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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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Why does not Shannon Chain Rule work in Quantum Mechanics

Shannon Entropy conditional variant is H(A/B)=H(AB)-H(B). Its chain rule implies: H(A/BC)=H(AB/C)-H(B/C). The Quantum ...
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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 ...
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What are examples of states saturating the strong subadditivity of the von Neumann entropy?

A well-known property of classical distribution is that they satisfy strong subadditivity, meaning that for any tripartite joint probability distribution $p(x,y,z)$, we have the inequality $$H(AB)+H(...
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On the use of $\log(P\otimes Q)= \log P\otimes I+I\otimes\log Q$ for relations between entropic quantities. What if $P,Q$ are only semidefinite?

Many properties of entropic quantities are shown by resorting to related properties of the relative entropy of suitable quantities. For instance, subadditivity of entropy may follow from non ...
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Maximum entanglement entropy of a random circuit

Consider the random quantum circuit below where the gates are randomly taken from SU(4) accordingly with the Haar measure. I am looking to determine an upper bound on the entanglement entropy between ...
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Exercise 11.7 in Nielsen & Chuang and basic properties of Shannon entropy

I apologize in advance if this question is trivial, I'm aware I'm a total beginner in this field. This is the exercise I would like to solve: As to the first point, what I get is that one should ...
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Is entanglement trainable?

There exists a famous result from Google that the gradients of the parameters of quantum neural networks (QNN) vanish exponentially with the number of qubits in the quantum circuit. Their result ...
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What is the classical cost of simulating an arbitrary quantum state?

The past couple of years has seen various groups claim quantum advantage/utility only to have their experiments efficiently simulated with classical methods, notably using tensor networks. My question ...
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Minimizing $1 - \text{Tr}(\Phi(\rho,U)^2)$

I am looking for a computationally efficient way to minimize the following function. Let $$\Phi(\rho, U) = \text{Tr}_2(U\rho U^\dagger)$$ be a reduced density matrix where $\rho = \overline{\rho}_1 \...
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Mutual information between Alice and Eve in a BB84 intercept resend attack

I'm new to information theory and i need to calculate $I(A,E)$. To calculate it I need conditional entropy $H(A|E)$. I assume the BB84 protocol standard states $\{ |0\rangle,|1\rangle \},\{|+\rangle,|-...
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Can post-measurement states have entropy larger than the original state?

Given a set of measurement operators $\{M_i\}$ that sum to unity, consider the post-measurement states on some $\rho$ as $\rho_i:=(\sqrt{M_i}\rho\sqrt{M_i})/p_i$ and $p_i:=\mathrm{Tr}(M_i\rho)$. It's ...
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Entropic uncertainty relations with measurements and memory [duplicate]

In entropic uncertainty relations involving measurements and memory, one has a quantum state $\rho_{AB}$. Alice holds register $A$ and performs one of two measurements denoted by observables $R$ and $...
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Does quantum mutual information encompass information only about quantum correlations, or does it encompass both classical and quantum correlations?

I am confused about what quantum mutual information gives us. Does it give all kinds of quantum correlations? Or does it give all kinds of quantum and classical correlations? If it consists of ...
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Is there a known deterministic counterexample for non-additivity of minimal output entropy?

Hastings has proved that the minimal output entropy is not additive: it may happen that $S_{\mathrm{min}}(\Phi_1 \otimes \Phi_2) < S_{\mathrm{min}}(\Phi_1)+S_{\mathrm{min}}(\Phi_2) $ for quantum ...
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Upper bound on entanglement entropy of a Product State for any possible partition of the Joint System

Let $|\psi\rangle$ be an $n$ qubit quantum state on a line with Von Neumann entanglement entropy at most $r$ with respect to any bipartition of the qubits (does not have to be a contiguous bipartition)...
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Is quantum mutual information an entanglement measure?

For a bipartite system, the quantum mutual information is defined via the Von Neumann entropy as follows: $$I(A:B)=S(A)+S(B)-S(AB).$$ It's always positive. Is it an entanglement measurement? Also how ...
reza's user avatar
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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 ...
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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 ...
QuestionEverything's user avatar
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Quantum relative entropy between pre- and post-measurement states

The quantum relative entropy between the states $\rho$ and $\sigma$ is defined by $$D(\rho||\sigma)= \textrm{tr}\Big(\rho \big(\log\rho - \log \sigma \big) \Big)\,,$$ as long as the support of $\rho$ ...
quantum_theo's user avatar
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Is Klein's inequality due to Klein?

You may be familiar with "Klein's inequality"; one form of it is $$ -\operatorname{tr}(\rho \log \sigma) + \operatorname{tr}(\rho \log \rho) \ge 0, $$ stating that relative entropy is ...
echinodermata's user avatar
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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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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}...
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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 Nielsen and Chuang's Box 11.2: Continuity of the entropy, in the process of proving the Fannes' inequality, it says: A moment’s thought shows that $\left|r_i − s_i\right| \le 1/2$ for all i, The ...
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What is the conditional min-entropy of a pure bipartite state?

In this paper, it is stated that the conditional min-entropy $H(A|B)_{\rho_{AB}}$ of $A$ conditioned on $B$ for any $\textbf{pure}$ quantum system $\rho_{AB}=|\psi_{AB} \rangle \langle \psi_{AB} |$ is ...
quantum_theo's user avatar
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Derive the Concavity of Quantum Conditional Entropy from Strong subadditivity

In Exercise 11.25, Page 522, Entropy and information, Quantum Computation and Quantum Information by Nielsen and Chuang, it is required to show that the concavity of the conditional entropy may be ...
Sooraj S's user avatar
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Find the minimal and maximal of $\widehat{S}_f (\rho \| U^* \sigma U)$

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^{...
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Quantum Relative entropy- the math and intuition

I am new to quantum information theory and have been reading Mark Wilde's notes on quantum relative entropy. http://www.markwilde.com/teaching/2015-fall-qit/lectures/lecture-19.pdf I have three basic ...
Newuser7's user avatar
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Does the quantum relative entropy have a direct operational interpretation?

Consider the quantum relative entropy, defined as $$D(\rho\|\sigma) = \operatorname{tr}(\rho\log\rho)-\operatorname{tr}(\rho\log\sigma),$$ for all $\rho,\sigma\ge0$ such that $\operatorname{im}(\rho)\...
glS's user avatar
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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 \...
Josh's user avatar
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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 \...
Josh's user avatar
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Prove that the conditional min-entropy is $H_{\rm min}(A|B)=\max_\sigma\sup\{\lambda:\,\rho\le 2^{-\lambda}(I\otimes\sigma)\}$

I have seen various definitions of quantum conditional min-entropy, which I believe are equivalent. $$H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \parallel I_A \otimes \sigma_B ...
Josh's user avatar
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Prove the equality conditions in the triangle inequality $S(A,B)\ge |S(A)-S(B)|$ for the von Neumann entropy

The triangle inequality or Araki-Lieb inequality of the von Neumann entropy is $$ S(A,B)\ge|S(A)-S(B)| $$ this is proven by introducing a system $R$ which purifies systems $A$ and $B$. Applying ...
Sooraj S's user avatar
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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, ...
Sooraj S's user avatar
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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(...
Sooraj S's user avatar
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Conditional entropy as relative entropy between probability distributions

Find the expression for the conditional entropy $H(Y|X)$ as a relative entropy between two probability distributions. Use this expression to deduce that $H(Y |X)≥0$, and to find the equality ...
Sooraj S's user avatar
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How is the von Neumann entropy of a state defined from its eigendecomposition?

The definition of the von Neumann entropy of a mixed state says that it can be calculated as the Shannon entropy of coefficients of the decomposition of the state into a sum of projectors. My question:...
eternalstudent's user avatar
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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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Qualitative meaning of the bond dimension of a matrix product state

Consider a matrix product state (MPS) with a bond dimension $D$. What is the physical intuition behind the bond dimension? Is it, in any way, related to the spatial geometry? In this note, it is ...
BlackHat18's user avatar
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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?
Hans's user avatar
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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 ...
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Difference between min/max-entropies and the von Neumann entropy

Consider the (smooth) min-entropy, max-entropy and von Neumann entropy of a given density operator $\rho_A$. Does a small gap between $H_{\max(\min)}(A)_\rho$ and $H(A)_\rho$ implies a small gap ...
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In what sense is the "conditional min-entropy" a conditional entropy?

$\newcommand{\H}{\mathsf{H}}\newcommand{\Hmin}{\H_{\rm min}}\newcommand{\D}{\mathsf{D}}\newcommand{\Dmax}{\D_{\rm max}}$Consider the conditional min-entropy $\Hmin(A|B)_\rho$, discussed e.g. in this ...
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Why are "smooth entropic quantities" useful/necessary?

Consider the $\epsilon$-smoothed relative max-entropy of $\rho$ with respect to $Q$, defined as (following Watrous' notation from these notes): $$\mathrm D_{\rm max}^{\epsilon}(\rho\|Q) = \min_{\xi\in ...
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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[\...
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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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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 \...
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Entanglement entropy for graph states defined on a tree graph

Consider a $k-\text{ary}$ tree $T$, for a constant $k$. Consider the corresponding graph state $|\mathsf{G}_T \rangle$ that is defined on $T$. Is it true that $|\mathsf{G}_T \rangle$ saturates the ...
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How to take Statevector for subsystem?

I want to calculate the 2nd Renyi entropy using the density matrix in Qiskit. To do this, I need to calculate the $Tr(\rho^2)$ for subsystem. The complete system consists of 12 qubits from which I ...
VladislavOkatev's user avatar
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2 answers
291 views

Derivation of the linear cross entropy

I'm looking at cross-entropy benchmarks and there's much that I'm reading at the moment but I'm stuck on one detail: how to derive the linear cross-entropy formula from the cross-entropy formula. The ...
James Whitfield's user avatar
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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 \...
Root's user avatar
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