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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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 $\...
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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 ...
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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, ...
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Quantum relative entropy with respect to 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 ...
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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}{...
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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 ...
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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 ...
user22003
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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:
$$S(\rho_A || \sigma_A) \leq S(\rho_{AB} || ...
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What is entropy quantum computing?
Recently there is news concerning some computational breakthroughs by using so-called "entropy quantum computing":
https://thequantuminsider.com/2022/07/20/qci-solves-3854-variable-problem-...
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Shannon entropy is least when Measurement basis = Mixture basis
For a one qubit system, take a basis.
Call this the mixture basis.
Consider only basis states and classical mixtures of these basis states.
Definition of Shannon Entropy used here: Defined with ...
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In the proof of the joint entropy theorem, why are $p_i\lambda_i^j$ the eigenvalues?
From section 11.3.2 of Nielsen & Chuang:
(4) let $\lambda_i^j$ and $\left|e_i^j\right>$ be the eigenvalues and corresponding eigenvectors of $\rho_i$. Observe that $p_i\lambda_i^j$ and $\left|...
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Questions about the relation between max-relative entropy $D_{\max}(\rho||\sigma)$ and max-information
The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$
where $\rho\leq \sigma$ should be read as $\sigma - \rho$ is ...
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Why can the max-relative entropy be written as $D_{\max}(\rho \parallel \sigma) = \inf \{ \lambda : \rho \leq 2^\lambda \sigma \}$?
The quantum conditional min-entropy is defined as
$$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 \...
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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 $$|S(\rho)-S(\sigma)|\leq T(...
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What is the conditional min-entropy for diagonal ("classical") matrices?
The conditional min-entropy, discussed e.g. in these notes by Watrous, as well as in this other post, can be defined as
$$\mathsf{H}_{\rm min }(\mathsf{X} \mid \mathsf{Y})_{\rho}\equiv -\inf _{\sigma \...
glS♦
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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 ...
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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, ...
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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 \...
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