# 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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### 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 ...
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### 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 ...
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### Why von Neumann entropy requires diagonalization and linear entropy doesn't?

The linear entropy for a state $\rho$ is defined as $S_L = 1 - Tr[\rho^2]$, while as von Neumann entropy as $S_{N} = -Tr[\rho \ln \rho]$. According to quantiki, the computation of $S_{N}$ requires ...
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### If information is increase in entropy, why does large entropy mean little information?

Excuse me since this is an elementary question in information theory. I am asking this question here since the statement "large entropy means little information" is mentioned in the first ...
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### Data processing inequality for relative entropy in the presence of an amplitude damping channel

Consider the single qubit quantum depolarizing channel, given by $$T(\rho) = (1- p)\rho + p \frac{\mathbb{I}}{2}.$$ For an $n$ qubit state $\rho$, according to Definition 6.1 of this paper, the ...
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### Proof of upper and lower bound (Gilbert-Varshamov bound) for linear code

I am trying to prove the following bounds for a $[n, k]$ code that can correct $t$ errors \begin{align} 1-H\left(\frac{t}{n}\right)\geq \frac{k}{n}\geq 1-H\left(\frac{2t}{n}\right) \end{align} where \...
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### Entanglement entropy and depth

I wanted to verify two intuitions about the entanglement entropy of quantum states. Consider an $n$ qubit quantum state, prepared by a depth $d$ circuit acting on $|0\rangle^{\otimes n}$ and a ...
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### Does entanglement entropy follow a volume or an area law for 2D cluster states?

Consider a 2D cluster state defined on a rectangular lattice, which is universal for one way quantum computers. For a description of the state, see for example question 2 in this problem set. Now, ...
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### Max-relative entropy quasi-convexity inequality under partial trace

The max-relative entropy between two states is defined as $$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\}.$$ It is known that the max-relative entropy is quasi-convex. ...
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### Quasi concavity of max-relative entropy?

The max-relative entropy between two states is defined as $$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\}.$$ It is known that the max-relative entropy is quasi-convex. ...
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### What does a quantum mutual information larger than its classical upper bound represent?

Let $\rho$ be a bipartite state. Its quantum mutual information is defined as $$\newcommand{\tr}{\operatorname{tr}}I(\rho) = S(\tr_B(\rho)) + S(\tr_A(\rho)) - S(\rho),$$ where $S(\sigma)$ is the von ...
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### Concavity of Conditional Quantum Entropy

Let's say I have a bipartite density operator $\gamma_{12} = (1 - \epsilon) \rho_{12} + \epsilon\sigma_{12}$, for $0 \le \epsilon \le 1$, i.e., a convex combination of $\rho_{12}$ and $\sigma_{12}$. I ...
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### How to understand intuitively the concavity of the binary entropy?

In Nielsen and Chuang's Quantum Computation and Quantum Information book, introducing the binary entropy, they gave an intuitive example about why binary entropy is concave: Alice has in her ...
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### How to prove that $\frac{| x_0 \rangle + | x_1 \rangle}{\sqrt{2}}$ hides one of $x_0$ or $x_1$?

I create a quantum state $| \psi \rangle = \frac{| x_0 \rangle + | x_1 \rangle}{\sqrt{2}}$ for a randomly chosen $x_0,x_1$ of 50 bits. I give this quantum state $|\psi \rangle$ to you and you return ...
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### Increasing entropy for projective LCPT mapping

Given a set of projectors $\{P_i\}$ acting on a space $\mathcal H_S$, let $\Phi$ be the LCPT map defined by $$\Phi(\rho)=\sum_i P_i\rho P_i.$$ The goal is to show that $S(\Phi(\rho))\ge S(\rho)$. The ...
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### Prove that for a cq-state $\rho_{XE}$, $H_\infty(X|E) \ge H_\infty(X) - \log|E|$

Given a classical-quantum(cq) state $\rho_{XE}$, where the $X$ register is classical, I want to prove the following: \begin{align} H_\infty(X|E) \ge H_\infty(X) - \log|E|,\tag{1} \end{align} i.e....
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### How can the entropy of quantum states increase after projective measurements?

I'm reading Nielsen and chuang 11.3.3 Measurements and Entropy. It says after measurement, one's entropy increases. How is this possible? Shouldn't measurement decrease one's uncertainty?
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### How to prove the positivity of the conditional quantum mutual information, $I(A;B|C)\ge0$?

I was reading Wilde's 'Quantum Information Theory' and saw the following theorem at chapter 11 $(11.7.2)$: $$I(A; B | C) \ge 0,$$ where, $$I(A;B|C) := H(A|C) + H(B | C) - H(AB|C).$$ I know that ...
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### Why does the entanglement entropy give the number of singlets required to create a given state?

I've read that, given a bipartite pure state $|\Phi\rangle$, its entanglement (equivalently here, von Neumann) entropy $E(\Phi)$ gives the asymptotic number of singlets required to create $n$ copies ...
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### Is data processing for relative entropy true when states are not normalized?

The data processing inequality for relative entropy states that $$D(\rho\|\sigma) \geq D(N(\rho)\|N(\sigma))$$ for some CPTP map $N$ where $\rho$ is a quantum state and $\sigma$ is a positive-...
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### 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 ...
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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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### Measuring entanglement entropy using a stabilizer circuit simulator

I'm trying to simulate stabilizer circuits using the Clifford tableau formalism that lets you scale up to hundreds of qubits. What I want to do is find the entanglement entropy on by splitting my ...
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### Continuity of Renyi entropies - limiting cases

The Renyi entropies are defined as $$S_{\alpha}(\rho)=\frac{1}{1-\alpha} \log \operatorname{Tr}\left(\rho^{\alpha}\right), \alpha \in(0,1) \cup(1, \infty)$$ It is claimed that this quantity is ...
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### Does the quantum Jensen-Shannon divergence appear in any quantum algorithms or texts on quantum computing?

The generalization of probability distributions on density matrices allows to define quantum Jensen–Shannon divergence (QJSD), which uses von Neumann entropy. Does QJSD appear in any quantum ...
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