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8 votes

What is the density matrix of a pure state?

You can determine whether a state is pure or mixed by considering the purity $\gamma$ which is defined as the trace (i.e. the sum of diagonal entries) of the density matrix squared. \begin{equation} \...
Callum's user avatar
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7 votes
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What are "completely positive" and "CPTP" quantum maps?

[A] States lie in Hilbert space $\mathcal{H_S}$. $|\psi\rangle \in \mathcal{H_S}\,.$ Operators, density operators lie in the bounded operator space of $\mathcal{H}_S$. $\rho \in \mathcal{B}(\...
FDGod's user avatar
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6 votes

Is $|A\rangle = \frac{1}{\sqrt2} |00\rangle + \frac{1}{\sqrt2} |01\rangle$ a valid quantum state?

Yes, the state $ |A\rangle = \frac{1}{\sqrt{2}} |00\rangle + \frac{1}{\sqrt{2}} |01\rangle $ is indeed a valid quantum state. In quantum mechanics, a valid quantum state can be any normalized linear ...
Yet another Random Guy's user avatar
5 votes
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How to eliminate the global phase of a state vector?

I think the best way to understand this is via projective geometry. The idea is that in quantum mechanics we always assume that our state vector is normalized and we don't care about global phases, so ...
smitke6's user avatar
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5 votes
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What is the expression for $|\psi\rangle\!\langle\psi|$ if $|\psi\rangle=\cos(\theta/2)|0\rangle+\sin(\theta/2)e^{i\phi}|1\rangle$?

You are missing the fact that $$ \langle \psi | = \bigg( |\psi\rangle \bigg)^{\dagger} \,.$$ "$\langle \psi|$" is conjugate transpose of "$|\psi\rangle$". So, if $$|\psi\rangle = \...
FDGod's user avatar
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4 votes
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Can you distinguish between $|0\rangle, |1\rangle$, and $\frac{1}{\sqrt 2} (|0\rangle + |1\rangle)$?

If two quantum states $|\psi\rangle$ and $|\phi\rangle$ are such that $\langle\phi|\psi\rangle \neq 0$ (i.e. they are not orthogonal), then it will not be possible to determine which was given with ...
Joseph Geipel's user avatar
4 votes

How would you draw the phase-estimation circuit for the eigenvalues of $U = \mathrm{diag}(1,1,\exp(\pi i/4),\exp(\pi i/8)) $?

TL;DR What does the circuit look like? See the diagram below. This one uses 3 measurement qubits and the eigenstate is $|11\rangle$. Here I prepare the $|11\rangle$ with two Pauli $X$ gates. You could ...
Callum's user avatar
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4 votes
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Prove that the eigenvectors of a Hermitian operator form a basis

Let's stick with finite-dimensional spaces. In infinite dimensions, you might deal with linear unbounded operators, and need to take much more care on how you define spectrum and what you mean with &...
glS's user avatar
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4 votes
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How do you write the Hadamard operator on two qubits in braket notation?

Hadamard in Dirac notation is $$ H=\frac{1}{\sqrt{2}}(|0\rangle\langle 0|+|0\rangle\langle 1|+|1\rangle\langle 0|-|1\rangle\langle 1|). $$ For two qubits, you take the tensor product $H\otimes H$. You ...
DaftWullie's user avatar
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How to prove the matrix identities $HXH = Z$ and $HZH = X$?

Consider the matrix representations of these operations and with little algebra, things will become obvious. $$H = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1\\ 1 & -1 \end{bmatrix}$$ $$X = \begin{...
FDGod's user avatar
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4 votes

What are "completely positive" and "CPTP" quantum maps?

Definitions Let $\mathcal{H}$ be a complex Hilbert space. It turns out that the set $L(\mathcal{H})$ of all linear operators on $\mathcal{H}$ is also a Hilbert space. Let $I_\mathcal{H}$ denote the ...
Adam Zalcman's user avatar
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4 votes
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How to find the eigenvectors and eigenvalues of a hermitian operator?

From your post and the comments therein, it seems that you would benefit in reading a little bit more about linear algebra. There are plenty of resources for that, including some excellent posts on ...
Tristan Nemoz's user avatar
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4 votes
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Prove that $\text{Tr}(M|ψ\rangle\langleϕ|)=\langleϕ|M|ψ\rangle$

Your proof is not general, it assumes implicitly that the operator $M|\psi\rangle\langle \phi|$ is diagonalizable (there is a basis of eigenvectors). You can instead just use basic fact of Trace, ...
Pierre-Paul T.'s user avatar
4 votes
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Why can't the eigenvalues of a unitary matrix have the form $e^{i\theta}$?

in case of $e^{2\pi\cdot i\cdot \theta}$ the values of $\theta\in [0,1]$, in the case of $e^{i\cdot \theta}$ the values of $\theta \in [0,2\pi]$. it's just a different convention but they are ...
Sezzart's user avatar
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3 votes
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Decomposing Hadamard gate

You got the H matrix, up to an unimportant global phase of $\frac{\left(1-i\right)}{\sqrt2}$.
Yaron Jarach's user avatar
3 votes
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In Shor's algorithm, why do we have ${\rm gcd}(x\pm 1, N) > 1$?

The assumptions $x\pm1\neq0\bmod N$ mean that neither of them is a multiple of (or equal to) $N$. That is, you cannot write $x+1=kN$ for some $k\in\mathbb{Z}$, and same for $x-1$. At the same time, $(...
glS's user avatar
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3 votes
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Can a density operator be written equivalently as $\rho=\sum_i p_i|\psi_i〉\!\langle\psi_i|$ and $\rho=\sum_i\lambda_i|\psi_i\rangle\!\langle\psi_i|$?

The statement $\rho=\sum_ip_i|\psi_i\rangle\langle \psi_i|$ is a very general way of writing down the density matrix. It must be noted that if you are simply presented with the matrix $\rho$, there ...
DaftWullie's user avatar
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3 votes

Exercise 4.16 in the Nielsen & Chuang book

In my copy, we are asked to consider this action $$ |x_1\rangle|x_2\rangle \rightarrow (I_1 \otimes H_2)|x_1\rangle|x_2\rangle $$ and to find the matrix representation of $I_1 \otimes H_2$. In matrix ...
banercat's user avatar
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3 votes

How do I find the reduced density matrix of a system where two people share one qubit and have one qubit of their own?

There are 4 qubits, and the total state can be written as $$ |\theta\rangle = |\phi\rangle \otimes |\beta_{00}\rangle \otimes |\psi\rangle = \frac{|\phi\rangle|00\rangle|\psi\rangle+|\phi\rangle|11\...
Danylo Y's user avatar
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3 votes

Does the state obtained flipping $a,b$ in the state $(a,b)^T$ have a name?

I agree that, in your context, "negate" doesn't mean to flip the sign of the vector but rather to flip the entries in the vector, much as the negation of logical ...
Mark Spinelli's user avatar
3 votes
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Why can $(0,0,3/5,0,0,0,4/5,0,0)$ be written as $\frac35|3\rangle+\frac45|7\rangle$?

A vector is a list of elements that represent the weights for the sum of some basis. This basis can in this case be the set $\{|1\rangle,|2\rangle,|3\rangle,|4\rangle,|5\rangle,|6\rangle,|7\rangle,|8\...
Quantum Brilliance's user avatar
3 votes

Why is the operator $M_a |x\rangle= |a \cdot x \pmod{N} \rangle $ unitary?

Assuming $\gcd(N,a)=1$, then the operator $M_a$ mapping $x\in\mathbb Z_N$ to $a\cdot x \pmod N$ is a permutation, which means that it’s reversible, ergo unitary.
Mark Spinelli's user avatar
3 votes
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Decomposition of a $4 \times 4$ unitary matrix

That paper appears to do their rotations in a very strange order. The method you're interested in is how to use Givens rotations to perform a QR decomposition (see, e.g. https://en.wikipedia.org/wiki/...
DaftWullie's user avatar
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3 votes
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Trace Distance in Bloch sphere, what is the vector of Pauli matrices?

The $\vec{\sigma}$ is a shorthand notation. It literally represents a vector of Pauli matrices. It is useful to denote things algebraically like this for conciseness. We usually use it to represent a ...
FDGod's user avatar
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3 votes

conditions for two hermitians operators same up to unitary

It holds precisely if they have the same spectrum: You already argue for necessity. For sufficiency: $A=VDV^\dagger$, $B=WDW^\dagger$, then $D=W^\dagger BW$ and thus $A=VW^\dagger DV^\dagger W$. Thus, ...
Norbert Schuch's user avatar
3 votes
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Bound on success Probability for Regev's factoring algorithm

This is just Markov's inequality that states that for a positive random variable $X$ and $a > 0$ then $$ P( X \geq a ) \leq \frac{E(X)}{a}. $$ see the wikipedia: https://en.wikipedia.org/wiki/...
Frederik Ravn Klausen's user avatar
2 votes

In general, what is feasible quantum computation?

I'm guessing that each 'feasible operation' is meant to refer to a specific quantum gate that can be easily implemented on your particular quantum computer of interest, e.g. with one or a couple of ...
Mark Spinelli's user avatar
2 votes

Why adding H-gate is referred as changing the basis of measurement?

There are many ways to see that (one can invoke group theory, for example), but the way that I find easier is the following. Let q denote a system of a single qubit. Consider then two situations: q ...
Bento Montenegro's user avatar
2 votes

The matrix norm $\|A\|=\max_{\langle u|u\rangle=1}|\langle u|A|u\rangle|$ in the proof of Lieb's theorem

The short answer is that there are two problems with your argument: The partial derivative of $u^TAu$ you state is wrong More gravely, the whole approach is flawed because you're treating $u$ as a ...
Frederik vom Ende's user avatar
2 votes

What Hamiltonians generate Hadamard and CNOT matrices?

To add to the other answer: multiple such Hamiltonians are possible, in general. A simple way to see it is to notice that you are looking for Hermitians $H$ such that $e^{iH}=U$ for a given unitary $U$...
glS's user avatar
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