Questions tagged [information-theory]
The tag is used for questions connected with information theory in classical and/or quantum sense.
260
questions
22
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4
answers
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Can we combine the square roots inside the definition of the fidelity?
The (Uhlmann-Jozsa) fidelity of quantum states $\rho$ and $\sigma$ is defined to be
$$F(\rho, \sigma) := \left(\mathrm{tr} \left[\sqrt{\sqrt{\rho} \sigma \sqrt{\rho}} \right]\right)^2.$$
However, as ...
16
votes
1
answer
863
views
Violation of the Quantum Hamming bound
The quantum Hamming bound for a non-degenerate $[[N,k,d]]$ quantum error correction code is defined as:
\begin{equation}
2^{N-k}\geq\sum_{n=0}^{\lfloor d/2\rfloor}3^n\begin{pmatrix}N \\ n\end{...
14
votes
0
answers
542
views
Relation between quantum entanglement and quantum state complexity
Both quantum entanglement and quantum state complexity are important in quantum information processing. They are usually highly correlated, i.e., roughly a state with a higher entanglement corresponds ...
13
votes
1
answer
2k
views
How can classical bits be copied if qubits cannot be copied?
The no-cloning theorem of quantum mechanics tells us there can be no general quantum circuit that can copy arbitrary qubit states, i.e. a quantum gate or circuit cannot send $|0\rangle |\psi\rangle\...
11
votes
2
answers
2k
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How many classical bits are needed to represent a qubit?
I have two question concerning information content of qubit.
Question 1: How many classical bits are needed to represent a qubit:
A qubit can be represented by a vector $q = \begin{pmatrix}\alpha \\\...
10
votes
1
answer
335
views
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{...
10
votes
0
answers
100
views
Entanglement-assisted hashing bound for asymmetric depolarizing channels
I reading the paper EXIT-Chart Aided Quantum Code Design
Improves the Normalised Throughput
of Realistic Quantum Devices, which proposes the use of QTCs in order to do quantum error correction for ...
9
votes
5
answers
2k
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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 ...
9
votes
1
answer
528
views
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 ...
8
votes
1
answer
560
views
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 $\...
7
votes
2
answers
758
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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 ...
7
votes
3
answers
3k
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Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$
$$Tr(\rho^{AB} (\sigma^A \otimes I/d)) = Tr(\rho^A \sigma^A)$$
I came across the above, but I'm not sure how it's true. I figured they first partial traced out the B subsystem, and then trace A, but ...
7
votes
2
answers
1k
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:
$$
...
7
votes
1
answer
390
views
Degradable channels and their quantum capacity
Note: I'm reposting this question as it was deleted by the original author, so that we do not lose out on the existing answer there, by Prof. Watrous. Further answers are obviously welcome.
I have ...
7
votes
1
answer
250
views
Can one quantify entanglement between different parts of a system?
Consider some state $|\psi\rangle$ of $n$ qubits. One can take any subsystem $A$ and compute its density matrix $\rho_A =Tr_{B} |\psi\rangle \langle\psi|$. The entanglement between subsystem $A$ and ...
7
votes
1
answer
1k
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Does the no-hiding theorem suggest that quantum information is never destroyed?
According to Wikipedia:
The no-hiding theorem proves that if information is lost from a system via decoherence, then it moves to the subspace of the environment and it cannot remain in the ...
7
votes
2
answers
763
views
Quantum relative entropy with respect to a pure state
I want to evalualte the quantum relative entropy $S(\rho|| \sigma)=-{\rm tr}(\rho {\rm log}(\sigma))-S(\rho)$, where $\sigma=|\Psi\rangle\langle\Psi|$ is a density matrix corresponding to a pure state ...
7
votes
1
answer
762
views
Quantum circuit for computing fidelity
Suppose we use Uhlmann-Jozsa fidelity
$$
F(\rho, \sigma):=\left(\mathrm{tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)^2.
$$
Can we construct a quantum circuit that helps us calculate the fidelity of ...
7
votes
1
answer
437
views
Understanding classical vs. quantum channel capacities
The classical channel capacity ($C_{ea}$) and the quantum channel capacity ($Q$) as defined here (eqs. 1 and 2) are given by
\begin{equation}
C_{ea} = \text{sup}_{\rho} \Big[S(\rho) + S(\Phi_t \rho) -...
7
votes
1
answer
770
views
How to derive the quantum Fisher information from the relative entropy?
The quantum relative entropy (QRE) between two states $\rho$ and $\sigma$ is given by
$$
S(\rho\|\sigma)=\operatorname{Tr}(\rho\ln\rho)-\operatorname{Tr}(\rho\ln\sigma)
$$
Now if $\rho$ and $\sigma$ ...
6
votes
1
answer
390
views
Can $2^n$ bits be sent with $n$ instances of quantum teleportation?
So, right now these are two pieces of information I've been told are correct:
Quantum teleportation can send a single qubit from Alice to Bob, with two classical bits
$n$ qubits can store $2^n$ ...
6
votes
1
answer
379
views
Schmidt decomposition for tripartite system $ABC$ with vanishing mutual information between $A$ and $C$
Suppose I have a tripartite system $ABC$ in a pure state $|\psi_{ABC}\rangle$ with mutual information $I(A:C)=0$. This implies that the reduced density matrix $\rho_{AC}$ factorizes as $\rho_{AC} = \...
6
votes
2
answers
476
views
In Bell nonlocality, why does $P(ab|xy)\neq P(a|x)P(b|y)$ mean the variables are not statistically independent?
I've been working through the paper Bell nonlocality by Brunner et al. after seeing it in user glS' answer here. Early on in the paper, the standard Bell experimental setup is defined:
Where $x, y \...
6
votes
2
answers
96
views
Prove that for one-qubit unitaries $\text{Tr}|U-V|=2\max_\psi\|(U-V)|\psi\rangle\|$
Given two 1-qubit rotations $U=R_n (\theta)$ and $V=R_m(\phi)$ with $n$ and $m$ vectors defining a rotation and $\theta, \phi$ angles, define $D(U,V)=Tr(|U-V|)$ where $|U-V|=\sqrt{(U-V)^\dagger (U-V)}$...
6
votes
1
answer
641
views
Why is the quantum Fisher information for pure states $F_Q[\rho,A]=4(\Delta A)^2$?
Assume that a density matrix is given in its eigenbasis as $$\rho = \sum_{k}\lambda_k |k \rangle \langle k|.$$ On page 19 of this paper, it states that the Quantum Fisher Information is given as $$F_{...
6
votes
1
answer
488
views
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 $\rho^{AB}$:
...
6
votes
1
answer
683
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 ...
6
votes
1
answer
165
views
Is it possible to extract $x_1$ and $x_2$ from $|\phi\rangle=\frac1{\sqrt2}(|x_1,0^n\rangle+|0^n,x_2\rangle)$ with non-negligible probability?
Let $\left\vert \phi\right\rangle=\frac 1{\sqrt2}\left\vert x_1,0^n\right\rangle+\frac1{\sqrt2}\left\vert 0^n,x_2\right\rangle$ be a $2n$-bit quantum state for some unknown $x_1,x_2\in\{0,1\}^n$. My ...
6
votes
0
answers
314
views
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 ...
6
votes
0
answers
126
views
Quantum channel Holevo information additivity: proof approach
I have an interesting idea for a proof approach that someone might find useful. Here it is.
Suppose we are given a quantum qubit channel $N$ (for example the amplitude damping channel) whose Holevo ...
5
votes
1
answer
228
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 $\...
5
votes
2
answers
340
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 ...
5
votes
1
answer
462
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=|\...
5
votes
1
answer
172
views
Does $\mathcal E^{\otimes n}$ admit a more efficient Stinespring dilation than the one used for $\mathcal E$?
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 $\...
5
votes
3
answers
300
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 $...
5
votes
1
answer
294
views
What is the root of the non-trace-preserving bit-flip map
I have a quantum channel defined by the Kraus operators:
$$
U_1 =
\begin{bmatrix}
p & 0 \\
0 & p
\end{bmatrix},\quad
U_2 =
\begin{bmatrix}
0 & p \\
p & 0
\end{bmatrix}
$$
i.e.
$$
U_1\...
5
votes
1
answer
865
views
No-cloning theorem and distinguishing between two non-orthogonal quantum states revisited
There are a couple of posts on this question, but I think they are not satisfactory. The question is Nielsen and Chuang's QCQI, Exercise 1.2, page 57, which asks "Explain how a device which, upon ...
5
votes
2
answers
308
views
Shannon entropy is least when Measurement basis = Mixture basis
For a one qubit system, take a basis.
Call this the mixture basis.
Consider only basis states and classical mixtures of these basis states.
Definition of Shannon Entropy used here: Defined with ...
5
votes
1
answer
110
views
Accessible information of system vs system, apparatus and environment
Suppose we have a quantum system $Q$ with an initial state $\rho^{(Q)}$. The measurement process will involve two additional quantum systems: an apparatus system $A$ and an environment system $E$. We ...
5
votes
2
answers
392
views
Tapering off qubits
Suppose you have a Hamiltonian of the form
$$ H = ZXXX + YXXX + XXXX $$
where $Z,X,Y$ are the usual Pauli matrices with $ZXXX = Z \otimes X \otimes X \otimes X$ and similar for the other two terms. ...
5
votes
1
answer
562
views
Positive conditional quantum entropy for entangled state
The quantum conditional entropy $S(A|B)\equiv S(AB)-S(A)$, where $S(AB)=S(\rho_{\rm AB})$ and $S(B)=S(\rho_{\rm B})$ is known to be non-negative for separable states. For entangled states, it is known ...
5
votes
1
answer
72
views
Does the quantum Jensen-Shannon divergence appear in any quantum algorithms or texts on quantum computing?
The generalization of probability distributions on density matrices allows to define quantum Jensen–Shannon divergence (QJSD), which uses von Neumann entropy. Does QJSD appear in any quantum ...
5
votes
1
answer
167
views
maximization of trace between two operators with respect to different norm constraints
I want to maximize $\text{Tr}(XY)$ over $X$ for fixed $Y$, where $X$ and $Y$ are both hermitian (but doesn't necessarily positive) operators, and $X$ is constrained by its p-norm bounded by $1$, i.e. $...
5
votes
2
answers
218
views
How to prove the positivity of the conditional quantum mutual information, $I(A;B|C)\ge0$?
I was reading Wilde's 'Quantum Information Theory' and saw the following theorem at chapter 11 $(11.7.2)$:
$$
I(A; B | C) \ge 0,
$$
where,
$$
I(A;B|C) := H(A|C) + H(B | C) - H(AB|C).
$$
I know that ...
5
votes
2
answers
211
views
Clarification on Watrous' proof of Alberti's theorem on the fidelity function
I am reading John Watrous' quantum information theory book. In the proof of Theorem 3.19 (practically the Alberti's theorem on the characterization of the fidelity function) he claims the following ...
5
votes
1
answer
345
views
How is the connection between Bures fidelity and quantum Fisher information derived?
I recently came to know that there is a connection between Bures Fidelity $(F_B)$ and Quantum Fisher Information $(F_Q)$ given by
$$[F_{B}(\rho, \rho_\theta)]^2 = 1 - \frac{\theta^2}{4} F_Q[\rho, A] + ...
5
votes
1
answer
395
views
Why does the bit flip channel produce a uniform contraction of $1-2p$?
Studying the bit flip channel using the Nielsen & Chuang's.
And ran into the picture with the caption stating $yz$ plane is uniformly contracted by a factor of $1-2p$. I don't quite understand how ...
5
votes
1
answer
82
views
What is the quantum analogue of $P_{XY} = P_{Y|X}P_X$
A standard trick in probability manipulation is to take some joint distribution $P_{XY}$ and express it as $P_{Y|X}P_X$. This trick is useful because when one looks at things like the ratio of $\frac{...
5
votes
1
answer
200
views
Is the composition of two extremal channels also extremal?
In this question, I follow the terminology and notation of the book of Watrous, most notably chapter two.
Extremal channels
An extremal channel $\Phi(X) \in C(\mathcal{X},\mathcal{Y})$ is a channel ...
5
votes
1
answer
612
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\...