\ge q_i$. We have $$ \begin{align} \sum_i(p_i-q_i)\mathrm{tr}(P\sigma_i)\le&\,\sum_{i\in A}(p_i-q_i)\mathrm{tr}(P\sigma_i)\tag1\\ \le&\,\sum_{i\...
Adam Zalcman's user avatar
  • 19.1k
2 votes

Under what conditions does entanglement distillation work?

My intuition says to me that, since pairs may be affected by any noise (X and Z i.i.d.), the protocol ends up in a loop where whenever you try to correct an error, you create another one. This only ...
Craig Gidney's user avatar
  • 30.1k
2 votes
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Comparison of Quantum Mutual Information and Coherent Information with Classical Mutual Information

You are asking about the quantum analogue of a classical quantity, but quantum information contains features that have no classical analogue and so you won't find an objective answer. Here are some ...
forky40's user avatar
  • 5,715
2 votes
Accepted

Physical interpretation of a 2-photon-qubit system occupying the antisymmetric Bell state

The confusion here arises because of not being careful about what the labels inside the kets mean. When we talk about the Bell states (or about the phenomena of entanglement), we consider two ...
Abdullah Khalid's user avatar
2 votes
Accepted

Quantum Relative entropy- the math and intuition

To answer your questions briefly: This is purely a matter of normalization. It doesn't matter really as you could define it for just positive semidefinite operators like Watrous and they will be the ...
Rammus's user avatar
  • 4,906
2 votes

Clarification about inverses in sandwiched Renyi divergence

Firstly, the sandwiched divergence can be infinite even when $\rho$ and $\sigma$ are not orthogonal. For example, consider $\rho = \frac{|0\rangle \langle 0| + |1\rangle\langle 1|}{2}$ and $\sigma = |...
Rammus's user avatar
  • 4,906
2 votes

Is it true that $|r_i-s_i| \le 1/2$ for all $i$, where $r_i$ and $s_i$ are the eigenvalues of density matrices $\rho$ and $\sigma$?

In this proof, we assume that the trace distance between $\rho$ and $\sigma$ is upper-bounded by $\frac{1}{\mathrm{e}}<\frac12$. As mentioned by glS in the comments, the trace distance is lower-...
Tristan Nemoz's user avatar
2 votes
Accepted

What are the antisymmetric terms in $\sigma_{mn}$ in the expression for the Fisher information?

Antisymmetric means that $\sigma_{nm}=-\sigma_{mn}$. Since the sum ranges over all values of $m$ and $n$, adding an antisymmetric term adds something proportional to $$|\langle \psi_m^\lambda|\...
Quantum Mechanic's user avatar
1 vote

Does the max-relative entropy satisfy $0 < D_{\max}(\rho \parallel I_A \otimes \sigma_B) < 1$?

No. As discussed e.g. in the second lecture of https://cs.uwaterloo.ca/~watrous/QIT-notes/, between pages 16 and 17, if $\sigma$ is a state, then $2^{-D_{\rm max}(\rho\|\sigma)}\in[0,1]$, or ...
glS's user avatar
  • 21.7k
1 vote

Why can the max-relative entropy be written as $D_{\max}(\rho \parallel \sigma) = \inf \{ \lambda : \rho \leq 2^\lambda \sigma \}$?

Assuming I'm reading the post correctly, the question seems to be "why/how is the standard notion of relative entropy related to the given expression for the max-relative entropy? Consider the ...
glS's user avatar
  • 21.7k
1 vote
Accepted

Prove that the conditional min-entropy is $H_{\rm min}(A|B)=\max_\sigma\sup\{\lambda:\,\rho\le 2^{-\lambda}(I\otimes\sigma)\}$

Observe that $$ H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \| I_A \otimes \sigma_B \right) = \sup\limits_{\sigma_B} [-D_{\max} \left( \rho_{AB} \| I_A \otimes \sigma_B \right)]...
glS's user avatar
  • 21.7k
1 vote
Accepted

How to measure an unknown state produced by a source of qubits?

I thought I'd elaborate a little on @Chris E's answer with some details about what kind of precision we can expect in our approximation as we increase the number of measurements we make. I'll be ...
Franklin Pezzuti Dyer's user avatar
1 vote
Accepted

Why is error correction very different for circuits compared to channels?

Error correction for circuits and channels works just the same (encode & decode from an error correcting code) provided the only thing that you are trying to correct for is what happens between ...
DaftWullie's user avatar
  • 52.3k
1 vote

How many measurements are needed to distinguish two fixed density matrices?

Settings In your setting, you assume you are given $n$ copies of some state $\rho$, which is either $\rho_1$ or $\rho_2$ with equal probability. The setting you have chosen is also symmetric in ...
rnva's user avatar
  • 530

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