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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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 ...
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Energy cost of quantum computation
A quantum computer can be modeled as a single unitary transition of a (large) effective quantum state to another. In order to get errors under control, quantum error correction is assumed. A logical ...
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Under what situation is $\sum_{i} p_{i}S(\rho_i)$ > 0
Concerning the Von Neumann Entropy $S(\rho) = H(pi) + \sum_{i}p_{i}S(\rho_{i})$, under what circumstances does $\sum_{i}piS(\rho_{i})$ become greater than 0? I am aware it occurs when $\rho_{i}$ is ...
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What are the thermodynamic limits of Shor's algorithm
The asymptotic time complexity of Grover's algorithm is the square root of the time of a brute force algorithm. However, according to Perlner and Liu, the thermodynamic behavior (theoretical minimum ...
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Increasing the von Neumann entropy despite the measurement?
Background
Assume we have a density matrix $\rho$ of a sub-ensemble. However, we have an imperfect measuring instrument. While it does perform a measurement, we do not know exactly when it performs ...
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Using entropy_mutual function in QuTiP
I am trying to calculate mutual entropies using QuTiP, but I am being unsuccessful so far. More specifically, I consider a 2^n x 2^n matrix representing the density operator of a n-qubit bipartite ...
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Schumacher compression - comparing with Shannon compression
Background
Shannon's source coding theorem tells us the following. We shall consider a binary alphabet for simplicity. Suppose Alice has $n$ independent and identically distributed instances of a ...
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What is the Von Neumann entropy of $\rho = \sum_ip_i|i\rangle\langle i| \otimes \rho_i$?
Let $\overline{p}$ be a probability distribution on $\{1,....,d\}$. Then let $\rho = \sum_ip_i|i\rangle\langle i| \otimes \rho_i$.
How should I take the Von-Neumann entropy of $\rho$? I know that ...
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Continuity bounds on $D_{\max}(\rho_{AB}\|\rho_A\otimes\rho_B)$
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 ...
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Questions about the relation between max-relative entropy $D_{\max}(\rho||\sigma)$ and 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 ...
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What are good resources for newcomers on entropy measurements? [closed]
I am starting on quantum computing and I got to entropy measurements, but that has me stuck because there seems to be a lack of resources for newcomers on those concepts and their utility. Does anyone ...
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How to calculate the Von Neuman entropy on qiskit with the module quantum_info?
I am trying to wrap my head around he quantum_info module on qiskit, since most of the functions on qiskit.tools are going to be ...
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Conditional version of the triangle inequality for Von Neumann entropy
I'm trying to solve problem 11.3 in Nielsen Chuang:
(3) Prove the conditional version of the triangle inequality:
$$
S(A,B|C)\geq S(A|C)-S(B|C)
$$
But the inequality seems incorrect. For example,...
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Forbidden/allowed outputs of a quantum channel
The coherent information of a channel $\mathcal{E}_{A'\rightarrow B}$ is defined as the maximum value obtained by the following function where the maximization is over all input states
$$I_{\rm{coh}}(...
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Quantum state discrimination and lower bound for conditional von Neumann entropy
Consider two quantum states $\rho_A$ and $\sigma_A$, and define the classical-quantum state over a classical binary system $B$ and $A$,
$$\omega_{AB}^\epsilon :=\epsilon \vert 0 \rangle \langle 0 \...
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Convexity of coherent information - erroneous argument!
Consider a state $\rho_{AB}$. Let it have purification $\psi_{A'AB}$. I am interested in the coherent information of this state which is given by
$$I(A\rangle B)_\rho = S(B)_\rho - S(AB)_\rho$$
I ...
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Entanglement entropy's role in quantum information
I am just new to the concepts of entanglement entropy and how it is used to measure the entanglement in systems. I want to know the role of entanglement entropy in quantum information, in general.
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Entropy of a shared state as measured by the individual parties
Suppose I prepare a Bell state $|\beta_{00}\rangle$, and distribute the product state $|\beta_{00}\rangle_{12}|\beta_{00}\rangle_{34}|\beta_{00}\rangle_{56}$ without telling them which state I ...
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How to measure entanglement in an algorithm?
Entanglement in Algorithms
Most algorithms in quantum computing find their strength in making use of entanglement.
I am interested in evaluating the amount of entanglement generated within an ...
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Quantum Supremacy: Some questions on cross-entropy benchmarking
I was skimming through the Google quantum supremacy paper but got stuck on this section:
For a given circuit, we collect the measured bit-strings $\{x_i\}$ and compute the linear XEB fidelity [24-26, ...
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How do I compute the Relative Entropy between pure and mixed states?
Let
$$ \rho = \begin{bmatrix} .7738 & -.0556 \\ -.0556 & .0040 \end{bmatrix} ,
\sigma = \begin{bmatrix} .9454 & -.2273 \\ -.2273 & .0546 \end{bmatrix} \\$$
As you can ...
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How to prove the following bosonic entanglement expression?
Based on the article given by J. L. Ball, I. Fuentes-Schuller, and F. P. Schuller, Phys. Lett. A 359, 550 (2006)
had used the following expression of von-Neumann entropy
\begin{equation}
S = - \...
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What is Landauer’s principle?
How does the act of erasing information increase the total entropy of the system? This goes by the name Landauer's principle. Some details are here. Can anyone shed more light on this?
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What property ensures that von Neumann entropy is conserved?
So I always had this idea in my mind that unitary evolution in quantum mechanics conserves information (or in other words von Neumann entropy) because unitary evolution preserves the trace. But this ...
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Degradable channels and their quantum capacity
Note: I'm reposting this question as it was deleted by the original author, so that we do not lose out on the existing answer there, by Prof. Watrous. Further answers are obviously welcome.
I have ...
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Prove that quantum channels cannot increase the Holevo information of an ensemble
I need to prove the fact that a quantum channel (a superoperator) cannot increase the Holevo information of an ensemble $\epsilon = \{\rho_x, p_x\}$. Mathematically expressed I need to prove
$$\begin{...
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In the proof of the joint entropy theorem, why are $p_i\lambda_i^j$ the eigenvalues?
From section 11.3.2 of Nielsen & Chuang:
(4) let $\lambda_i^j$ and $\left|e_i^j\right>$ be the eigenvalues and corresponding eigenvectors of $\rho_i$. Observe that $p_i\lambda_i^j$ and $\left|...
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Superoperator cannot increase relative entropy
Note: Cross-posted on Physics SE.
So I have to show that a superoperator $\$$ cannot increase relative entropy using the monotonicity of relative entropy:
$$S(\rho_A || \sigma_A) \leq S(\rho_{AB} || ...
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Can an isometry leave entropy invariant?
Consider two finite dimensional Hilbert spaces $A$ and $B$. If I have an isometry $V:A\rightarrow A\otimes B$, under what condition can I find a unitary $U:A\otimes B\to A\otimes B$ such that $$U\rho_{...
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Understanding classical vs. quantum channel capacities
The classical channel capacity ($C_{ea}$) and the quantum channel capacity ($Q$) as defined here (eqs. 1 and 2) are given by
\begin{equation}
C_{ea} = \text{sup}_{\rho} \Big[S(\rho) + S(\Phi_t \rho) -...
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How do I calculate the von Neumann entropy of a pure one-qubit density matrix?
Let's say I have a pure state of the form:
$$\psi = \sqrt{\frac{3}{9}} \lvert 0 \rangle + \sqrt{\frac{6}{9}} \lvert 1 \rangle$$
Then the density matrix representation would be:
$$\rho = \psi \otimes \...
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What do entanglement cost and distillable entanglement have to do with measuring entanglement?
So far what I have learned is that von-Neumann entropy is a tool to measure or quantify information and therefore entanglement for a given pure state system. However, similar concepts emerge from the ...
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Where does the Xmon simulator from Googles cirq framework its entropy from?
Measurements create entropy as we all know. But computers themselves are deterministic machines. Most devices use processor heat as a source for random number generation as far as I know - which has ...
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Building Intuition for Relative Von Neumann Entropy
This is how I think about classical relative entropy: There is a variable that has distribution P, that is outcome $i$ has probability $p_i$ of occuring, but someone mistakes it to be of a ...
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Is the set of all states with negative conditional Von Neumann entropy convex?
I have read somewhere / heard that the set of all states that have non-negative conditional Von Neumann entropy forms a convex set. Is this true? Is there a proof for it?
Can anything be said about ...
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Shannon entropy is least when Measurement basis = Mixture basis
For a one qubit system, take a basis.
Call this the mixture basis.
Consider only basis states and classical mixtures of these basis states.
Definition of Shannon Entropy used here: Defined with ...
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Does computing the quantum mutual information $I(\rho^{AB})$ require full tomographic information of $\rho^{AB}$?
In the discussions about quantum correlations, particularly beyond entanglement (discord, dissonance e.t.c), one can often meet two definitions of mutual information of a quantum system $\rho^{AB}$:
...
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Maximally mixed states for more than 1 qubit
For 1 qubit, the maximally mixed state is $\frac{\mathrm{I}}{2}$.
So, for two qubits, I assume the maximally mixed state is the maximally mixed state is $\frac{\mathrm{I}}{4}$?
Which is:
$\frac{1}{...
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How does the conditional min-entropy $H_{\rm min}(A|B)_\rho$ relate to the conditional entropy $H(X|Y)_\rho$?
Suppose we have a classical quantum state $\sum_x |x\rangle \langle x|\otimes \rho_x$, one can define the smooth-min entropy $H_\min(A|B)_\rho$ as the best probability of guessing outcome $x$ given $\...
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Proof of an Holevo information inequality for a classical-classical-quantum channel
Suppose I have a classical-classical-quantum channel $W : \mathcal{X}\times\mathcal{Y} \rightarrow \mathcal{D}(\mathcal{H})$, where $\mathcal{X},\mathcal{Y}$ are finite sets and $\mathcal{D}(\mathcal{...
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Accessible information of system vs system, apparatus and environment
Suppose we have a quantum system $Q$ with an initial state $\rho^{(Q)}$. The measurement process will involve two additional quantum systems: an apparatus system $A$ and an environment system $E$. We ...
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