Questions tagged [entropy]

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

Filter by
Sorted by
Tagged with
0 votes
1 answer
20 views

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 ...
user avatar
0 votes
1 answer
57 views

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 ...
user avatar
2 votes
2 answers
51 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 ...
user avatar
1 vote
0 answers
30 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 \...
user avatar
  • 449
7 votes
1 answer
536 views

What is "linear" in linear entropy?

Why is the linear entropy, defined by $S_L = 1 - \textrm{Tr} \rho^2$, called linear?
user avatar
  • 371
1 vote
1 answer
54 views

Data Processing equality variation

Let $\rho_{AB}$ be a state and $T: B \rightarrow C$ be a CPTP map with $\sigma_{AC}= T(\rho_{AB})$. It is well known that $H_{\infty}(A \vert B)_{\rho} \geq H_{\infty}(A \vert C)_{\sigma}$ (aka data ...
user avatar
  • 449
5 votes
3 answers
2k views

What is a "maximally mixed state"?

What is meant by maximally mixed states? Does this mean that there are partially mixed states? For example, consider $\rho_{GHZ} = \left| {GHZ} \right\rangle \left\langle {GHZ} \right|$ and $\rho_W =...
user avatar
  • 255
2 votes
1 answer
63 views

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 ...
user avatar
  • 496
2 votes
1 answer
46 views

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 ...
user avatar
  • 165
4 votes
1 answer
79 views

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 ...
user avatar
3 votes
0 answers
28 views

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 \...
user avatar
  • 830
3 votes
1 answer
129 views

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 ...
user avatar
4 votes
1 answer
130 views

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, ...
user avatar
3 votes
0 answers
21 views

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. ...
user avatar
  • 2,109
3 votes
1 answer
35 views

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. ...
user avatar
  • 2,109
2 votes
0 answers
211 views

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 ...
user avatar
  • 18.8k
1 vote
0 answers
38 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: $...
user avatar
0 votes
0 answers
101 views

Coherence measurement for density matrix

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 ^...
user avatar
2 votes
1 answer
30 views

Showing that $S(\rho_{XB}||\sigma_{XB})=\sum_{x}p(x)D(\rho_{B}^{x}||\sigma_{B}^{x})$ for classical-quantum states

Having some trouble showing that $S(\rho_{XB}||\sigma_{XB})=\sum_{x}p(x)D(\rho_{B}^{x}||\sigma_{B}^{x})$ for $\rho_{XB}=\sum_{x}p(x)|x\rangle\langle x|\otimes\rho_{B}^{x}$ and $\sigma_{XB}=\sum_{x}p(x)...
user avatar
  • 1,123
2 votes
1 answer
57 views

What can be said about the non-negativity of the relative entropy of $S(\rho_{AB}||\rho_{B})$?

Taking $\rho_{AB}=\rho_{A}\otimes \rho_{B}$, where $S(\rho_{A})$ and $S(\rho_{B})$ aren't 0, it's easy to see that $$S(\rho_{AB}||I \otimes \rho_{B})=-S(\rho_{A})-S(\rho_{B})+S(\rho_{B})=-S(\rho_{A}).$...
user avatar
  • 1,123
4 votes
1 answer
112 views

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 ...
user avatar
2 votes
2 answers
250 views

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 ...
user avatar
  • 515
4 votes
1 answer
139 views

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 ...
user avatar
  • 103
1 vote
1 answer
35 views

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 ...
user avatar
2 votes
1 answer
85 views

Prove the subadditivity for the von Neumann entropy of a bipartite state

I want to prove the subadditivity relation $S(\rho_{AB})\le S(\rho_A)+S(\rho_B)$ for the Von Neumann entropy. The tip is to use the Klein inequality $S(\rho_{AB}\Vert \rho_A\otimes \rho_B)\ge 0$: $$S(\...
user avatar
2 votes
0 answers
162 views

How to prove that the mutual information is subadditive?

Let $\mathbf x=(x_1,...,x_n)$ and $\mathbf y=(y_1,...,y_n)$ be two vectors of random variables. To make things concrete, assume that Alice sends each component $x_j$ through a noisy channel to Bob, ...
user avatar
2 votes
1 answer
57 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 |\...
user avatar
2 votes
1 answer
101 views

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....
user avatar
3 votes
1 answer
111 views

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?
user avatar
5 votes
2 answers
89 views

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 ...
user avatar
2 votes
1 answer
130 views

Understanding the definition of entropy in the joint entropy theorem derivation

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|...
user avatar
  • 21
2 votes
1 answer
58 views

Computing $H(Z|B)$ in a bipartite density matrix $\rho_{AB}$

Let's say Bob prepares a bipartite quantum state $\rho_{AB}$ to be shared between him and Alice. Bob sends Alice's part to her lab. Alice measures her subsystem $A$ in the computational basis $\...
user avatar
3 votes
0 answers
84 views

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 ...
user avatar
  • 125
3 votes
2 answers
96 views

How is the additivity of accessible information, $\frac{1}{n} I_{\rm acc}(\rho^{\otimes n})=I_{\rm acc}(\rho)$, proved?

Let $\rho^{XA}$ be a classical-quantum state of the form $$ \rho^{XA} = \sum_{x\in X} p_x |x\rangle\langle x|\otimes \rho_x^A, $$ and let the accessible information be given by $$ I_{acc}(\rho^{XA}) = ...
user avatar
  • 113
2 votes
0 answers
55 views

Partial trace instead of trace in definition of entropy

For a bipartite quantum state $\rho_{AB}$, we have that the von Neumann entropy is $$S(\rho_{AB}) = -\text{Tr}(\rho_{AB}\log\rho_{AB})$$ If instead, one took the partial trace above and obtained $$\...
user avatar
  • 2,109
2 votes
1 answer
60 views

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-...
user avatar
  • 395
1 vote
0 answers
111 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 ...
user avatar
  • 2,109
2 votes
1 answer
69 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 ...
user avatar
  • 2,109
3 votes
2 answers
380 views

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 ...
user avatar
  • 155
3 votes
1 answer
65 views

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 ...
user avatar
  • 2,109
1 vote
1 answer
294 views

Calculate the von Neumann Entropy of a two-qubit entangled state

After working through an exercise I got a confusion answer/solution that either may be because I've made a mistake or I'm not understanding von Neumann Entropy. I have the two qubit system $$ | \psi \...
user avatar
2 votes
2 answers
142 views

How do Rényi entropies act under unitary time evolution?

I am trying to find information/ help on Rényi entropies given by $$ S_n(\rho) = \frac{1}{1-n} \ln [Tr(\rho^n)] $$ and how it acts under unitary time evolution? Is the entropy independent on the state ...
user avatar
4 votes
0 answers
118 views

Calculating the entropy of a quantum state

Let $\rho_{AR}$ be some $d-$dimensional pure quantum state. Consider a channel $N_{A\rightarrow B}$ that outputs a constant state in $B$. We now consider the Stinespring dilation of this channel with ...
user avatar
  • 2,109
5 votes
1 answer
180 views

Prove that the conditional entropy of a classical-quantum state is non-negative

Let $\rho_{XA}$ be a classical-quantum state, i.e., $\rho_{XA} = \sum_{x} p(x) |x\rangle \langle x| \otimes \rho_A^x$. How to prove that the conditional von Neumann entropy $S(X|A) = S(\rho_{XA}) - ...
user avatar
3 votes
1 answer
53 views

Do we know the limits of the quantum Tsallis entropy?

From the two main generalizations of the von Neumann entropy: \begin{equation} S(\rho)=-\operatorname{Tr}(\rho \log \rho) \end{equation} meaning Rényi: \begin{equation} R_{\alpha}(\rho)=\frac{1}{1-\...
user avatar
  • 73
2 votes
1 answer
120 views

Prove the additivity of the Renyi entropy: $H_{\beta}(p \times r) = H_{\beta}(p) + H_{\beta}(r)$

The Renyi entropy of order $\beta$, for a discrete probability distribution $p$ is given by \begin{equation} H_{\beta}(p) = \frac{1}{1 - \beta} ~\log \left( \sum_{i \in S} p(i)^{\beta} \right), \end{...
user avatar
3 votes
1 answer
178 views

Positive conditional quantum entropy for entangled state

The quantum conditional entropy $S(A|B)\equiv S(AB)-S(A)$, where $S(AB)=S(\rho_{\rm AB})$ and $S(B)=S(\rho_{\rm B})$ is known to be non-negative for separable states. For entangled states, it is known ...
user avatar
1 vote
1 answer
59 views

Prove that for a general tri-partite state $\rho_{ABE}$, $H(AB) = H(E)$

How do I prove that for a general tri-partite state $\rho_{ABE}$, the following holds: $$ H(\rho_{AB}) = H(\rho_{E}), H(\rho_{AE}) = H(\rho_{B}), $$ where, $H$ is the Von Neumann entropy. Would ...
user avatar
5 votes
1 answer
65 views

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 ...
user avatar
2 votes
0 answers
44 views

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 ...
user avatar