Questions tagged [information-theory]
The tag is used for questions connected with information theory in classical and/or quantum sense.
138
questions
2
votes
1answer
44 views
Does ${\rm tr}(\Pi_z\rho\Pi_z)\le p$ imply $\cal E(\rho)$ and $\cal E(\Pi_{-z}\rho\Pi_{-z})$ are close in trace distance?
Suppose I have a quantum operation $\mathcal{E}$ and a state $\rho$ such that:
$$ \operatorname{tr}(\Pi_z \rho \Pi_z) \le p $$
for some probability $p$ and some projection $\Pi_z$ onto some subspace ...
1
vote
1answer
41 views
In Schumacher’s noiseless channel coding theorem, how do we get the exponents in $|0\rangle ^{\otimes n(1−p)/2}|1\rangle ^{\otimes n(1−p)/2}$?
On pg. 55 in Nielsen and Chuang, it's said that:
the $|0\rangle + |1\rangle$ product can be well approximated by a superposition of states of the form $|0\rangle ^{\otimes n(1−p)/2}|1\rangle ^{\...
4
votes
0answers
57 views
What is the quantum capacity of the combined amplitude and phase damping channel?
Quantum capacity for the amplitude damping channel and the pure dephasing channel have closed-form formulas as it can be seen in section 24.7.2 of From Classical to Quantum Shannon Theory. However, I ...
4
votes
1answer
52 views
What is a maximal entangled multipartite state?
We know the four Bell states are the maximal entangled states for two-qubit states, and we know if a state cannot be written as the tensor product by its subsets, then it is a entangled state, so is ...
1
vote
1answer
25 views
How to take the limits of the sandwiched Renyi divergences?
The sandwiched Renyi divergence is defined as
$$\begin{equation}
\tilde{D}_{\alpha}(\rho \| \sigma):=\frac{1}{\alpha-1} \log \operatorname{tr}\left[\left(\sigma^{\frac{1-\alpha}{2 \alpha}} \rho \...
2
votes
1answer
42 views
What is the difference between having a single-qubit state and knowing a result of a measurement you want to perform on it?
In the quantum teleportation protocol Alice can send Bob an unknown quantum state $|\psi\rangle$. If the only thing Bob does with $|\psi\rangle$ is to measure it in some basis, I guess it would be ...
3
votes
0answers
29 views
Uhlmann's theorem analogue for channels
Let the stabilized channel fidelity between two channels $M_{A\rightarrow B}$ and $N_{A\rightarrow B}$ be defined as
$$F(M,N) = \min\limits_{\vert\psi\rangle_{AR}} F\left((M\otimes I_R)\vert\psi\...
2
votes
1answer
40 views
What are the "higher moments" of the gate fidelity?
Reading the paper Gate fidelity fluctuations and quantum process invariants I came across the concept of higher moments of the gate fidelity, for example in the following excerpt from the introduction:...
4
votes
1answer
122 views
Can all mixed states be written as a convex combination $\rho=\sum_j p_j |\psi_j\rangle\langle \psi_j|$?
States belonging to some space $\mathcal H$ can be described by density operators $\rho\in L(\mathcal H)$ that are positive and have trace one. Pure states are the ones that can be written as $\rho=|\...
1
vote
0answers
72 views
Prove that autocompatible LCPT maps are antidegradable
A LCPT map $\Phi$ is antidegradable if and only if it is compatible with itself.
The proof that an antidegradable map (satisfying $\Phi=\Lambda_E\circ \tilde\Phi$ where $\tilde\Phi$ is the ...
2
votes
1answer
100 views
How does the Kraus decomposition imply the Stinespring representation?
To show that the Kraus decomposition $\Phi(\rho)=\sum_{k=1}^D M_k\rho_S M_k^\dagger$ implies the Stinespring form $$\Phi(\rho)=\text{tr}_E[U_{SE}(\rho_S\otimes|0\rangle\langle 0|_E)U_{SE}^\dagger]$$ ...
0
votes
1answer
39 views
Why do we want the no error limit to be 1?
In a textbook by Nielsen and Chuang, there's the following paragraph:
The idea of quantum data compression is that the compressed data should be recovered with very good fidelity. Think of the ...
2
votes
1answer
51 views
What can be said about the non-negativity of the relative entropy of $S(\rho_{AB}||\rho_{B})$?
Taking $\rho_{AB}=\rho_{A}\otimes \rho_{B}$, where $S(\rho_{A})$ and $S(\rho_{B})$ aren't 0, it's easy to see that
$$S(\rho_{AB}||I \otimes \rho_{B})=-S(\rho_{A})-S(\rho_{B})+S(\rho_{B})=-S(\rho_{A}).$...
4
votes
1answer
56 views
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 ...
1
vote
0answers
11 views
Optimality of teleportation for entanglement-assisted quantum channel coding
Setting:
Consider a quantum channel $N_{A\rightarrow B}$ between Alice and Bob. Let Alice and Bob also share arbitrary amounts of entanglement assistance through the state $\phi_{A_EB_E}$ (these can ...
3
votes
0answers
46 views
How to translate performance of two classical codes to a quantum CSS code?
I have a database of classical codes with simulation results in binary symmetric channel (BSC).
The codes are defined by their parity check matrices $H$. I can pick pairs of codes and call them $H_x$ ...
3
votes
1answer
58 views
Tradeoff between error and rates of quantum communication
Suppose Alice and Bob share $n$ copies of a noiseless quantum channel $I_{A\rightarrow B}$ which can be used to send quantum states and $H_A\cong H_B$ i.e. the input and output Hilbert spaces are the ...
2
votes
2answers
196 views
How to understand intuitively the concavity of the binary entropy?
In Nielsen and Chuang's Quantum Computation and Quantum Information book, introducing the binary entropy, they gave an intuitive example about why binary entropy is concave:
Alice has in her ...
0
votes
1answer
101 views
How to describe a known quantum state using classical information?
In Nielsen and Chuang, it's said that to describe a known quantum state precisely takes an infinite amount of classical information since $|\psi\rangle$ takes values in a continuous space (from the ...
1
vote
1answer
32 views
Increasing entropy for projective LCPT mapping
Given a set of projectors $\{P_i\}$ acting on a space $\mathcal H_S$, let $\Phi$ be the LCPT map defined by $$\Phi(\rho)=\sum_i P_i\rho P_i.$$ The goal is to show that $S(\Phi(\rho))\ge S(\rho)$. The ...
2
votes
1answer
69 views
Prove the subadditivity for the von Neumann entropy of a bipartite state
I want to prove the subadditivity relation $S(\rho_{AB})\le S(\rho_A)+S(\rho_B)$ for the Von Neumann entropy. The tip is to use the Klein inequality $S(\rho_{AB}\Vert \rho_A\otimes \rho_B)\ge 0$: $$S(\...
2
votes
0answers
90 views
How to prove that the mutual information is subadditive?
Let $\mathbf x=(x_1,...,x_n)$ and $\mathbf y=(y_1,...,y_n)$ be two vectors of random variables. To make things concrete, assume that Alice sends each component $x_j$ through a noisy channel to Bob, ...
2
votes
1answer
39 views
Is the quantum mutual information variance bounded from above?
The relative entropy variance between two quantum states $\rho$ and $\sigma$ is defined to be
$$V(\rho\|\sigma) = \text{Tr}(\rho(\log\rho - \log\sigma)^2) - D(\rho\|\sigma)^2,$$
where $D(\rho\|\sigma)$...
3
votes
1answer
84 views
How can the entropy of quantum states increase after projective measurements?
I'm reading Nielsen and chuang 11.3.3 Measurements and Entropy.
It says after measurement, one's entropy increases.
How is this possible? Shouldn't measurement decrease one's uncertainty?
7
votes
1answer
99 views
Can one quantify entanglement between different parts of a system?
Consider some state $|\psi\rangle$ of $n$ qubits. One can take any subsystem $A$ and compute its density matrix $\rho_A =Tr_{B} |\psi\rangle \langle\psi|$. The entanglement between subsystem $A$ and ...
5
votes
2answers
69 views
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 ...
2
votes
1answer
124 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
145 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
107 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
70 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
92 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
$$ I_{acc}(\rho^{XA}) = ...
3
votes
0answers
42 views
Minmax theorem for optimization over isometries and states
I have the following minmax problem and I am wondering if the order of the minimum and maximum can be interchanged and if yes, why?
Let $\|\cdot\|_1$ be the trace norm defined as $\|\rho\|_1 = \text{...
2
votes
0answers
93 views
Does the von Neumann entropy equal the smallest accessible Shannon entropy?
I've been reading about the von Neumann entropy of a state, as defined via $S(\rho)=-\operatorname{tr}(\rho\ln \rho)$. This equals the Shannon entropy of the probability distribution corresponding to ...
5
votes
1answer
81 views
Trace distance between mixed state and pure state vs trace distance between their purifications
Let $\rho$ be a mixed state and $\vert\psi\rangle\langle\psi\vert$ be a pure state on some Hilbert space $H_A$ such that
$$\|\rho - \vert\psi\rangle\langle\psi\vert \|_1 \leq \varepsilon,$$
where $\|A\...
2
votes
1answer
34 views
How does $F(\psi, \phi) = [\sum_{x}\sqrt{p(x)q(x)}]^{2}$ [duplicate]
From Quantum Information Theory by Mark Wilde, pg 243 asks to show that $F(\psi, \phi) = [\sum_{x}\sqrt{p(x)q(x)}]^{2}$, which is described as the Bhattacharyya overlap, or classical fidelity, between ...
4
votes
0answers
184 views
What was the meaning of Lossless Quantum compression?
I was reading few questions regarding lossless quantum compression on stack exchange, then out of curiosity, I started reading this article. After reading I end up being confused about what does ...
4
votes
0answers
107 views
States used in lossless quantum compression?
I was reading about quantum compression in this article and have some doubts regarding an example mentioned. Specifically, I have two questions:
In example they represented $|a\rangle = \dfrac{1}{\...
2
votes
1answer
37 views
How can we prove that the covariance satisfies $\mathrm{Cov}_\rho(X,Y)=\mathrm{Cov}_\rho(Y,X)$?
While attempting to prove the Cauchy Schwarz Inequality I came across this problem. First of all, if we are given a $\rho$ density matrix and two matrix of obserables $X,Y$, after defining the ...
2
votes
1answer
38 views
Is data processing for relative entropy true when states are not normalized?
The data processing inequality for relative entropy states that
$$D(\rho\|\sigma) \geq D(N(\rho)\|N(\sigma))$$
for some CPTP map $N$ where $\rho$ is a quantum state and $\sigma$ is a positive-...
1
vote
0answers
96 views
Continuity of relative entropy variance
Related question here - copying over the definitions.
The relative entropy between two quantum states is given by $D(\rho\|\sigma) = \operatorname{Tr}(\rho\log\rho -\rho\log\sigma)$. It is known that ...
1
vote
1answer
64 views
Does the relative entropy variance $V(\rho_{AB}\|\rho_A\otimes\sigma_B)$ satisfy an ordering for different $\sigma_B$?
The relative entropy between two quantum states is given by $D(\rho\|\sigma) = \operatorname{Tr}(\rho\log\rho -\rho\log\sigma)$. It is known that for any bipartite state $\rho_{AB}$ with reduced ...
4
votes
1answer
53 views
Upper bounding a permutation invariant state
Let $\rho_{A^n}$ be a permutation invariant quantum state on $n$ registers i.e. $\pi(A^n)\rho_{A^n}\pi(A^n) = \rho_{A^n}$ for any permutation $\pi$ among the $n$ registers.
If we trace out $n-1$ ...
3
votes
1answer
41 views
Continuity of Renyi entropies - limiting cases
The Renyi entropies are defined as
$$S_{\alpha}(\rho)=\frac{1}{1-\alpha} \log \operatorname{Tr}\left(\rho^{\alpha}\right), \alpha \in(0,1) \cup(1, \infty)$$
It is claimed that this quantity is ...
3
votes
1answer
129 views
Is there an identity for the partial transpose of a product of operators?
The partial transpose of an operator $M$ with respect to subsystem $A$ is given by
$$
M^{T_A} := \left(\sum_{abcd} M^{ab}_{cd} \underbrace{|a\rangle \langle b| }_{A}\otimes \underbrace{|c \rangle \...
2
votes
2answers
136 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
79 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
57 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
67 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 $|00\rangle$...
5
votes
1answer
103 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)]^2 = 1 - \frac{\theta^2}{4} F_Q[\rho, A] + ...
4
votes
1answer
50 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 ...
