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Questions tagged [quantum-information]

This tag is related to Quantum information Theory.

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1answer
30 views

How to prove Shannon entropy inequality with something that seems to be some sort of taylor expansion

I'm slightly confused about some sort of "proof" (probably not a real proof since it's physics math) I have the formula $f(x) = f(y) + (x-y)f'(y) + \frac{f''(y)}{2\epsilon}, \quad \epsilon \in (x,y)$...
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0answers
18 views

Prove that the operator of the Grover's algorithm 'inverts about the mean'

I'm trying to solve the following problems that I have found in the book An Introduction to Quantum Computing by Phillip Kaye, but I don't know exactly where should I start. I would appreciate any ...
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0answers
14 views

Inequality between independence numbers(one-shot zero-error capacity)

I had a question when I was studying the Entanglement-Assisted Independence Number $\tilde{\alpha}(S)$, where $S$ is a operator system. In the inserted figure, the high-lighted sentence said that the ...
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0answers
47 views

Help with Inductive Proof - Grover Search

I have a question about an inductive proof. First of all, let me explain the problem. The algorithm starts in $|\psi\rangle$ and applies $O_x$ $k$-times, with some unitary operators. We now define: $...
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1answer
27 views

Why can input states $\rho_m$ be chosen to be rank-one operators in Zero-error communication?

I have one problem in Quantum Information Theory, specially in Zero-error communication. I am reading a paper : Duan, Runyao; Severini, Simone; Winter, Andreas, Zero-error communication via quantum ...
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0answers
89 views

Quantum Info in math departments

I'm currently a graduate student in mathematics getting my master's degree. I am interested in a bit probability and partial differential equations, and, secondarily, a bit of mathematical physics; ...
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1answer
35 views

A exercise with Dirac notation

I am dealing with the above exercise and not sure about my solution is correct or not, It is possible for me to find another state that the payoff would be larger than 50? Or it is possible for me to ...
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1answer
61 views

Conditional probability given only an average

I’ve been working on this question, which I found on physics.SE. Unfortunately it was closed because it’s a homework question, but I’d like to get more of a hint than the original poster got. I know ...
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1answer
58 views

Clarification on notation from a probabilty/quantum mechanics question

I'm working through Stephen Barnett's book on quantum information and have come across the following question (1.5, for anyone keeping track at home) A particle counter records counts with an ...
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1answer
31 views

Partial trace and linear operators

Trying to solve the following: Consider a linear map as follows $$vec: L(X,Y)\rightarrow Y \otimes X $$ $$ vec: E_{b,a} \mapsto e_b \otimes e_{a}$$ which can be looked as a change of a basis map. ...
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1answer
30 views

Linear Operator to vector corrrespondanc through change of basis

Consider a linear map as follows $$vec: L(X,Y): Y \otimes X $$ $$ vec: E_{b,a} \mapsto e_b \otimes e_{a}$$ which can be looked as a change of a basis map. Where $E_{b,a}$ is usual basis for $L(X,Y)$ ...
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2answers
168 views

What is meant by “symplectic Fourier transform”?

I've recently come across the term symplectic Fourier transform (see this paper, first page, second column), but googling didn't lead me to any satisfactory explanation of what is meant with this term....
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1answer
46 views

Partial trace of action on density matrix

This topic is from chapter 11 of Kitaev, Shen and Vyalyi's Classical and Quantum Computation. If $V$ is an isometric (i.e. preserves the inner product) embedding $V: \mathcal{N} \rightarrow \mathcal{N}...
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0answers
70 views

Continuity of matrix trace functionals

Let $P$ be a positive $n\times n$ matrix over $\mathbb{C}$, let $f:(0,\mathbb{R})\rightarrow\mathbb{R}$ be a continuous function, and define a function on positive definite $n\times n$ matrices as $$ ...
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0answers
78 views

Simplify bra-ket notation with kronecker product and kronecker sum

I am taking a quantum informatics and communication course, this is the first time I have faced with Dirac's Bra-ket notation. I have the following equation(Swap gate with 3 cnot): First equation $|...
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2answers
302 views

Linear combination of Pauli matrices and projectors

Premise: this is exercise 2.60 of Quantum Computation and Quantum Information, by Nielsen and Chuang, where I'm currently stuck. Suppose $\vec{v}$ is any real three-dimensional unit vector, and $\...
2
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1answer
125 views

Prove that the Schmidt number of a state is equal to the rank of the reduced density matrix

Suppose $|\psi\rangle$ is a pure state of a composite system with components $A$ and $B.$ Prove that the Schmidt number of $|\psi\rangle$ is equal to the rank of the reduced density matrix $\...
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0answers
47 views

Correlations in classical-quantum states

Suppose that $X$ and $Y$ are finite-dimensional Hilbert spaces with $\mathrm{dim}(X)=n$ and $\mathrm{dim}(Y)=m$. Let $\rho$ be a density operator on $X \otimes Y$ such that the reduced density ...
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1answer
43 views

Image of a sum of positive operators contains the images of each individual operator?

In the proof of Proposition 2.52 here: https://cs.uwaterloo.ca/~watrous/TQI/TQI.2.pdf, there is the statement that $\text{im}(\eta(a))\subset\text{im}(\rho)$, where $\rho=\sum_{i=1}^{N}\eta(i)$ is ...
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0answers
112 views

Partition function inequality for Gibbs states associated with graphs

Suppose I have two undirected graphs $G_1$ and $G_2$ with the same vertex set $V$ and let $A_1$ and $A_2$ denote their respective adjacency matrices. Define the intersection of the two graphs $G_\cap$ ...
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2answers
51 views

Summation over set notation

I am reading some lecture notes on quantum computing and ran into the following expression that I cannot seem to understand: For example, if we apply the Hadamard gate $H$ to each bit in a register ...
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1answer
488 views

Bitwise inner product and orthogonality

I am confused about the definition of bitwise inner product used in quantum algorithms. For example, bitwise inner product of 01111 with itself (in mod2) gives us 0. But they are not orthogonal to ...
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0answers
89 views

Lower bound for quantum relative entropy

In my research this summer, I have become interested in lower bounds on the standard "Umegaki quantum relative entropy". For two non-negative matrices $X$ and $Y$, the Umegaki quantum relative ...
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1answer
70 views

Pure states on subalgebras of $\mathcal{B}(\mathcal{H})$ in finite dimensions.

I consider only finite-dimensional Hilbert spaces. We know that pure states on $\mathcal{B}(\mathcal{H})$ are exactly the vector states or in terms on density matrices, the rank one projections. My ...
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0answers
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How can one define the quantum interferometric power in the case of a multiparametric system using the quantum Fisher information matrix?

please, was the quantum interferometric power defined in the case of a multiparametric system? I know that in the case of a single parameter, the quantum interferometric power is defined as the ...
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2answers
328 views

How do unitary matrices preserve the magnitude of unit vectors?

In quantum mechanics and quantum computing, quantum particles evolve in a unitary manner. That is to say at any point in time the particle/system (represented as a vector) has a magnitude of 1, ...
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1answer
82 views

Intuition for Kitaev's geometrical lemma

Background Kitaev's geometrical lemma is a linear algebraic result which is very useful within Hamiltonian complexity theory. Under certain conditions, it allows one to bound the spectrum of the sum ...
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2answers
152 views

The Kraus representation of a completely depolarising channel

I have an answer, but I want to know how to get from the left hand size to the right hand side: $$ \frac{1}{d^2} \sum_{i=1}^{d^2} U_i \rho U_i^{\dagger} = \operatorname{Tr}[\rho] \frac{I}{d}. $$ ...
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234 views

What is the essential difference between classical and quantum information geometry?

This question may be a little subjective, but I would like to understand, from a geometric perspective, how the structure of quantum theory differs from that of classical probability theory. I have a ...
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1answer
50 views

Problem in quantum information theory

I am reading the following paper: on quantum information theory Does anybody understand the estimate at the bottom of page $7$ $$\left\lVert \rho-\rho_n \right\rVert \le 2 \left\lVert (id_A-P_n)\...
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1answer
41 views

Difference between operator defined on a space and operator represented in a space

I am quite confused about linear operators that are defined as acting on a Hilbert space $\mathcal{H}$ and their representations. The operators form an operator algebra and as such can be represented ...
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2answers
147 views

Does $\mathcal{B}(\mathcal{H})=H\otimes H^*$ in infinite dimensions?

In quantum information contexts (where in most cases finite dimensions are considered) I have often seen the statement that the space of bounded linear operators $\mathcal{B}(\mathcal{H})$ on a ...
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1answer
47 views

Question concerning Stirling’s Approximation

From Nielsen & Chuang, page 55: The basic intuition for this decrease in resources required can be understood quite easily. Suppose the source emitting states $|0\rangle$ with probability $p$ ...
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1answer
108 views

Characterizing families of $p^2$ orthogonal $p \times p$ unitaries?

Suppose I walk up with $2$-by-$2$ unitary matrices $U_1,U_2,U_3,U_4 \in \mathbb{C}^{2 \times 2}$ with the property that $\mathrm{Tr}(U_i U_j^\dagger) = 0$ for all $i \neq j$. One can show that there ...
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0answers
84 views

Characterize set of matrices that are orthogonal in two particular senses

Does there exist an analytical characterization of the set of matrices $\Gamma_k\in\mathbb{R}^{m\times n}$ such that both $$ \sum_{k=1}^K\Gamma_k^T\Lambda_0\Gamma_k=I $$ and $$ \sum_{k=1}^K\Gamma_k\...
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0answers
32 views

Optimal measurement by projectors for density operators!

Theorem: Let $\{\rho_i,1\leq i\leq m\}$ be a quantum state ensemble consisting of linearly independent density operators $\rho_i$ with prior probabilities $p_i$. Then the optimal measurement is a von ...
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206 views

How to optimally distinguish between linearly independent vectors in higher dimensional complex/real space?

I am to distinguish between 4 linearly independent vectors belonging to $\mathbb{C^{16}}$ space by creating a set of Positive Operator Valued Measurements (POVM) that will act on these vectors. I have ...
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1answer
33 views

Notation on quantum information notes

I'm currently reading An Introduction to Quantum Computing, Without the Physics and on page 10 the following notation is used repeatedly $$\sum_{j:(jB_q)_{k}=0} $$ or $$\sum_{j:(jB_q)_{k}=1} $$ $...
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1answer
66 views

Derivative of fidelity with respect to time

Consider the quantum fidelity between two states defined as $$ F(\rho(t),\sigma(t)):=\text{Tr}\left(\sqrt{\sqrt{\rho(t)}\sigma(t)\sqrt{\rho(t)}}\right)^2 $$ Does $dF/dt$ have a closed form equation?
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2answers
48 views

Equivalence of matrices and operators

I'm super new to studying functional analysis and I'm currently reading https://arxiv.org/pdf/1410.7188.pdf (The Functional Analysis of Quantum Information Theory based on lectures by Paulsen, Pisier,...
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74 views

Is entropy being maximized and the norm being minimized by the same (unique) probability vector somehow related?

Consider the set of probability vectors $$ \mathcal P_n=\Big\lbrace x\in[0,1]^n\,\Big|\,\sum_{i=1}^n x_i=1\Big\rbrace\subset\mathbb C^n $$ for any $n\in\mathbb N$ where $x_i$ is the $i$-th component ...
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1answer
606 views

Why does this solution to the Bernstein-Vazirani problem use $ (-1)^{f(x)} $

I am reading a solution to the Bernstein-Vazirani problem. For those unaware, the issue is to find a randomly selected $ 0 \leq a \lt 2^n $ given only a function $ f(x) = a_0x_0 \oplus a_1x_1 \oplus .....
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0answers
38 views

Question on the outer product used to construct this 2-Qbit state

I am reading a book on quantum computing. The author is constructing an arbitrary 2-Qbit state from unitary transformations. I need help understanding on step in his logic. For those unfamiliar with ...
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2answers
78 views

What does the following equation say about the conditional distribution $P(ab|xy)$?

What does the equation below say about the conditional distribution $P(ab|xy)$? $$ P(ab|xy) = \int{ d\lambda \ \rho(\lambda)\ P(a|x\lambda)\ P(b|y\lambda) }$$ What does $\lambda$ denote here? Could ...
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0answers
42 views

Existence of an ensemble of pure states for a prescribed density matrix

I managed to prove (a), but am stuck at (b). Given a density matrix, I can't find a way to construct an ensemble of pure states for the matrix. I think I may have to diagonalize $\rho$.... Could ...
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1answer
291 views

What are the eigenvalues and eigenvectors of the Quantum Fourier Transform?

In general, $\operatorname{QFT}_m$ is used to denote the QFT defined on basis states $|0\rangle, |1\rangle, \dotsc, |m-1\rangle$ according to $$ \operatorname{QFT}_m \colon |x\rangle ...
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1answer
123 views

an example of stochastic matrix not derived from a unitary matrix

I solved (a) but cannot find an example for (b). Could anyone show me a stochastic matrix not derived from a unitary matrix?
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0answers
40 views

How to compute the average value of an observable

This is from the p.116 of Nielsen and Chuang's quantum computation and quantum information book. The (2.229) equations are the ones I can't see how to calculate. In (2.227) and (2.228), the index 1 ...
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1answer
113 views

solving “Cascaded measurements are single measurements”

This is an exercise from p. 86 Quantum Computation and Quantum Information by Michael A. Nielsen and Isaac L. Chuang. Here I think I have to show that $p\left(l\right)p\left(m\right)=p\left(lm\right)$...
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1answer
28 views

what does it mean by “relabelling of the measurement outcomes”

This is an exercise from p.52 of An Introduction to Quantum Computing by P. Kaye, R. Laflamme and M. Mosca. The oberservable Z denotes Pauli Z matrix. What does it mean by relabelling of the ...