Questions tagged [information-theory]

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

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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 ...
tparker's user avatar
  • 2,711
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{...
Josu Etxezarreta Martinez's user avatar
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 ...
XXDD's user avatar
  • 333
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\...
Maximal Ideal's user avatar
11 votes
2 answers
2k views

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 \\\...
Martin Vesely's user avatar
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{...
Stephen Diadamo's user avatar
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 ...
Josu Etxezarreta Martinez's user avatar
9 votes
5 answers
2k 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 ...
KamWoh Ng's user avatar
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 ...
Mahathi Vempati's user avatar
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 $\...
john_smith's user avatar
7 votes
2 answers
758 views

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 ...
Mahathi Vempati's user avatar
7 votes
3 answers
3k views

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 ...
Mahathi Vempati's user avatar
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: $$ ...
the mmmPodcast's user avatar
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 ...
Sanchayan Dutta's user avatar
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 ...
Nikita Nemkov's user avatar
7 votes
1 answer
1k 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 ...
Nick ODell's user avatar
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 ...
Confinement's user avatar
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 ...
raycosine's user avatar
  • 860
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) -...
Tobias Fritzn's user avatar
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 $\sigma$ ...
m1rohit's user avatar
  • 93
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 $2^n$ ...
Radvylf Programs's user avatar
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} = \...
nervxxx's user avatar
  • 520
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 \...
ahelwer's user avatar
  • 4,108
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 $|U-V|=\sqrt{(U-V)^\dagger (U-V)}$...
Apo's user avatar
  • 545
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_{...
John Doe's user avatar
  • 821
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 $\rho^{AB}$: ...
Ilya's user avatar
  • 163
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 ...
GaussStrife's user avatar
  • 1,115
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 ...
Henry's user avatar
  • 117
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 ...
user1936752's user avatar
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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 ...
user120404's user avatar
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 $\...
user1936752's user avatar
  • 2,835
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 ...
Hari krishnan S V's user avatar
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=|\...
Balter 90s's user avatar
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 $\...
user1936752's user avatar
  • 2,835
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 $...
Woka's user avatar
  • 85
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\...
Johny Dow's user avatar
  • 157
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 ...
Anna Naden's user avatar
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 ...
Mahathi Vempati's user avatar
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 ...
John Doe's user avatar
  • 821
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. ...
KAJ226's user avatar
  • 13.8k
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 ...
Confinement's user avatar
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 ...
develarist's user avatar
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. $...
Jon Megan's user avatar
  • 497
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 ...
QuestionEverything's user avatar
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 ...
adabb's user avatar
  • 71
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] + ...
Mike's user avatar
  • 191
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 ...
John Parker's user avatar
  • 1,071
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{...
user1936752's user avatar
  • 2,835
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 ...
JSdJ's user avatar
  • 5,419
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\...
user1936752's user avatar
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