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Proof of an Holevo information inequality for a classical-classical-quantum channel

It appears that the statement is not true in general. Suppose $X = Y = \{0,1\}$, $\mathcal{H}$ is the Hilbert space corresponding to a single qubit, and $W$ is defined as \begin{align} W(0,0) & = |...
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10 votes
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Is the set of all states with negative conditional Von Neumann entropy convex?

The conditional von Neumann entropy is a concave function: if $\rho$ and $\sigma$ are states of a pair of registers $(\mathsf{X},\mathsf{Y})$ and $\lambda\in[0,1]$ is a real number, then $$ \mathrm{H}(...
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10 votes
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What is a "maximally mixed state"?

The maximally mixed state is a quantum state whose density matrix is proportional to the identity matrix. Physically, it may be interpreted as a uniform mixture of states in an orthonormal basis. The ...
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9 votes

Is the set of all states with negative conditional Von Neumann entropy convex?

Geometric characterization (as any other characterization) of subsets of the quantum state space in relation with their locality and entanglement properties becomes very complicated as the number of ...
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9 votes

Maximally mixed states for more than 1 qubit

For two probability distributions, there is a clear notion how to say which one is more mixed: $\vec p$ is more mixed than $\vec q$ if it can be obtained from $\vec p$ by a mixing process, this is, a ...
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7 votes
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Degradable channels and their quantum capacity

A channel $\Phi$ is said to be degradable if there exists another channel $\Xi$ such that $\Xi\Phi$ is complementary to $\Phi$. The idea here is as follows. Suppose $\Phi$ is a channel and $\Psi$ is ...
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7 votes
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What is the Von Neumann entropy of $\rho = \sum_ip_i|i\rangle\langle i| \otimes \rho_i$?

Operator $\rho$ is not a tensor product, it's a sum of tensor products $$ p_1|1\rangle\langle 1| \otimes \rho_1 + p_2|2\rangle\langle 2| \otimes \rho_2 + \dots + p_d|d\rangle\langle d| \otimes \rho_d. ...
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7 votes
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How to prove that $\frac{| x_0 \rangle + | x_1 \rangle}{\sqrt{2}}$ hides one of $x_0$ or $x_1$?

We can bound the amount of information that can be retrieved from $|\psi\rangle$ using Holevo's bound. Alice and Bob Let us first reformulate the situation in the terms usually employed in the context ...
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7 votes

What is "linear" in linear entropy?

$S_L$ called linear because it's obtained from the usual definition of von Neumann entropy $S = -\mathrm{Tr}(\rho \ln \rho)$ by taking a linear approximation for the natural log $\ln \rho = \rho - 1$. ...
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6 votes
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Accessible information of system vs system, apparatus and environment

For density matrices $\rho_A$ and $\rho_B$ having eigenvalues $\lambda^{\left(A\right)}$ and $\lambda^{\left(B\right)}$, \begin{align}S\left(\rho_A\otimes\rho_B\right) &= -\rho_A\otimes\rho_B\ln\...
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6 votes

Relating min-entropy with conditional entropy

The conditional min-entropy $\text{H}_{\text{min}}(A | B)_{\rho}$ can be defined for an arbitrary state $\rho$ of a pair of registers $(A,B)$ as $$ - \inf_{\sigma} \,\text{D}_{\text{max}}(\rho \| \...
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6 votes

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

You are calculating the entropy of one of the marginal states and so you would not expect the answer to be independent of $\theta$, except in the case that $|\psi\rangle = |\phi_A\rangle \otimes |\...
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  • 4,212
6 votes

What is a "maximally mixed state"?

If you want to describe one part of a many-qubit state, such as a GHZ state, then usually, that one qubit cannot be said to be in a state $|\phi\rangle$. That only works if the overall state is ...
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5 votes
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Does computing the quantum mutual information $I(\rho^{AB})$ require full tomographic information of $\rho^{AB}$?

The mutual information can be written in terms of the relative entropy, please see Nielsen and Chuang (the entropy Venn diagram figure 11.2). I am writing the equation in the question's notation: $$I(...
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5 votes
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Understanding classical vs. quantum channel capacities

These are not really the definitions of classical and quantum capacity, as I will explain. Before doing that, let me adjust the notation being used slightly: let $\Phi:\text{L}(\mathcal{X}) \...
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5 votes
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Where does the Xmon simulator from Googles cirq framework its entropy from?

Cirq uses numpy's pseudo random number generator to pick measurement results, e.g. here is code from XmonStepper.simulate_measurement: ...
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5 votes
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Does the quantum Jensen-Shannon divergence appear in any quantum algorithms or texts on quantum computing?

That quantity appears to be identical to Holevo information, which turns out to be the upper bound on how much classical information you can transmit using a quantum channel [1]. More generally the ...
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5 votes
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Prove the additivity of the Renyi entropy: $H_{\beta}(p \times r) = H_{\beta}(p) + H_{\beta}(r)$

This only holds if the two distributions are independent. In this case $$ \begin{aligned} H_{\beta}(p \times q) &= \frac{1}{1-\beta} \log\left( \sum_{i,j}(p(i) q(j))^{\beta} \right) \\ &= \...
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  • 4,212
5 votes

Measuring entanglement entropy using a stabilizer circuit simulator

I think stim is the right tool for the job here, because it gives you access to the stabilizer generators and also it defines stim.PauliString which you can use to ...
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5 votes

Understanding the definition of entropy in the joint entropy theorem derivation

It is not just the binary entropy that is denoted $H(p_i)$. The quantity that is relevant here is the Shannon entropy of the distribution $\{p_i\}$ which is defined as $$ H(p_i) = - \sum_i p_i \log ...
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  • 4,212
4 votes

Shannon entropy is least when Measurement basis = Mixture basis

My favourite way of proving that the Shannon entropy is minimized for a measurement in the qubit basis is through the notion of majorizaion (see Nielsen and Chuang or the book on Matrix Analyis by ...
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4 votes
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Building Intuition for Relative Von Neumann Entropy

Posting an answer because I realised what my issue was: What I didn't realise then: When a density matrix is written in any basis, the diagonal elements correspond to the probabilities of the density ...
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4 votes
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Maximally mixed states for more than 1 qubit

The Von Neumann entropy of $1/2 (|00\rangle \langle 00| + |11\rangle \langle 11|)$ is one bit. For $I/4$ it's two bits of entropy instead. The entropy of states that only have entries on the ...
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4 votes
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Can an isometry leave entropy invariant?

You don't need any additional conditions beyond those already stated in the question. That is, for any isometry $V: A \rightarrow A\otimes B$ and any unit vector $|\psi\rangle_B$, there will always be ...
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  • 4,568
4 votes

Quantum Supremacy: Some questions on cross-entropy benchmarking

That seems to restrict the output probability distributions of all quantum circuits to rather high entropy distributions. The output of a typical randomly chosen quantum circuit is rather high ...
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4 votes

What is Landauer’s principle?

It means that if you lose information from your system, that information must have been transferred to the system's surroundings. This shows up as an increase in the entropy in the surroundings. This ...
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  • 549
4 votes

Prove that quantum channels cannot increase the Holevo information of an ensemble

Suppose that $\mathsf{X}$ is a register that can store each possible choice for $x$, as a classical state, while $\mathsf{Y}$ is a register that can store each possible state $\rho_x$. It is then ...
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  • 4,568
4 votes

How to measure entanglement in an algorithm?

This is not much of an answer, but is probably too long for a comment... I don't believe that there's a canonical way of doing this. You'd be best off understanding why you're asking the question, ...
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4 votes
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Under what situation is $\sum_{i} p_{i}S(\rho_i)$ > 0

Mathematically when is $\sum_i p_i S(\rho_i) > 0$? I am assuming that $\{p_i\}$ form a probability distribution (and that none of the $p_i = 0$) and each $\rho_i$ is a normalised state. As $p_i \...
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4 votes
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Prove that Shannon and von Neumann entropies satisfy $H(P)\ge S(\rho)$ with $P$ diagonal of $\rho$

This can be seen through "twirling" with a bunch of unitaries. Call your density operator $\rho$. Let $U_i$ be a unitary with $\pm 1$ on the diagonal, and zeros everywhere else when ...
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