Questions tagged [information-theory]
The tag is used for questions connected with information theory in classical and/or quantum sense.
110
questions
2
votes
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
votes
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
votes
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
votes
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
$$ I_{acc}(\rho^{XA}) = ...
3
votes
0answers
42 views
Minmax theorem for optimization over isometries and states
I have the following minmax problem and I am wondering if the order of the minimum and maximum can be interchanged and if yes, why?
Let $\|\cdot\|_1$ be the trace norm defined as $\|\rho\|_1 = \text{...
2
votes
0answers
84 views
Does the von Neumann entropy equal the smallest accessible Shannon entropy?
I've been reading about the von Neumann entropy of a state, as defined via $S(\rho)=-\operatorname{tr}(\rho\ln \rho)$. This equals the Shannon entropy of the probability distribution corresponding to ...
4
votes
1answer
54 views
Trace distance between mixed state and pure state vs trace distance between their purifications
Let $\rho$ be a mixed state and $\vert\psi\rangle\langle\psi\vert$ be a pure state on some Hilbert space $H_A$ such that
$$\|\rho - \vert\psi\rangle\langle\psi\vert \|_1 \leq \varepsilon,$$
where $\|A\...
2
votes
1answer
28 views
How does $F(\psi, \phi) = [\sum_{x}\sqrt{p(x)q(x)}]^{2}$ [duplicate]
From Quantum Information Theory by Mark Wilde, pg 243 asks to show that $F(\psi, \phi) = [\sum_{x}\sqrt{p(x)q(x)}]^{2}$, which is described as the Bhattacharyya overlap, or classical fidelity, between ...
4
votes
0answers
177 views
What was the meaning of Lossless Quantum compression?
I was reading few questions regarding lossless quantum compression on stack exchange, then out of curiosity, I started reading this article. After reading I end up being confused about what does ...
4
votes
0answers
105 views
States used in lossless quantum compression?
I was reading about quantum compression in this article and have some doubts regarding an example mentioned. Specifically, I have two questions:
In example they represented $|a\rangle = \dfrac{1}{\...
2
votes
1answer
37 views
How can we prove that the covariance satisfies $\mathrm{Cov}_\rho(X,Y)=\mathrm{Cov}_\rho(Y,X)$?
While attempting to prove the Cauchy Schwarz Inequality I came across this problem. First of all, if we are given a $\rho$ density matrix and two matrix of obserables $X,Y$, after defining the ...
2
votes
1answer
37 views
Is data processing for relative entropy true when states are not normalized?
The data processing inequality for relative entropy states that
$$D(\rho\|\sigma) \geq D(N(\rho)\|N(\sigma))$$
for some CPTP map $N$ where $\rho$ is a quantum state and $\sigma$ is a positive-...
1
vote
0answers
93 views
Continuity of relative entropy variance
Related question here - copying over the definitions.
The relative entropy between two quantum states is given by $D(\rho\|\sigma) = \operatorname{Tr}(\rho\log\rho -\rho\log\sigma)$. It is known that ...
1
vote
1answer
59 views
Does the relative entropy variance $V(\rho_{AB}\|\rho_A\otimes\sigma_B)$ satisfy an ordering for different $\sigma_B$?
The relative entropy between two quantum states is given by $D(\rho\|\sigma) = \operatorname{Tr}(\rho\log\rho -\rho\log\sigma)$. It is known that for any bipartite state $\rho_{AB}$ with reduced ...
4
votes
1answer
51 views
Upper bounding a permutation invariant state
Let $\rho_{A^n}$ be a permutation invariant quantum state on $n$ registers i.e. $\pi(A^n)\rho_{A^n}\pi(A^n) = \rho_{A^n}$ for any permutation $\pi$ among the $n$ registers.
If we trace out $n-1$ ...
3
votes
1answer
32 views
Continuity of Renyi entropies - limiting cases
The Renyi entropies are defined as
$$S_{\alpha}(\rho)=\frac{1}{1-\alpha} \log \operatorname{Tr}\left(\rho^{\alpha}\right), \alpha \in(0,1) \cup(1, \infty)$$
It is claimed that this quantity is ...
3
votes
1answer
109 views
Is there an identity for the partial transpose of a product of operators?
The partial transpose of an operator $M$ with respect to subsystem $A$ is given by
$$
M^{T_A} := \left(\sum_{abcd} M^{ab}_{cd} \underbrace{|a\rangle \langle b| }_{A}\otimes \underbrace{|c \rangle \...
2
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 $z_{i} \in \{0, 1\}^{n}$,
$$ |\langle 0|\psi\rangle|^{2}, |\langle 1|\psi\...
2
votes
1answer
66 views
What is the Stinespring representation of the adjoint of a channel?
For any completely positive trace nonincreasing map $N_{A\rightarrow B}$, the adjoint map is the unique completely positive linear map $N^\dagger_{B\rightarrow A}$ that satisfies
$$\langle N^\dagger(\...
2
votes
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
votes
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.
$$ \| \Phi_0 - \Phi_1 \|_{\diamond} = \sup_{\rho} \: \| (\Phi_0 \otimes \operatorname{Id}_k)(\rho) - (\Phi_1 ...
3
votes
1answer
77 views
Positive conditional quantum entropy for entangled state
The quantum conditional entropy $S(A|B)\equiv S(AB)-S(A)$, where $S(AB)=S(\rho_{\rm AB})$ and $S(B)=S(\rho_{\rm B})$ is known to be non-negative for separable states. For entangled states, it is known ...
1
vote
1answer
52 views
Prove that for a general tri-partite state $\rho_{ABE}$, $H(AB) = H(E)$
How do I prove that for a general tri-partite state $\rho_{ABE}$, the following holds:
$$
H(\rho_{AB}) = H(\rho_{E}), H(\rho_{AE}) = H(\rho_{B}),
$$
where, $H$ is the Von Neumann entropy. Would ...
3
votes
1answer
73 views
Teleportation followed by measurement: Lowering communication cost
Suppose Alice and Bob have access to shared entanglement and a classical channel and wish to simulate the following quantum protocol. Alice sends over to Bob an $n$-qubit state which is not known to ...
4
votes
1answer
56 views
Does the quantum Jensen-Shannon divergence appear in any quantum algorithms or texts on quantum computing?
The generalization of probability distributions on density matrices allows to define quantum Jensen–Shannon divergence (QJSD), which uses von Neumann entropy. Does QJSD appear in any quantum ...
2
votes
1answer
44 views
Pseudoinverse of a quantum state
The max-relative entropy between two states is defined as $D_{\max}(\rho\|\sigma) = \log\lambda$, where $\lambda$ is the smallest real number that satisfies $\rho\leq \lambda\sigma$, where $A\leq B$ ...
2
votes
0answers
42 views
Quantum analogues of information theoretic measures: are log probabilities replaced with the density matrix?
Below is a question and an answer.
How does quantum information relate to, diverge from or reduce to
Shannon information, which used log probabilities?
What people are more often interested in are ...
2
votes
1answer
62 views
Relating quantum max-relative entropy to classical maximum entropy
The quantum max-relative entropy between two states is defined as
$$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
votes
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, ...
