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Questions tagged [entropy]

For questions about the various kinds of entropies --- as defined in the context of quantum information theory and quantum statistical mechanics.

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8 votes
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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 $\...
john_smith's user avatar
5 votes
1 answer
196 views

How to measure entanglement in an algorithm?

Entanglement in Algorithms Most algorithms in quantum computing find their strength in making use of entanglement. I am interested in evaluating the amount of entanglement generated within an ...
Jake Xuereb's user avatar
2 votes
2 answers
208 views

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

The proof of the Fannes' inequality replies on the formula $T(ρ, σ)≥\sum_i|r_i − s_i|$, where $r_i,s_i$ are the eigenvalues of $\rho,\sigma$, in the descending order. In the proof given in Box 11.2, ...
Sooraj S's user avatar
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7 votes
2 answers
873 views

How is the quantum relative entropy $S(\rho\|\sigma)$ defined when $\sigma$ is a pure state?

I want to evalualte the quantum relative entropy $S(\rho|| \sigma)=-{\rm tr}(\rho {\rm log}(\sigma))-S(\rho)$, where $\sigma=|\Psi\rangle\langle\Psi|$ is a density matrix corresponding to a pure state ...
Confinement's user avatar
7 votes
4 answers
1k views

Maximally mixed states for more than 1 qubit

For 1 qubit, the maximally mixed state is $\frac{\mathrm{I}}{2}$. So, for two qubits, I assume the maximally mixed state is the maximally mixed state is $\frac{\mathrm{I}}{4}$? Which is: $\frac{1}{...
Mahathi Vempati's user avatar
6 votes
1 answer
421 views

Thermodynamic Speed Limit to Quantum Computing

There's a lot of mystifying jargon in the field of quantum computation, so I would like to examine some elementary physics to maybe help clarify the assumptions being made. Is it not true that the ...
user avatar
6 votes
1 answer
750 views

How can the Holevo bound be used to show that $n$ qubits cannot transmit more than $n$ classical bits?

The inequality $\chi \le H(X)$ gives the upper bound on accessible information. This much is clear to me. However, what isn't clear is how this tells me I cannot transmit more than $n$ bits of ...
GaussStrife's user avatar
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5 votes
1 answer
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What is entropy quantum computing?

Recently there is news concerning some computational breakthroughs by using so-called "entropy quantum computing": https://thequantuminsider.com/2022/07/20/qci-solves-3854-variable-problem-...
QGK's user avatar
  • 51
5 votes
1 answer
151 views

Superoperator cannot increase relative entropy

Note: Cross-posted on Physics SE. So I have to show that a superoperator $\$$ cannot increase relative entropy using the monotonicity of relative entropy: $$S(\rho_A || \sigma_A) \leq S(\rho_{AB} || ...
CFRedDemon's user avatar
5 votes
1 answer
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Questions about the relation between max-relative entropy $D_{\max}(\rho||\sigma)$ and max-information

The max-relative entropy between two states is defined as $$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$ where $\rho\leq \sigma$ should be read as $\sigma - \rho$ is ...
user1936752's user avatar
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5 votes
2 answers
326 views

Shannon entropy is least when Measurement basis = Mixture basis

For a one qubit system, take a basis. Call this the mixture basis. Consider only basis states and classical mixtures of these basis states. Definition of Shannon Entropy used here: Defined with ...
Mahathi Vempati's user avatar
4 votes
2 answers
299 views

In the proof of the joint entropy theorem, why are $p_i\lambda_i^j$ the eigenvalues?

From section 11.3.2 of Nielsen & Chuang: (4) let $\lambda_i^j$ and $\left|e_i^j\right>$ be the eigenvalues and corresponding eigenvectors of $\rho_i$. Observe that $p_i\lambda_i^j$ and $\left|...
CFRedDemon's user avatar
3 votes
2 answers
565 views

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

The conditional min-entropy, discussed e.g. in these notes by Watrous, as well as in this other post, can be defined as $$\mathsf{H}_{\rm min }(\mathsf{X} \mid \mathsf{Y})_{\rho}\equiv -\inf _{\sigma \...
glS's user avatar
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3 votes
0 answers
46 views

Proof that the relative entropy satisfies $S(\rho\|\sigma)=S(T\rho\|T\sigma)$ iff $\hat TT\rho=\rho$, $\hat TT\sigma=\sigma$ for some $\hat T$

To prove the saturation condition for the strong subadditivity of the von Neumann entropy, the authors of [HJPW2004] make use of the following characterisation of when the monotonicity of the ...
glS's user avatar
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3 votes
0 answers
235 views

Prove $|η(r) − η(s)| ≤ η(|r − s|)$ when $|r − s| ≤ 1/2$ [closed]

Background If $\rho$ and $\sigma$ are density matrices such that the trace distance between them satisfies $T(\rho,\sigma)\leq1/e$. Then the Fannes' inequality states that $$|S(\rho)-S(\sigma)|\leq T(...
Sooraj S's user avatar
  • 831
3 votes
1 answer
116 views

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

The quantum conditional min-entropy is defined as $$H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \parallel I_A \otimes \sigma_B \right),$$ where $$D_{\max}(\rho \parallel I_A \...
Josh's user avatar
  • 417
2 votes
1 answer
128 views

Prove that for a pure tripartite state $\rho_{ABE}$, $H(RB) = H(RE)$

Let's say we have a pure tripartite state $\rho_{ABE}$ and a completely positive map $\mathcal{R}$, which is defined as: $$ \mathcal{R} : \rho \rightarrow \sum_j \langle\psi_j|\rho |\psi_j \rangle |\...
QuestionEverything's user avatar
2 votes
1 answer
102 views

Does the relative entropy variance $V(\rho_{AB}\|\rho_A\otimes\sigma_B)$ satisfy an ordering for different $\sigma_B$?

The relative entropy between two quantum states is given by $D(\rho\|\sigma) = \operatorname{Tr}(\rho\log\rho -\rho\log\sigma)$. It is known that for any bipartite state $\rho_{AB}$ with reduced ...
user1936752's user avatar
  • 3,085
2 votes
1 answer
90 views

Is "classical information" the same as "Shannon information"?

does Shannon meet Feynman? Bits underlie classical information measurements in information theory, while qubits underlie quantum information measurements in, what I can only assume to be called, ...
develarist's user avatar
1 vote
1 answer
36 views

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

The quantum conditional min-entropy is defined as $$H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \parallel I_A \otimes \sigma_B \right)$$ where in general $$D_{\max}(\rho \...
Josh's user avatar
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