Questions tagged [entropy]
For questions about the various kinds of entropies --- as defined in the context of quantum information theory and quantum statistical mechanics.
141
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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:
$...
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0
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146
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Coherence measurement for density matrix
I have a density matrix of the form:
$$\rho(t)=\left[
\begin{array}{ccc}
\frac{1}{4}+\frac{1}{12} e^{-\frac{2 \tau ^{2 H+2}}{2 H+2}} & \frac{1}{3} & \frac{1}{4}+\frac{1}{12} e^{-\frac{2 \tau ^...
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Showing that $S(\rho_{XB}||\sigma_{XB})=\sum_{x}p(x)D(\rho_{B}^{x}||\sigma_{B}^{x})$ for classical-quantum states
Having some trouble showing that $S(\rho_{XB}||\sigma_{XB})=\sum_{x}p(x)D(\rho_{B}^{x}||\sigma_{B}^{x})$ for $\rho_{XB}=\sum_{x}p(x)|x\rangle\langle x|\otimes\rho_{B}^{x}$ and $\sigma_{XB}=\sum_{x}p(x)...
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What can be said about the non-negativity of the relative entropy of $S(\rho_{AB}||\rho_{B})$?
Taking $\rho_{AB}=\rho_{A}\otimes \rho_{B}$, where $S(\rho_{A})$ and $S(\rho_{B})$ aren't 0, it's easy to see that
$$S(\rho_{AB}||I \otimes \rho_{B})=-S(\rho_{A})-S(\rho_{B})+S(\rho_{B})=-S(\rho_{A}).$...
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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 ...
- 103
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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 ...
- 165
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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 $S(\rho_{AB}\Vert \rho_A\otimes \rho_B)\ge 0$: $$S(\...
- 165
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How to prove that the mutual information is subadditive?
Let $\mathbf x=(x_1,...,x_n)$ and $\mathbf y=(y_1,...,y_n)$ be two vectors of random variables. To make things concrete, assume that Alice sends each component $x_j$ through a noisy channel to Bob, ...
- 165
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Prove that for a pure tripartite state $\rho_{ABE}$, $H(RB) = H(RE)$
Let's say we have a pure tripartite state $\rho_{ABE}$ and a completely positive map $\mathcal{R}$, which is defined as:
$$
\mathcal{R} : \rho \rightarrow \sum_j \langle\psi_j|\rho |\psi_j \rangle |\...
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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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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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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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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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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
$$ I_{acc}(\rho^{XA}) = ...
- 123
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Partial trace instead of trace in definition of entropy
For a bipartite quantum state $\rho_{AB}$, we have that the von Neumann entropy is
$$S(\rho_{AB}) = -\text{Tr}(\rho_{AB}\log\rho_{AB})$$
If instead, one took the partial trace above and obtained
$$\...
- 2,597
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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 ...
- 2,597
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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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139
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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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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
$$ | \psi \...
- 39
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2
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How do Rényi entropies act under unitary time evolution?
I am trying to find information/ help on Rényi entropies given by
$$ S_n(\rho) = \frac{1}{1-n} \ln [Tr(\rho^n)] $$
and how it acts under unitary time evolution? Is the entropy independent on the state ...
- 39
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Calculating the entropy of a quantum state
Let $\rho_{AR}$ be some $d-$dimensional pure quantum state. Consider a channel $N_{A\rightarrow B}$ that outputs a constant state in $B$. We now consider the Stinespring dilation of this channel with ...
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Prove that the conditional entropy of a classical-quantum state is non-negative
Let $\rho_{XA}$ be a classical-quantum state, i.e., $\rho_{XA} = \sum_{x} p(x) |x\rangle \langle x| \otimes \rho_A^x$.
How to prove that the conditional von Neumann entropy $S(X|A) = S(\rho_{XA}) - ...
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Do we know the limits of the quantum Tsallis entropy?
From the two main generalizations of the von Neumann entropy:
\begin{equation}
S(\rho)=-\operatorname{Tr}(\rho \log \rho)
\end{equation}
meaning Rényi:
\begin{equation}
R_{\alpha}(\rho)=\frac{1}{1-\...
- 73
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Prove the additivity of the Renyi entropy: $H_{\beta}(p \times r) = H_{\beta}(p) + H_{\beta}(r)$
The Renyi entropy of order $\beta$, for a discrete probability distribution $p$ is given by
\begin{equation}
H_{\beta}(p) = \frac{1}{1 - \beta} ~\log \left( \sum_{i \in S} p(i)^{\beta} \right),
\end{...
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Positive conditional quantum entropy for entangled state
The quantum conditional entropy $S(A|B)\equiv S(AB)-S(A)$, where $S(AB)=S(\rho_{\rm AB})$ and $S(B)=S(\rho_{\rm B})$ is known to be non-negative for separable states. For entangled states, it is known ...
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Prove that for a general tri-partite state $\rho_{ABE}$, $H(AB) = H(E)$
How do I prove that for a general tri-partite state $\rho_{ABE}$, the following holds:
$$
H(\rho_{AB}) = H(\rho_{E}), H(\rho_{AE}) = H(\rho_{B}),
$$
where, $H$ is the Von Neumann entropy. Would ...
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Does the quantum Jensen-Shannon divergence appear in any quantum algorithms or texts on quantum computing?
The generalization of probability distributions on density matrices allows to define quantum Jensen–Shannon divergence (QJSD), which uses von Neumann entropy. Does QJSD appear in any quantum ...
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Quantum analogues of information theoretic measures: are log probabilities replaced with the density matrix?
Below is a question and an answer.
How does quantum information relate to, diverge from or reduce to
Shannon information, which used log probabilities?
What people are more often interested in are ...
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Relating quantum max-relative entropy to classical maximum entropy
The quantum max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$
where $\rho\leq \sigma$ should be read as $\sigma - \...
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Is "classical information" the same as "Shannon information"?
does Shannon meet Feynman?
Bits underlie classical information measurements in information theory, while qubits underlie quantum information measurements in, what I can only assume to be called, ...
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von Neumann entropy in a limiting case
I am stuck with a question from the book Quantum theory by Asher Peres.
Excercise (9.11):
Three different preparation procedures of a spin 1/2 particle are represented by the vectors $\begin{pmatrix}
...
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What is the relationship between these two definitions for the max-entropy?
On Wikipedia, the max-entropy for classical systems is defined as
$$H_{0}(A)_{\rho}=\log \operatorname{rank}\left(\rho_{A}\right)$$
The term max-entropy in quantum information is reserved for the ...
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Quantum relative entropy with respect to a pure state
I want to evalualte the quantum relative entropy $S(\rho|| \sigma)=-{\rm tr}(\rho {\rm log}(\sigma))-S(\rho)$, where $\sigma=|\Psi\rangle\langle\Psi|$ is a density matrix corresponding to a pure state ...
- 147
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Relative entropy inequality for many copies of a channel
Suppose we have two quantum channels $\mathcal{E}_{A\rightarrow B}, \mathcal{F}_{A\rightarrow B}$ that satisfy
$$D(\mathcal{E}(\rho_A)\|\mathcal{E}(\sigma_A))\geq D(\mathcal{F}(\rho_A)\|\mathcal{F}(\...
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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 ...
- 429
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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 ...
- 2,597
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124
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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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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 $...
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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 ...
- 2,597
3
votes
1
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206
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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? ...
- 33
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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 ...
- 1,115
1
vote
1
answer
124
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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 $S(\rho^{'})=S(\rho)-S(\rho,\varepsilon)$....
- 1,115
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votes
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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 ...
- 1,115
2
votes
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114
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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-...
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