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Questions tagged [quantum-computation]

Quantum Computation deals with considering computation as fundamentally physical, as well as replacing the classical binary digit (bit) with the quantum binary digit (qubit). While the classical bit is either 0 or 1, the qubit can be in a superposition of these states. Computation systems that use quantum phenomena, such as superposition and entanglement, can solve certain complex problems very quickly.

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How to lower bound the quantum conditional entropy?

I am trying to lower bound the quantum conditional entropy $H(X|Y)$ when $X$ and $Y$ are two quantum systems. Classically, it can be done as follows: $$ H(X|Y) = \sum_{y}P_Y(y) H(X|Y=y) \geq \sum_{Y \...
Jaswanthi Mandalapu ee19d700's user avatar
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Is the decomposition $|0\rangle\langle0| \otimes \rho =( |0\rangle\otimes I_n)( I_A\otimes \rho)( \langle0|\otimes I_n) $ correct?

$ %\newcommand{\ketbra}[2]{\mathinner{|{#1}\rangle\,\langle{#2}|}} \newcommand{\ketbra}[1]{\mathinner{|{#1}\rangle\,\langle{#1}|}} \newcommand{\bra}[1]{\langle{#1}|} \newcommand{\ket}[1]{|{#1}\rangle}$...
Coco's user avatar
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If $U $ is a unitary linear operator, how can I show that any matrix representation of $U$ must be a unitary matrix?

Nielsen / Chuang "Quantum Computation and Quantum Information" states on p. 70: "A matrix $ U$ is said to be unitary if $U^\dagger U = I$. Similarly, an operator $U$ is unitary if $U^\...
mchk's user avatar
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Commutation relation between exponentials of Pauli matrices

Define $P_\phi := e^{-iP\phi}$, where $P$ is a Pauli matrix with some overall phase factor and $\phi\in[0,2\pi)$. It is claimed (see Page 1 of this paper) that if $P'P = -PP'$ i.e. we have two ...
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Some properties about the equation $P(z)\phi(z)=0$

Let $\phi(z)=\sum_{m=1}^\infty (\phi_{m}z^m-\psi_{-m}z^{-m})$ where $\phi_m, \psi_{-m}\in \mathbb{C}$. If we can find one polynomial $P(z)\in \mathbb{C}[z]$ such that $P(z)\phi(z)=0$, how can I get ...
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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 ...
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Approximating an unitary matrix to $\mathbb{Z}[i,1/\sqrt2]$ while keeping unitarity

In this paper (lemma 3) it is claimed that any column of a unitary matrix that has two entries at zero can be approximated to $\mathbb{Z}[i,1/\sqrt2]$ by solving the Diophantine equation $a^2 + b^2 + ...
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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 ...
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Why does a summation vanish for a constant $f(x)$ in the Deutsch–Jozsa algirthm?

In Quantum Computing: From Linear Algebra to Physical Realizations, pg. 103, it states: Let us consider the summation $$ \frac{1}{2^n}\sum_{x=0}^{2^n-1}(-1)^{x \cdot y} $$ with a fixed $y \in S_n$ ...
John Hippisley's user avatar
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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 ...
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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
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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 $|0⟩⟨0|$ ...
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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
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3 answers
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Why are $|0 ⟩$ and $|1 ⟩$ perpendicular to each other? [closed]

"The special states $|0 ⟩$ and $|1 ⟩$ are known as computational basis states, and form an orthonormal basis for this vector space." Orthonormal means that both the qubits are perpendicular ...
knightshadies's user avatar
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How to obtain the following equality: $EE' = (-1)^{a.b' + a'.b}E'E$

I am having difficulty understanding why: $$EE' = (-1)^{a.b' + a'.b}E'E$$ $E$ and $E'$ are error operators of the form: $$E = i^{\lambda} X(a)Z(b)$$ $$E'=i^{\lambda '} X(a')Z(b')$$ where $\lambda \in ...
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Size of an $\epsilon$-net on n-qubit unitaries

I am currently reading through Computational Quantum Entanglement paper and there is a following statement there in proof of Lemma 4.1 We then use that an $\eta$-net (this is an $\epsilon$-net with ...
Piotr Lewandowski's user avatar
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1 answer
291 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
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Good book on (Quantum) Complexity and Computability Theories to start learning the theorem $MIP^* = RE$ as an operator algebraist

I am looking for some greatest references that could help me understand the theorem $MIP^* = RE$ ($MIP*=RE$) step by step. The paper (The Connes Embedding Problem: A guided tour) covers various ...
Kadi Harouna Illia's user avatar
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1 answer
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The action of the error subgroups $S$ and $S'$ on the encoded space

From https://arxiv.org/pdf/quant-ph/9608006.pdf $E$ is the group of possible errors in $n$ qubits $S'$ is a subgroup of $E$ consisting of undetectable errors. These are errors $e$, in which ...
am567's user avatar
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0 answers
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Trace norm of matrices in SU(2)

This thesis states ... the $L^2$ or trace norm, [is] defined to be $||M|| = Tr \sqrt{M M^\dagger}$. If elements of SU(2) are associated with points on the surface of the 3-sphere, then this is simply ...
Chris Henson's user avatar
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1 answer
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search for specific value out of data set

I am looking for a function or formula how to determine the data set for a specific value (sum) out of a set of data in a table. The function should be the fastest possible search to achieve the ...
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1 answer
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How do we know that the quotient group $\bar{E} = E/Z(E)$ is an elementary abelian group

My question is: How do we know that the quotient group $\bar{E}=E/Z(E)$ is an elementary abelian group? Please find below some background information on the different relevant groups involved from ...
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Proof that you can solve the discrete logarithm problem given the period of a certain function.

Given $q$ a prime number , $a$ a primitive root modulo $q$ and $b=a^x \pmod q$. The discrete logarithm problem is to find $x$ (specifically the smallest positive integer $x$ for which the previous ...
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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
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1 answer
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Suppose $tr(A(P\otimes Q))\geq0$ for all semi-definite positive matrices P and Q, does it implies that A is semi-definite positive? [closed]

Suppose $\text{tr}(A(P\otimes Q))\geq0$ for all semi-definite positive matrices P and Q, does it imply that A is semi-definite positive? If it is not true, please provide some ideas on restricting $A$ ...
Unicode's user avatar
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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
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1 answer
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operation rules involving tensor product of perturbed wavefunctions

Let $i, j$ represent a wavefunction. In Bra - Ket notation, I have an expression like $\langle ij | g | i j \rangle $ which is also just $(\langle i | \otimes \langle j | ) | g | (|i \rangle \otimes |...
Mathematicing's user avatar
3 votes
1 answer
157 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
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1 answer
41 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
1 vote
1 answer
92 views

Existence of unitary operator affecting partial trace? [closed]

Let $\rho$ and $\phi$ be any two different density matrices on $H_A \otimes H_B$ such that $Tr_B (\rho) = Tr_B (\phi)$. Does there always exist a unitary $U$ on $H_A \otimes H_B$, $U\rho U^{\dagger} = ...
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0 answers
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inner product (Bra - Ket) involving projection operator

In quantum mechanics, the action of a projection operator $\hat{P}$ acting on a quantum mechanical system, prepared in a state $| \psi \rangle$, is described by the eigenvector equation $\hat{P} | \...
Mathematicing's user avatar
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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
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5 votes
0 answers
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Prove that set of matrices is dense in $U(2)$

Consider the group of matrices $B$ generated by taking products of the matrices \begin{equation} \rho_1 = \begin{pmatrix}\exp(-4\pi i/5) & 0\\ 0 & \exp(3\pi i/5)\end{pmatrix}\\ \rho_2 = \begin{...
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1 answer
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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
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Given a modulus p, two primitive roots and the two products of the modular exponentiation attained by swapping the exponents: find the exponents.

Given a prime $p$, two of its primitive roots $(h_1, h_2)$, $A_1=(h_1^xh_2^y) \mod p,$ $A_2=(h_1^yh_2^x) \mod p$, find $(x,y)$. How can it be solved? Example: $p=23$, $(h_1=2, h_2=3)$, $A_1=(2^x3^y) \...
Fabbrini's user avatar
3 votes
1 answer
102 views

Justification of bra ket notation for operators

Let ${\lvert e_1 \rangle, \ldots, \lvert e_m \rangle}$ be a basis for the first arbitary space $V$, and write $A = \sum_{i,j=1}^{m} a_{ij} \lvert e_i \rangle\langle e_j \rvert$ (the $(i,j)$-th element ...
Sarah's user avatar
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3 votes
2 answers
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What is the result of $\ell^1 \otimes \ell^1$?

I recently stumbled upon the concept of a tensor product while studying quantum computing, and felt that my understanding of it from preliminary readings was incomplete. I challenged myself to ...
kipawaa's user avatar
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"The value of y is handled as parameter of the function" What does it mean?

It's a statement from an optimization paper, where we have a random variable $X$ taking up values $\in \Omega_X = [0,7]$ and a decision variable $y \in \mathbb{R}^d$ , and a function $f: \Omega_X \...
Another Random Guy's user avatar
2 votes
0 answers
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Generalisation of Bloch-sphere rotations to higher dimensional Bloch-spheres

Background Our question concerns the generalisation of Bloch sphere rotations to higher-dimensional Bloch spheres. We note the connection between states of a Hilbert space represented on $S^2$ with ...
fintallrik's user avatar
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0 answers
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Can projective measurements of a qubit be written as conditional probabilities of a distribution on the sphere?

Let $\Psi :=p_1|\psi_1 \rangle \langle \psi_1 | + p_2|\psi_2 \rangle \langle \psi_2|$ be a density matrix/operator representing a (possibly mixed) qubit state. A projective measurement corresponds to ...
hasManyStupidQuestions's user avatar
1 vote
1 answer
108 views

Can a qutrit ($\mathbb{CP}^2$) be "simulated" by two qubits (two copies of $\mathbb{CP}^1$)?

Classical observation: Given a "trinoulli" random variable ("trit") $X$ with possible states $0, 1, 2$, there always exist two Bernoulli random variables ("bits") $Y_1, ...
hasManyStupidQuestions's user avatar
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0 answers
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Are Convex Linear Functions of Density Matrices Actually Linear Functions?

I feel like I often see passing comments about this without proof in quantum info literature, but I wasn't able to show it myself! For instance p17/p18 of this classic Lets say that we have some ...
Matt_Wilson's user avatar
1 vote
1 answer
86 views

Usefulness of a Hermitian Matrix

A Hermitian Matrix is defined as a self-adjoint matrix (so the complex conjugate transpose of M is equal to M). But the usefulness of Hermitian Matrices seems to be that they necessarily have ...
buzzword12's user avatar
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0 answers
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Derivative of $A^x$ w.r.t $x$ inside Trace Operation

In my reference Page 520, Entropy and information, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that The relative entropy $S(ρ||σ)$ is jointly convex in its ...
Sooraj S's user avatar
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1 vote
1 answer
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Mystery constant in an approximation of the number of stabilizer states

The number of n-qubit stabilizer states is: $$|\text{StabilizerStates(n)}| = 2^n \prod_{k=0}^{n-1} (2^{n-k}+1)$$ This is a bit cumbersome, so it's nicer to use this approximation that you get by ...
Craig Gidney's user avatar
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Validity of Commutation relations on Superoperators

Lieb's theorem states that, let $X$ be a matrix and $0\le t\le 1$ then the function $f(A,B)\equiv tr(X^\dagger A^tXB^{1-t})$ is jointly concave in positive matrices $A$ and $B$, ie., $$ tr\Big[X^\...
Sooraj S's user avatar
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1 vote
1 answer
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Show that if $\mu,\eta\in I\implies \dfrac{\mu+\eta}{2}\in I$ and $\{0,1,1/2\}\subset I$ implies $[0,1]\subset I$

Let $R_1,R_2,S_1,S_2,T_1,T_2$ be positive operators such that $[R_1,R_2]=[S_1,S_2]=[T_1,T_2]=0$, and $R_1\ge S_1+T_1$ $,R_2\ge S_2+T_2$ then for all $0\le t\le 1$, \begin{align} R_1^tR_2^{1-t}\ge S_1^...
Sooraj S's user avatar
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0 votes
1 answer
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How to shift the eigenvalues of a quantum Hermitian operator G to ±r?

Consider a gate $\mathcal{G}(\mu)=e^{-i \mu G}$ generated by a Hermitian operator G. If G has just two distinct eigenvalues(which can be repeated) we can, without loss of generality, shift the ...
ZLL's user avatar
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1 answer
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Measuring in Different Bases - Deterministic versus Random outcomes [closed]

The Qiskit Textbook on https://qiskit.org/textbook/ch-states/single-qubit-gates.html in section 4: Digression: Measuring in Different Bases, says – Z-basis is not intrinsically special, and that ...
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Explicitly verify the quantum error-correction conditions for the Shor code, for the error set $\{I, X_{i}, Y_{i}, Z_{i}\}$ for i ranging from 1 to 9

I came across this question when attempting exercise 10.10 in Nielson and Chaung's Quantum Computation and Quantum Information. The exercise was "Explicitly verify the quantum error-correction ...
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