Questions tagged [information-theory]

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

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Shared entanglement and classical resources

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 ...
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38 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 ...
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93 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 ...
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32 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 ...
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58 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\...
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2answers
49 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)\...
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1answer
85 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}) - ...
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1answer
64 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{...
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1answer
63 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 $\...
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1answer
86 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'})\...
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1answer
143 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 $\...
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94 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\...
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1answer
48 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(\...
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1answer
37 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 ...
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1answer
61 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 ...
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1answer
49 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 ...
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1answer
42 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 ...
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41 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 some $n-$qubit state to Bob. The state is not ...
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1answer
53 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 ...
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1answer
36 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 $A\leq B$ ...
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41 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 ...
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1answer
58 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 - \...
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1answer
57 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 ...
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1answer
48 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, ...
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53 views

Alternative definition of the coherent information of a quantum channel

Let $T: M_n \to M_n$ be a quantum channel. If I understand Definition 13.5.1 of the book "Quantum information theory" of Wilde, the coherent information $Q(T)=\max_{\phi_{AA'}} I(A \rangle B)...
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1answer
72 views

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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1answer
29 views

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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1answer
87 views

Quantum marginal problem - constructing a global state from reduced states

Consider a bipartite state $\rho_{AB}$ with reduced states $\rho_A = \text{Tr}_B(\rho_{AB})$ and $\rho_B = \text{Tr}_A(\rho_{AB})$. Suppose one obtains states $\rho'_{A}$ and $\rho'_{B}$ such that $\|\...
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2answers
87 views

Checking whether a state is almost orthogonal to permutation invariant states

Let us consider \begin{equation} |T\rangle = |\psi \rangle^{\otimes m} \end{equation} for an $n$-qubit quantum state $|\psi\rangle$. Let $\mathcal{V}$ be the space of all $(m + 1)$-partite states that ...
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81 views

What is the difference between no signaling and non locality at operational and ontological level?

I understand the basic definitions. Locality means Alice's measurements do not affect Bob's and system and that no-signalling means a party can't send information faster than light. I also know that ...
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93 views

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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30 views

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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1answer
57 views

The proof of monotonicity of fidelity for channels and its meaning

I have two questions regarding the exercise 9.2.8 of Quantum information by Wilde, which is as follows: Let $\rho,\sigma \in \mathcal{D}(\mathcal{H}_A)$ and let $\mathcal{N: L(H}_A)\rightarrow \...
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1answer
61 views

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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3answers
163 views

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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91 views

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 ...
3
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1answer
53 views

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? ...
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643 views

Can classical linear algebra solvers implement quantum algorithms with similar speed-ups?

A quantum algorithm begins with a register of qubits in an initial state, a unitary operator (the algorithm) manipulates the state of those qubits, and then the state of the qubits is read out (or at ...
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1answer
149 views

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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How is a bit field represented in Quantum Computing?

For example, a computer represent a variable named "A" as 01000001. How does a quantum computer represent "A"? I am a newbie having difficulty understanding quantum computers. I ...
4
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1answer
73 views

Definition of quantum min-relative entropy

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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1answer
97 views

Mutual information of Choi state=0, what would that imply about the quantum channel?

Classically, if the mutual information between the input and output of some channel or circuit $= 0$, it means the output is independent of the input, and the circuit is in a way 'useless'. For the ...
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2answers
53 views

Prove that for one-qubit unitaries $\text{Tr}|U-V|=2\max_\psi\|(U-V)|\psi\rangle\|$

Given two 1-qubit rotations $U=R_n (\theta)$ and $V=R_m(\phi)$ with $n$ and $m$ vectors defining a rotation and $\theta, \phi$ angles, define $D(U,V)=Tr(|U-V|)$ where $|U-V|=\sqrt{(U-V)^\dagger (U-V)}$...
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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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1answer
104 views

Can every process in nature be simulated by a Turing Machine or a quantum computer?

Given any initial condition or value A, A leads to B after a procedure of physics or nature P. Now is there any turing machine or quantum computer that can simulates P,converting A into B? In other ...
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1answer
162 views

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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2answers
229 views

Nielsen and Chuang ex 2.73

I've been trying to solve exercise 2.73 (p.g 105), and I'm not sure if i'v been overthinking it and the answer is as simple as i've described below or if I am missing something, or i'm just wrong! Ex ...
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2answers
162 views

What does it mean to take the Choi-Jamiolkowski of a quantum channel?

The Choi-Jamiolkowski of a channel $\newcommand{\on}[1]{\operatorname{#1}}\Lambda : \on{End}(\mathcal{H_A}) \xrightarrow{} \on{End}(\mathcal{H_B})$ is obtained through an isomorphism of the form: $$ ...
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1answer
70 views

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 ...
1
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0answers
35 views

Impossibility of QRAC with random question, or more general proof of impossibility

TL;DR: Is it known that QRAC where the "question" is random instead of chosen are impossible? More generally what are the methods to prove impossibility games? I often face problems for which I want ...