Questions tagged [information-theory]

The tag is used for questions connected with information theory in classical and/or quantum sense.

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2
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1answer
111 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|...
4
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2answers
126 views

Clarification on Watrous' proof of Alberti's theorem on the fidelity function

I am reading John Watrous' quantum information theory book. In the proof of Theorem 3.19 (practically the Alberti's theorem on the characterization of the fidelity function) he claims the following ...
2
votes
1answer
100 views

Permutation covariant channels and their Stinespring dilations

I am interested in a quantum channel from $A^{\otimes n}$ to $B^{\otimes n}$ denoted as $N_{A^{\otimes n} \rightarrow B^{\otimes n}}(\cdot)$. Let $\pi(\cdot)$ be a permutation operation among the $n$ ...
3
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0answers
60 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 ...
3
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2answers
89 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}) = ...
3
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0answers
42 views

Minmax theorem for optimization over isometries and states

I have the following minmax problem and I am wondering if the order of the minimum and maximum can be interchanged and if yes, why? Let $\|\cdot\|_1$ be the trace norm defined as $\|\rho\|_1 = \text{...
2
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0answers
84 views

Does the von Neumann entropy equal the smallest accessible Shannon entropy?

I've been reading about the von Neumann entropy of a state, as defined via $S(\rho)=-\operatorname{tr}(\rho\ln \rho)$. This equals the Shannon entropy of the probability distribution corresponding to ...
4
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1answer
54 views

Trace distance between mixed state and pure state vs trace distance between their purifications

Let $\rho$ be a mixed state and $\vert\psi\rangle\langle\psi\vert$ be a pure state on some Hilbert space $H_A$ such that $$\|\rho - \vert\psi\rangle\langle\psi\vert \|_1 \leq \varepsilon,$$ where $\|A\...
2
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1answer
28 views

How does $F(\psi, \phi) = [\sum_{x}\sqrt{p(x)q(x)}]^{2}$ [duplicate]

From Quantum Information Theory by Mark Wilde, pg 243 asks to show that $F(\psi, \phi) = [\sum_{x}\sqrt{p(x)q(x)}]^{2}$, which is described as the Bhattacharyya overlap, or classical fidelity, between ...
4
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0answers
177 views

What was the meaning of Lossless Quantum compression?

I was reading few questions regarding lossless quantum compression on stack exchange, then out of curiosity, I started reading this article. After reading I end up being confused about what does ...
4
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0answers
105 views

States used in lossless quantum compression?

I was reading about quantum compression in this article and have some doubts regarding an example mentioned. Specifically, I have two questions: In example they represented $|a\rangle = \dfrac{1}{\...
2
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1answer
37 views

How can we prove that the covariance satisfies $\mathrm{Cov}_\rho(X,Y)=\mathrm{Cov}_\rho(Y,X)$?

While attempting to prove the Cauchy Schwarz Inequality I came across this problem. First of all, if we are given a $\rho$ density matrix and two matrix of obserables $X,Y$, after defining the ...
2
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1answer
37 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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0answers
93 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 ...
1
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1answer
59 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 ...
4
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1answer
51 views

Upper bounding a permutation invariant state

Let $\rho_{A^n}$ be a permutation invariant quantum state on $n$ registers i.e. $\pi(A^n)\rho_{A^n}\pi(A^n) = \rho_{A^n}$ for any permutation $\pi$ among the $n$ registers. If we trace out $n-1$ ...
3
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1answer
32 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 ...
3
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1answer
109 views

Is there an identity for the partial transpose of a product of operators?

The partial transpose of an operator $M$ with respect to subsystem $A$ is given by $$ M^{T_A} := \left(\sum_{abcd} M^{ab}_{cd} \underbrace{|a\rangle \langle b| }_{A}\otimes \underbrace{|c \rangle \...
2
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2answers
133 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 ...
2
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1answer
75 views

Is the quantum state discrimination success probability always $\lambda_0\langle\mu(0),\rho_0\rangle+\lambda_1\langle\mu(1),\rho_1\rangle$?

Consider the standard quantum state discrimination setup: Alice sends Bob either $\rho_0$ or $\rho_1$. She picks $\rho_0$ and $\rho_1$ with probabilities $\lambda_0$ and $\lambda_1$, respectively. Bob ...
0
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1answer
40 views

In classical state discrimination, why does the trace distance quantify the probability of success?

Consider the following task: we are given a probability distribution $p_y:x\mapsto p_y(x)$ with $y\in\{0,1\}$ (e.g. we are given some black box that we can use to draw samples from either $p_0$ or $...
2
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2answers
56 views

Deriving the depolarizing channel

Consider a circuit built as follows: take two ancilla states and an operator $U$ made of a series of controlled gates which act on a pure state $\rho$ as follows: $X$ if the ancilla is in $|00\rangle$...
3
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1answer
61 views

How is the connection between Bures fidelity and quantum Fisher information derived?

I recently came to know that there is a connection between Bures Fidelity $(F_B)$ and Quantum Fisher Information $(F_Q)$ given by $$F_{B}(\rho, \rho_\theta) = 1 - \frac{\theta^2}{4} F_Q[\rho, A] + \...
4
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1answer
48 views

Why do probablity distribution with orthogonal suppor have maximal Kolmogorov distance?

Can anyone explain why the $l_1$ distance has the property that probability distributions $P,Q$ with orthogonal support (meaning that the product $p_iq_i$ vanishes for each value of $i$) are at a ...
7
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1answer
469 views

Does the no-hiding theorem suggest that quantum information is never destroyed?

According to Wikipedia: The no-hiding theorem proves that if information is lost from a system via decoherence, then it moves to the subspace of the environment and it cannot remain in the ...
9
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5answers
938 views

If quantum computing always return random measurement (or uncertain measurement), why do we still need it?

I am very new to quantum computing and currently studying quantum computing on my own through various resources (Youtube Qiskit, Qiskit website, book). As my mindset is still "locked" with ...
2
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2answers
76 views

If Alice and Bob share only classical communication resources, is shared entanglement always equivalent to shared randomness?

If Alice and Bob share only classical communication resources such as noisy or perfect classical channels, is shared entanglement always equivalent to shared randomness? In other words, must there be ...
4
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0answers
56 views

Average secret key rate calculation

I am working on a problem related to average secret key rate calculation in satellite quantum key distribution (QKD). The problem is similar to a paper on arXiv:1712.09722 (P. No. 21). In a single ...
4
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0answers
107 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 ...
3
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0answers
34 views

Pure state ensembles achieving the Holevo $\chi$-quantity with at most $d^2$ states

Theorem 8.10 in Chapter 8 of Theory of Quantum Information asserts that the Holevo capacity of a quantum channel (between density operators on $\mathbb{C}^d$) can be achieved by an ensemble consisting ...
4
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0answers
68 views

A question in classical and quantum information

Let $\rho, \sigma \in \mathfrak{D}(A)$ with $\operatorname{supp}(\rho) \subseteq \operatorname{supp}(\sigma),$ and spectral decomposition $$ \rho=\sum_{x} p_{x}\left|\psi_{x}\right\rangle\left\langle\...
2
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2answers
85 views

When is the Choi matrix of a channel pure?

For a quantum channel $\mathcal{E}$, the Choi state is defined by the action of the channel on one half of an unnormalized maximally entangled state as below: $$J(\mathcal{E}) = (\mathcal{E}\otimes I)\...
5
votes
1answer
116 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}) - ...
2
votes
1answer
73 views

Additivity of Renyi entropy

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{...
5
votes
1answer
85 views

Stinespring dilation: Size of environment

Let $\mathcal{E}_{A\rightarrow B}$ be a quantum channel and consider its $n-$fold tensor product $\mathcal{E}^{\otimes n}_{A^n\rightarrow B^n}$. Any isometry $V_{A\rightarrow BE}$ that satisfies $\...
3
votes
1answer
96 views

Fidelity of extensions of states

Given two states $\rho_A, \sigma_A$, Uhlmann's theorem states that the fidelity between them is achieved in the following way $$F(\rho_A, \sigma_A) = \max_{U_{R'}}F(\rho_{AR'}, (I\otimes U_{R'})\...
5
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1answer
150 views

Closeness of purifications of states

Uhlmann's theorem states that if two states $\rho_A, \sigma_A$ satisfy $F(\rho_A, \sigma_A)\geq 1 - \varepsilon$, then there for any purification $\Psi_{AR}$ of $\rho_A$, one can find a purification $\...
4
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1answer
119 views

Question about Haar random quantum states

Let $|\psi\rangle$ be a $n$ qubit Haar-random quantum state. I am trying to show that in the limit of large $n$, for each $z_{i} \in \{0, 1\}^{n}$, $$ |\langle 0|\psi\rangle|^{2}, |\langle 1|\psi\...
2
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1answer
66 views

What is the Stinespring representation of the adjoint of a channel?

For any completely positive trace nonincreasing map $N_{A\rightarrow B}$, the adjoint map is the unique completely positive linear map $N^\dagger_{B\rightarrow A}$ that satisfies $$\langle N^\dagger(\...
2
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1answer
53 views

What is meant with “reconciliation” in CV QKD?

I am working on a paper about Continuous Variable QKD. (https://arxiv.org/abs/1711.08500v2) I read about direct and reverse reconciliation in this paper. I don't understand what exactly Reconciliation ...
2
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1answer
72 views

Diamond norm distance bound on Stinespring dilations of channels

The diamond distance between two channels $\Phi_0$ and $\Phi_1$ is defined in this answer. $$ \| \Phi_0 - \Phi_1 \|_{\diamond} = \sup_{\rho} \: \| (\Phi_0 \otimes \operatorname{Id}_k)(\rho) - (\Phi_1 ...
3
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1answer
77 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 ...
1
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1answer
52 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 ...
3
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1answer
73 views

Teleportation followed by measurement: Lowering communication cost

Suppose Alice and Bob have access to shared entanglement and a classical channel and wish to simulate the following quantum protocol. Alice sends over to Bob an $n$-qubit state which is not known to ...
4
votes
1answer
56 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 ...
2
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1answer
44 views

Pseudoinverse of a quantum state

The max-relative entropy between two states is defined as $D_{\max}(\rho\|\sigma) = \log\lambda$, where $\lambda$ is the smallest real number that satisfies $\rho\leq \lambda\sigma$, where $A\leq B$ ...
2
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0answers
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
62 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 - \...
2
votes
1answer
62 views

Wheeler's “information theoretic” derivation of quantum information

1980s. John Wheeler at the University of Texas would tell his students, “Give an information theoretic derivation of quantum theory!” Information theoretic is an adjective for Claude Shannon's ...
2
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1answer
49 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, ...