# 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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### Increasing the von Neumann entropy despite the measurement?

Background Assume we have a density matrix $\rho$ of a sub-ensemble. However, we have an imperfect measuring instrument. While it does perform a measurement, we do not know exactly when it performs ...
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### Convexity of coherent information - erroneous argument!

Consider a state $\rho_{AB}$. Let it have purification $\psi_{A'AB}$. I am interested in the coherent information of this state which is given by $$I(A\rangle B)_\rho = S(B)_\rho - S(AB)_\rho$$ I ...
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### General structure of the state with $I(A:B|C)_{\rho}{=}2 \log_2 \{\min (d_A, d_B)\}$

The conditional quantum mutual information (CQMI) of a state $\rho^{ABC}$ respects the dimension bound $I(A:B|C)_{\rho}{\leq}2 \log_2 \{\min (d_A, d_B)\}$ (Mark Wilde's book, exercise 11.7.9). One ...
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### Can't understand how $D_{A,B} + D_{A,C} \ge E_{A,B}+E_{A,C}$ is proved used the subadditivity of entropy

I am reading Monogamy properties of quantum and classical correlations. Eq.10 states that $$D_{A,B} + D_{A,C} \ge E_{A,B}+E_{A,C},$$ where $D_{i,j}$ is the quantum discord, and $E_{A,B}$ is the ...
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### Quantum analogues of information theoretic measures: are log probabilities replaced with the density matrix?

Below is a question and an answer. How does quantum information relate to, diverge from or reduce to Shannon information, which used log probabilities? What people are more often interested in are ...
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### Conditional Time Evolution increases entropy?

Question Does the below calculation conclusively show the idea of conditional time evolution (if state measured is $x$ I do $y$ else I do $z$ ) increases the Von Neumann entropy? Has this already ...
1 vote
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### Why are all Rényi entropies equal for Clifford dynamics?

In this paper, by Adam Nahum et al., the authors trivially states that "For Clifford dynamics all Rényi entropies are equal ... " which is not trivial to me. Is there a paper or lecture ...
1 vote
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### States for Tight Maassen-Uffink Uncertainty Relation

I was reading this paper titled "Entropic Uncertainty Relations and their Applications". There,at equation (47) we have the Maassen-Uffink uncertainty relation which states that for a pair ...
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I have been study the minimal (maximal) of a $f-$divergence. Fumio Hiai introduced the $\widehat{S}_f (\rho \| \sigma)$ divergence in his article. $$\widehat{S}_f (\rho \| \sigma) := \text{Tr} \sigma^{... 1 vote 0 answers 64 views ### von Neumann entropy of an arbitrary composition system I understand the von Neumann entropy of a 2-composite system is that of the reduced density matrix? What is the von Neumann entropy of an entanglement of a more than 3 composite system? 1 vote 0 answers 16 views ### What is the idea behind compressibility results in terms of Renyi entropies? In (Tomamichel 2015), in (1.2) the author mentions the result that a source X with probability distribution \rho\equiv\rho_X admits an (\varepsilon,m)-code as long as there is some \alpha\in[\... 1 vote 0 answers 42 views ### Understanding conditional L_2 distances I see that conditional L_2 distances from uniform are defined in the following way: L_2(\rho_{AB}\vert \sigma_B)= \text{tr}\left(((\rho_{AB}- \mu_{A} \otimes \rho_{B}) (\mathbb{I}_A \otimes \... 1 vote 0 answers 66 views ### Linear and Logarithmic Constraint in Semidefinite Programming I am trying to minimize the largest component of a vector x = [x_1, x_2, x_3, x_4], where x_1 \ge x_2 ... \ge x_4, such that it satisfies a set of linear inequalities A, b in the following way: ... 1 vote 0 answers 164 views ### Continuity of relative entropy variance Related question here - copying over the definitions. The relative entropy between two quantum states is given by D(\rho\|\sigma) = \operatorname{Tr}(\rho\log\rho -\rho\log\sigma). It is known that ... 0 votes 0 answers 57 views ### Proof of the Lieb's theorem Lemma A6.2: Let R1 , R2 , S1 , S2 , T1, T2 be positive operators such that 0 = [R1, R2 ] = [S1, S 2 ] = [T1, T2 ], and$$ R1 ≥ S1 + T1\\ R2 ≥ S2 + T2 $$Then for all 0 ≤ t ≤ 1,$$ R_1^t R_2^{1−t}...
I have a density matrix of the form: \rho(t)=\left[ \begin{array}{ccc} \frac{1}{4}+\frac{1}{12} e^{-\frac{2 \tau ^{2 H+2}}{2 H+2}} & \frac{1}{3} & \frac{1}{4}+\frac{1}{12} e^{-\frac{2 \tau ^...