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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73 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 ^...
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1answer
27 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)...
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1answer
49 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}).$...
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56 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 ...
2
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2answers
196 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 ...
4
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1answer
135 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 ...
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1answer
32 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 ...
2
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1answer
68 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(\...
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88 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, ...
2
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1answer
49 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 |\...
2
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1answer
82 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....
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1answer
83 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?
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69 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 ...
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1answer
124 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|...
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1answer
55 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 $\...
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70 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 ...
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2answers
92 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}) = ...
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41 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 $$\...
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1answer
38 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-...
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96 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 ...
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1answer
63 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 ...
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2answers
179 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 ...
3
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1answer
41 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 ...
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1answer
189 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 \...
2
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2answers
135 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 ...
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113 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 ...
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1answer
134 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}) - ...
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1answer
42 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-\...
2
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1answer
93 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{...
3
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1answer
100 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 ...
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1answer
58 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 ...
5
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1answer
58 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 ...
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42 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 ...
2
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1answer
68 views

Relating quantum max-relative entropy to classical maximum entropy

The quantum 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 - \...
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1answer
52 views

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, ...
2
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1answer
82 views

von Neumann entropy in a limiting case

I am stuck with a question from the book Quantum theory by Asher Peres. Excercise (9.11): Three different preparation procedures of a spin 1/2 particle are represented by the vectors $\begin{pmatrix} ...
2
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1answer
33 views

What is the relationship between these two definitions for the max-entropy?

On Wikipedia, the max-entropy for classical systems is defined as $$H_{0}(A)_{\rho}=\log \operatorname{rank}\left(\rho_{A}\right)$$ The term max-entropy in quantum information is reserved for the ...
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2answers
169 views

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 ...
5
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0answers
31 views

Relative entropy inequality for many copies of a channel

Suppose we have two quantum channels $\mathcal{E}_{A\rightarrow B}, \mathcal{F}_{A\rightarrow B}$ that satisfy $$D(\mathcal{E}(\rho_A)\|\mathcal{E}(\sigma_A))\geq D(\mathcal{F}(\rho_A)\|\mathcal{F}(\...
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0answers
251 views

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 ...
4
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1answer
68 views

Non-lockability of quantum max-entropy

Lockability and non-lockability are explained in this paper. A real valued function of a quantum state is called non-lockable if its value does not change by too much after discarding a subsystem. The ...
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1answer
47 views

Semi-definite program for conditional smooth max-entropy

I am aware of a SDP formulation for smooth min-entropy: question link. That program for smooth min-entropy was found in this book by Tomachiel: page 91. However, I am yet to come across a semi-...
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3answers
177 views

Prove that Shannon and von Neumann entropies satisfy $H(P)\ge S(\rho)$ with $P$ diagonal of $\rho$

Suppose there is some $n$-qubit state $\rho$. It is well known fact that, given some orthonormal basis $U = \{|u_i\rangle\}$, if $p_i = \langle u_i| \rho |u_i \rangle$ (that is, measuring $\rho$ with $...
5
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0answers
99 views

Connection between smooth max-relative entropy and smooth 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 ...
3
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1answer
75 views

How can one estimate the von Neumann entropy of an unknown quantum state?

Given many copies of some unknown quantum state $\rho$, I would like to compute its von Neumann entropy $S(\rho)$. What algorithm could be used for this that minimizes the number of copies required? ...
2
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1answer
75 views

How is $S(\rho)=H(p_{i})+\sum_{i}p_{i}S(\rho_{i})\le \log(d)$ possible if $\rho_{i}$ are not pure states?

I know how this can be proved using the quantum relative entropy. However, even with this proof, and am still confused about how this emerges. Say I have a source that produces two states $\rho_1$ and ...
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1answer
66 views

Proof of quantum data processing inequality in N&C on pg 566

On page 566, it states that using $S(\rho^{'})-S(\rho,\varepsilon) \ge S(\rho)$ and combining this with $S(\rho) \ge S(\rho^{'})-S(\rho,\varepsilon))$, we get $S(\rho^{'})=S(\rho)-S(\rho,\varepsilon)$....
5
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1answer
235 views

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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1answer
83 views

How well defined is $\log(P)$ for $P$ projection?

Whenever we calculate entropy we make use, for example, $\log(P)$ for $P$ a projection defined for some arbitrary finite dimensional Hilbert space. But for projection operators this is not well-...
5
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1answer
91 views

Is the quantum min-relative entropy $D_{\min}(\rho\|\sigma)=-\log(F(\rho, \sigma)^2)$ or $D_{\min}(\rho\|\sigma)=-\log(tr(\Pi_\rho\sigma))$?

In John Watrous' lectures, he defines the quantum min-relative entropy as $$D_{\min}(\rho\|\sigma) = -\log(F(\rho, \sigma)^2),$$ where $F(\rho,\sigma) = tr(\sqrt{\rho\sigma})$. Here, I use this ...