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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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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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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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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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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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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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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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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What is "linear" in linear entropy?
Why is the linear entropy, defined by $S_L = 1 - \textrm{Tr} \rho^2$, called linear?
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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 ...
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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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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 =...
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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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Thermodynamic Speed Limit to Quantum Computing
There's a lot of mystifying jargon in the field of quantum computation, so I would like to examine some elementary physics to maybe help clarify the assumptions being made.
Is it not true that the ...
user22003
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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 ...
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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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What is entropy quantum computing?
Recently there is news concerning some computational breakthroughs by using so-called "entropy quantum computing":
https://thequantuminsider.com/2022/07/20/qci-solves-3854-variable-problem-...
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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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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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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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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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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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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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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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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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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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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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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:
$$S(\rho_A || \sigma_A) \leq S(\rho_{AB} || ...
- 525
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Relative entropy inequality for many copies of a channel
Suppose we have two quantum channels $\mathcal{E}_{A\rightarrow B}, \mathcal{F}_{A\rightarrow B}$ that satisfy
$$D(\mathcal{E}(\rho_A)\|\mathcal{E}(\sigma_A))\geq D(\mathcal{F}(\rho_A)\|\mathcal{F}(\...
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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 ...
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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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2
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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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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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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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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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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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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, ...
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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 ...
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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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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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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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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 ...
- 2,597
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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 ...
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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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Is there a known deterministic counterexample for non-additivity of minimal output entropy?
Hastings has proved that the minimal output entropy is not additive: it may happen that $S_{\mathrm{min}}(\Phi_1 \otimes \Phi_2) < S_{\mathrm{min}}(\Phi_1)+S_{\mathrm{min}}(\Phi_2) $ for quantum ...
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Does the quantum relative entropy have a direct operational interpretation?
Consider the quantum relative entropy, defined as
$$D(\rho\|\sigma) = \operatorname{tr}(\rho\log\rho)-\operatorname{tr}(\rho\log\sigma),$$
for all $\rho,\sigma\ge0$ such that $\operatorname{im}(\rho)\...
glS♦
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Prove the equality conditions in the triangle inequality $S(A,B)\ge |S(A)-S(B)|$ for the von Neumann entropy
The triangle inequality or Araki-Lieb inequality of the von Neumann entropy is
$$
S(A,B)\ge|S(A)-S(B)|
$$
this is proven by introducing a system $R$ which purifies systems $A$ and $B$. Applying ...
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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
\...
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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. ...
- 2,597
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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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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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