Skip to main content

Questions tagged [quantum-information]

This tag is related to Quantum information Theory.

Filter by
Sorted by
Tagged with
-1 votes
0 answers
12 views

Inequality for square root of partial trace

Some non-trivial problem involving partial trace, any help is appreciated! Consider a system consisting of subsystems 1 and 2, and ${\rm Tr}_1$ (${\rm Tr}_2$) is a partial trace over subsystem 1 (2), ...
quirksman's user avatar
0 votes
0 answers
19 views

The trace distance between two gaussian states [closed]

If I know two gaussian states' means and covariance matrixes (both of them are two-mode gaussian states), how can I compute the trace distance between them?
John James's user avatar
1 vote
2 answers
37 views

Why does block coding via typical strings give messages longer than $nH(p)$?

This semester, I am taking a course on quantum information and quantum computing. Since I am rather new to information theory I have a problem with understanding a paragraph in my lecture notes. The ...
luki luk's user avatar
  • 139
0 votes
0 answers
18 views

Queues wait for other queues: A communication problem

I am working on a problem which involves a single server that requires multiple inputs to do a computation. Each of these inputs arrive as a Poisson process with rate $\lambda$. Hence, a situation ...
Ishan's user avatar
  • 26
2 votes
1 answer
60 views

Integration by substitution in Selberg's Integral

I am reading the article "Hilbert--Schmidt volume of the set of mixed quantum states" (https://arxiv.org/abs/quant-ph/0302197). I do not understand the step in which we start from (4.2) and ...
Zsombor's user avatar
  • 730
0 votes
0 answers
45 views

Isometric maps between spaces with different dimensions

I have the following problem: Consider complex Euclidean spaces H and H′ such that $dim(H) ⩽ dim(H′)$. Let ${|v_i⟩}^{k}_{i=1} ⊂ H$ and ${|w_i⟩}^{k}_{i=1} ⊂ H′$ denote orthonormal sets of vectors. Show ...
Pink Elephants's user avatar
2 votes
1 answer
39 views

Sum of entangled operators is entangled

Let $\mathcal{H},\mathcal{K}$ be Hilbert spaces ($\cong \mathbb{C}^2$ if necessary) and let $h_1,h_2$ denote semi-positive $\ge 0$ operators on $\mathcal{H}\otimes \mathcal{K}$. If $h_1,h_2$ are ...
Andrew Yuan's user avatar
  • 2,900
1 vote
1 answer
209 views

Range is product then so must the operator

Consider an operator $\eta$ on the (tensor) product of Hilbert spaces $\mathcal{H}_1\otimes \mathcal{H}_2$ such that its range is of the form $V_1\otimes V_2$. Does this imply that $\eta$ must be a ...
Andrew Yuan's user avatar
  • 2,900
0 votes
0 answers
27 views

Different ways of defining Entanglement

Consider (finite-dim) Hilbert spaces $\mathcal{H},\mathcal{K}$ and let $A$ denote an operator on $\mathcal{H}\otimes \mathcal{K}$. There are then two ways of defining the ``entanglement" of ...
Andrew Yuan's user avatar
  • 2,900
2 votes
1 answer
34 views

A proof of no-cloning theorem in the case of pures states in a qubit system

I am working on this problem Consider a qubit $\scr H =\Bbb C^2$ and pure states. Prove the no-clonning theorem ( hint: Use the linearity of the channel to arrive to a contradiction) I wonder if the ...
darkside's user avatar
  • 637
1 vote
1 answer
36 views

If $\rho_{AB}$ is a separable then the partial transpose w.r.t to A is PSD

Def: The partial transpose of a linear operator $\rho_{AB}$ over a Hilbert space $H_A \otimes H_B$ w.r.t A is defined for a linear operator $\rho_{AB}=\rho_A \otimes\rho_B$ as $\rho^{T_A}_{AB}=\rho_A^...
some_math_guy's user avatar
1 vote
1 answer
25 views

Noisy Quantum state exercise regarding trace of matrix.

Prove the following equality: $$\text{Tr}\{A\}= \langle \Gamma \vert_{RS} I_R \otimes A_S \vert \Gamma \rangle_{RS}.$$ Where $A$ is a square operator acting on a Hilbert space $\mathbf{H}_S$, $I_R$ is ...
homosapien's user avatar
  • 4,173
1 vote
1 answer
35 views

Writing a positive semi-definite operator as a non-negative combination of rank-1 projectors with the same 'phase profile'

Let $e_1, \ldots, e_D$ be a given orthonormal basis for $\mathbb{C}^D$. We define the phase profile of a vector $u \in \mathbb{C}^D$ to be $( \text{arg} (e_1^* u), \ldots \text{arg} (e_D^* u) )$. We ...
Mohammad Alhejji's user avatar
2 votes
1 answer
53 views

If $ρ_{AB} ∈ D(H_A \otimes H_B)$ such that $ρ_{A}$ is pure. $\implies ρ_{AB} = ρ_{A} \otimes ρ_{B}$

Let $H_A, H_B, H_C$ be arbitrary Hilbert spaces. Let $ρ_{AB} ∈ D(H_A \otimes H_B)$ such that $ρ_{A}$ is pure. Prove that $ρ_{AB} = ρ_{A} \otimes ρ_{B}$ ( Hint: One way could be to prove it before for ...
darkside's user avatar
  • 637
1 vote
1 answer
63 views

Prove that a change-of-basis map is an isometry between complex hilbert spaces, to prove uniqueness of purifications

I am trying to prove the following For this I should use the following fact: Let $ρ_A = \sum_{i=1}^r p_i|e_i⟩⟨e_i|$, where $p_i$ are the nonzero eigenvalues of $ρ_A$ and $|e_i⟩$ corresponding ...
some_math_guy's user avatar
0 votes
1 answer
50 views

trouble with a change of basis

I have two O.N sets $\{|e_i\rangle\}_{i=1}^r$ and $\{|\tilde e_i\rangle\}_{i=1}^r$ Then there is gotta be a change of basis matrix C, such that $|\tilde e_i\rangle = \sum_{j=1}^rc_{j,i}|e_j\rangle$ I ...
some_math_guy's user avatar
1 vote
2 answers
78 views

Show that a state $\rho=\sum_i p_i|e_i\rangle\!\langle e_i|$ has purifications of the form $\sum_i s_i |e_i\rangle\otimes|f_i\rangle$

Let $ρ_A = \sum_{i=1}^r p_i|e_i⟩⟨e_i|$, where $p_i$ are the nonzero eigenvalues of $ρ_A$ and $|e_i⟩$ corresponding orthonormal eigenvectors. If some eigenvalue appears more than once then this ...
some_math_guy's user avatar
1 vote
1 answer
62 views

Find an ensemble consisting of three distinct pure states that correspond to the same mixed state

I am trying to solve part b) of the following problem. I solved part a here e Compute a quantum state $\rho$ corresponding to an ensemble Consider the ensemble consisting of the qubit states $|0⟩⟨0|$ ...
darkside's user avatar
  • 637
1 vote
0 answers
57 views

Compute a quantum state $\rho$ corresponding to an ensemble

I am trying to solve the following problem Consider the ensemble consisting of the qubit states $|0⟩⟨0|$ and $|1⟩⟨1|$ occuring with probabilities 2/3 and 1/3, respectively. Compute the quantum state $\...
darkside's user avatar
  • 637
1 vote
1 answer
54 views

Schmidt decomposition in larger Hilbert spaces

Consider a bipartite quantum system described by the density operator, $\hat{\rho}$, an operator acting on the Hilbert space $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. This matrix will be a ...
Oscarcillo's user avatar
1 vote
1 answer
67 views

Outer product as an operator in an infinite Hilbert space

The outer product between a bra-ket $|a\rangle\langle a|$ where if $|a\rangle\in\mathcal{H}$ and $\langle a|\in\mathcal{H}_{dual}$ is a vector in the tensor vector space formed by the Hilbert space ...
Oscarcillo's user avatar
0 votes
1 answer
53 views

Going between two sets of vectors in $\mathbb{C}^d$

Let $|\psi_1 \rangle, \dots, |\psi_n \rangle, |\phi_1 \rangle, \dots, |\phi_n\rangle \in \mathbb{C}^m$. Show that if $\langle \psi_i | \psi_j \rangle = \langle \phi_i | \phi_j \rangle$, then there ...
Eulerian's user avatar
  • 109
0 votes
1 answer
106 views

Properties of a Product of Commuting Projections

Let $\Pi_1$ and $\Pi_2$ be two positive-semidefinite projections which commute, and let $\Pi=\Pi_1\Pi_2=\Pi_2\Pi_1$. Are the following statements correct? $\quad\Pi\:\preceq\:\Pi_1\,$ and $\,\Pi\:\...
Nick Cooper's user avatar
1 vote
1 answer
55 views

Change of basis of a matrix - what am i doing wrong?

I want to change the basis from: $$(|00\rangle,|01\rangle,|10\rangle,|11\rangle)$$ to $$(|00\rangle,|u_2\rangle,|u_3\rangle,|11\rangle)$$ , where $|u_2\rangle, |u_3\rangle = \frac{1}{\sqrt{2}}(|01\...
Kobamschitzo's user avatar
0 votes
0 answers
11 views

Concentration around $L_2$ average

An exercise from Aubrun-Szarek book: Alice and Bob Meet Banach The Interface of Asymptotice Geometric Analysis and Quantum Information Theory: I want to know how to prove this: Exercise 5.46 - Let $f$ ...
Dan David's user avatar
2 votes
1 answer
289 views

How to find expectation value $p_y$ from the Bloch sphere?

Consider an arbitrary state: $$|\psi\rangle = a|0\rangle+b|1\rangle,$$ where $a=cos\left(\frac{\theta}{2}\right), b=sin\left(\frac{\theta}{2}\right)e^{i\phi}$ (neglecting global phase), $\phi$ is the ...
Curious's user avatar
  • 105
1 vote
0 answers
35 views

Example of statistical data with the property of being contextual that is generated by quantum mechanics

Definition for the specifics of the question as well as an example of contextuality in quantum mechanics I have a set of measurements acting on a 2 qubit state for whom the statistics of the ...
TheStressTensor's user avatar
1 vote
1 answer
87 views

Computing expectation value of Pauli-$x$ operator for a single qubit, given a Hamiltonian matrix.

I am working through some introductory quantum mechanics materials, and I am stuck on the following problem... Suppose we are working on a single qubit and we are given the Hamiltonian to be $$H=\...
2307's user avatar
  • 334
0 votes
0 answers
39 views

How to map a Bitwise inner product in finite space to an inner product in a Hilbert space.

Consider the vector space $\mathbb{Z}_2^n$ over the finite field. For vectors $a,b\in \mathbb{Z}_2^n$, we define the bitwise inner product $a\cdot b$ as follows: $$a \cdot b = \sum_{i=1}^{n} a_ib_i (\...
Calpis 50's user avatar
0 votes
1 answer
78 views

How to diagonalize an infinte dimensional operator

I want to take logarithm of an infinite dimensional operator given by $\rho = \int\int dx_1 dx_1'C(x_1,x_2)C^*(x_1',x_2)|x_1\rangle\langle x_1' |$, where $C(x_1,x_2)$ is a gaussian function in $x_1$ ...
QuantumOscillator's user avatar
2 votes
1 answer
56 views

Predual of von Neumann algebra M is closed in the dual $M*$ [closed]

I am reading Stephane Attals Book Open Quantum Systems 1, chapter Elements of Operator Algebras and Modular Theory. And have issues understanding the proof given here. First of all in the "...
y4nik's user avatar
  • 193
1 vote
0 answers
83 views

about a quantum circuit

Consider the following circuit : where $|\psi\rangle$ is a qubit in $\mathbb{C}^2$, $|0\rangle= \begin{pmatrix}1 \\ 0 \end{pmatrix}$, $T= \begin{pmatrix}1 & 0\\ 0 & e^{i\pi/4} \end{pmatrix}$ ...
NotaChoice's user avatar
0 votes
0 answers
25 views

quantum guesswork of an ensemble

I am reading an article about quantum guesswork [ Guesswork of a quantum ensemble by michele Dall'Arno ] and i am wondering why he considers ensembles whose traces sum up to 1 in the introduction. I ...
yosh's user avatar
  • 73
0 votes
0 answers
48 views

Fidelity as bregman divergence

Fidelity is defined as follows: $$F(\rho, \sigma) = \left( \operatorname{tr} \sqrt{\sqrt{\rho} \sigma \sqrt{\rho}} \right)^2$$ A Bregman divergence $D_B$ is defined as: $$D_B(\rho||\sigma) = f(\rho) - ...
Assaf Katz's user avatar
0 votes
0 answers
58 views

What does "coarse" mean in the context of probability bounds and Markov's inequality?

In John Watrous's The Theory of Quantum Information the author introduces Markov's inequality as follows: The first theorem to be stated in this subsection is Markov’s inequality, which provides a ...
latin-noob's user avatar
3 votes
1 answer
150 views

Decomposing the unitary Haar measure as product of unit vector Haar measures

Let $\mu_{D}(U)$ be the Haar measure on the D-dimensional Unitary group $U(D)$, where $U \in \mathrm{SU}(D)$ or $U(D)$. Can we think of this measure as picking first a unit vector according to the ...
Soham Ghosh's user avatar
1 vote
1 answer
67 views

Why should the symmetric logarithmic derivative be linear?

I recently learned about the symmetric logarithmic derivative $L_\rho(A)$, defined implicitly by $$i\left[\rho,A\right]=\frac 1 2\left\{\rho,L_\rho(A)\right\}.$$ Here $\rho$ is a Hermitian, positive ...
AccidentalTaylorExpansion's user avatar
-1 votes
1 answer
40 views

Conditions for inverse of a matrix [closed]

Matrix inverse and its properties The matrix M = ∑piσi ⊗ σi∗ where pi is probability and σi is Pauli matrix has an inverse(its determinant det M is not equal to 0), i.e. the following three conditions ...
Mushahid Khan's user avatar
2 votes
1 answer
43 views

2nd order Commutant of permutation matrices

Let $d\ge 2$ denote an integer, let $S_d$ denote the group of permutations of $d$ elements, and let $L(\mathbf{C}^d)$ denote the space of linear operators on the Hilbert space of $d$-dimensional ...
ffff's user avatar
  • 41
0 votes
2 answers
47 views

Quantum mechanics [closed]

When we want to calculate the probability of finding a particle in a small region of space given the time-dependent Schrödinger equation, it should be equal to $\vert\psi(x)\vert^2$ times $dx$ times ...
ayah Eyad's user avatar
0 votes
1 answer
45 views

Are two linear mappings $M$ and $N$ on $\mathcal{S}(H_A \otimes H_B)$ equal if they have same behaviour on product states? [closed]

Let $M$ and $N$ be mappings from $\mathcal{S}(H_A \otimes H_B)$ to itself, where $\mathcal{S}(H)$ denotes the set of density operators over the Hilbert space $H$. If the following two conditions hold: ...
vfx01's user avatar
  • 55
2 votes
0 answers
60 views

Prove that $\xi (\rho) = tr_{env} \left[ U (\rho \otimes \rho_{env}) U^{\dagger}\right] = \sum_{k} E_{k}\rho E_{k}^{\dagger}$

In section 8.2.3 of Nielsen and Chuang, there is a derivation of the operator-sum representation as follows: $\xi (\rho) = tr_{env} \left[ U (\rho \otimes \rho_{env}) U^{\dagger}\right] \tag{1}$ And $\...
JiQing's user avatar
  • 21
0 votes
0 answers
131 views

Exercise 4.9 Nielsen and Chuang's "Quantum Computation and Quantum Information

I am trying to work through an online solution to Exercise 4.9 in Nielsen and Chuang's "Quantum Computation and Quantum Information: The question is: Explain why any single qubit unitary operator ...
am567's user avatar
  • 309
0 votes
1 answer
129 views

A Property of the Trace-Norm

Let, $\mathcal{E}$ be a Completely-Positive Trace-Preserving Map, i.e, for linear operators $\rho$ on a Hilbert-Space, $$ \mathcal{E}(\rho) = \sum_i E_i\rho E_i^{\dagger}\qquad {\rm s.t.}\qquad \sum_i ...
Sowmitra Das's user avatar
1 vote
0 answers
93 views

How quickly is a Lie algebra generated using commutators?

Let $\mathcal{S}=\{ s_i,\dots,s_n \}$ be a set of elements of the matrix Lie algebra $su(d)$, and let $\mathcal{L}(\mathcal{S})$ be the Lie subalgebra of $su(d)$ that is obtained by closing $\mathcal{...
David T's user avatar
  • 377
0 votes
1 answer
32 views

Proof that adjoint is equal to complex conjugate in a quantum information theory viewpoint

† While trying to prove that $$(A+B)^†=A^†+B^†$$ I have stumbled accross a self proof that seems to validly suggest that $A^† = A^*$ This intuitively seems false but I cannot find where in my proof my ...
Matthew Ediz Beadman's user avatar
0 votes
0 answers
121 views

About the kernel/null space of a quantum channel as a linear map

Consider the CPTP map $\Phi: M_n(\mathbb{C})\rightarrow M_n(\mathbb{C})$ that is not injective, which means the null space of $\Phi$ is non-trivial. My main question is, what can we say about the ...
Thinkpad's user avatar
  • 383
1 vote
2 answers
105 views

About density matrices as both the vertices of a convex hull and the basis of a matrix space

Given $n$ density matrices $D_1, \dots, D_n$, that is, $D_i$ is positive semi-definite and $\operatorname{Tr}(D_i)=1$ for all $1\leq i\leq n$. Suppose that $D_1, \dots, D_n$ are linearly independent. ...
Thinkpad's user avatar
  • 383
0 votes
0 answers
24 views

Question about the toric code

I am reading this paper on toric code for my research and has some questions about it. $\sigma_j^Z$ appears in the equation for $A_S$ (last but five line) and in the last but four line, it’s defined ...
Coco's user avatar
  • 623
0 votes
2 answers
118 views

Dilation for several POVMs

Let $E$, $F$ be two finite sets. Let $\mathcal{H}$ be a finite-dimensional Hilbert space, and let $(A^{f}_e)_{e,f}$ be a family of positive operators on $\mathcal{H}$ such that for all $e$, $\sum_{f} ...
Plop's user avatar
  • 2,647

1
2 3 4 5 6