16 votes
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General parametrisation of an arbitrary $2 \times 2$ unitary matrix

What is the proof that any given unitary matrix can be converted as above? Let $U$ be an arbitrary $2\times 2$ unitary matrix. This is equivalent to the rows/columns of $U$ being orthonormal bases. ...
glS's user avatar
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13 votes
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What is the difference between "code space", "code word" and "stabilizer code"?

Code spaces and code-words A quantum error correcting code is often identified with the code-space (Nielsen & Chuang certainly seem to do so). The code space $\mathcal C$ of e.g. an $n$-...
Niel de Beaudrap's user avatar
12 votes
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Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$

The equation at the top of the question is not correct: there is a missing factor of $1/d$ on the right-hand side. Let's eliminate this factor from the left-hand side to make it simpler, so that the ...
John Watrous's user avatar
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12 votes
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How to perform quantum state tomography on two qubits?

Preliminary I would like to rewrite the equation that you have in a slightly different manner. Since a density matrix can be written as a matrix, we can also write it down as a linear combination of ...
JSdJ's user avatar
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10 votes
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Nielsen and Chuang, Exercise 6.5: How to simulate oracle for n+1 qubits using one oracle gate for n qubits and one extra qubit?

The simplest solution is to use an ancilla in the $|+\rangle$ state. Swap that ancilla for the oracle's output qubit, conditioned on the control qubit being false, before and after applying the oracle....
Craig Gidney's user avatar
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10 votes
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How to perform Quantum Process Tomography for three qubit gates?

I am sure that since you are asking this question you probably already understand this, but for future & other's reference let me give a quick recap of what we are trying to achieve. Quantum ...
JSdJ's user avatar
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9 votes
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Connection between stabilizer generators and parity check matrices in the Steane code

There are a few conventions and intuition here, which perhaps it would help to have spelled out — $\def\ket#1{\lvert#1\rangle}\def\bra#1{\!\langle#1\rvert}$ Sign bits versus {0,1} bits The ...
Niel de Beaudrap's user avatar
9 votes
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How can we be sure that for every $A$, $A^\dagger A$ has a positive square root?

A matrix is positive if and only if it is Hermitian (and thus unitarily diagonalizable) and all its eigenvalues are positive (that they are real follows automatically from it being Hermitian). If this ...
glS's user avatar
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9 votes
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Confusion on the definition of the phase-damping channel

Let $\mathcal{N}$ be the channels which subscripts for which conventions. $$ \mathcal{N}_{N.C.} (\rho) = \begin{pmatrix} \rho_{00} & \rho_{01} \sqrt{1-\lambda}\\ \rho_{10} \sqrt{1-\lambda} & \...
AHusain's user avatar
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9 votes
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Are three POVM measurements on a single qubit physically realizable?

Three outcomes amounts to more than one bit if the outcomes are all deterministic, and give you information about the original qubit. But suppose I have a coin (that is either heads or tails). I ...
Peter Shor's user avatar
9 votes
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Correct Formulation of N&C Exercise 4.11 and other textbooks misquoting

The errata is correct. I had a project student who erroneously took one of these mis-quotes and she spent ages working with it, realising it didn't make sense, and subsequently proving that the stated ...
DaftWullie's user avatar
9 votes

Procedures and intuition for designing simple quantum circuits?

Here are three strategies for learning to make this kind of circuit. They all involve being initially loose with what is allowed and gradually tightening constraints until everything is accounted for. ...
Craig Gidney's user avatar
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9 votes
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What's the 'physical consistency' in the partial trace scenario?

Measurement average Measurement average $\langle M \rangle_\rho$ of observable (a Hermitian operator) $M$ on the state $\rho$ is the average of measurement outcomes $m$ in the limit of infinite number ...
Adam Zalcman's user avatar
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9 votes
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Why does quantum distinguishability ensure no faster-than-light communication?

In the situation described in the book, Alice and Bob share the state $$ |\psi\rangle = \frac{|00\rangle+|11\rangle}{\sqrt{2}}. $$ Using the definition $|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt2$ and ...
Adam Zalcman's user avatar
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8 votes
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How to find the operator sum representation of the depolarizing channel?

This really depends where you want to start from. For instance, you can construct the Choi state of $\mathcal E$, i.e., $$ \sigma = (\mathcal E \otimes \mathbb I)(|\Omega\rangle\langle\Omega|)\ , $$ ...
Norbert Schuch's user avatar
8 votes
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Premise of the proof of the No-Cloning Theorem

The proof does not seem to rule out the case that there exists a specific U that can clone only the specific state |ψ⟩. That's because you can clone specific states. Cloning is only impossible if the ...
Craig Gidney's user avatar
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8 votes

Procedures and intuition for designing simple quantum circuits?

Here are the actions for the given transformation on the computational basis states: $$|000\rangle \rightarrow |000\rangle \qquad |001\rangle \rightarrow |010\rangle \qquad |010\rangle \rightarrow |...
Davit Khachatryan's user avatar
8 votes
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Is there any 'official' list of errata for Nielsen & Chuang?

Michael Nielsen said the book is not maintained. I asked him on Twitter .
Victory Omole's user avatar
7 votes

Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$

Here the important fact is that the maximally mixed state is in fact an identity matrix. Let me rewrite the expression on the left in index notation (the summation sign is omitted according to the ...
Alexey Uvarov's user avatar
7 votes
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Proving the inequality $|\mathrm{tr}(AU)|\le \mathrm{tr}|A|$ in Uhlmann's theorem

I'll give a couple of methods to do this: (Using matrix inequalities) The idea is to use CS inequality in the form $$\newcommand{\tr}{\operatorname{Tr}}\lvert \sum_{ij}A_{ij}^* B_{ij}\rvert\le\sqrt{\...
glS's user avatar
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7 votes
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Why is Deutsch's gate universal?

I have thoughts on a couple of different approaches, although I'm sure there'll be simpler options. Firstly, imagine you start from a two-qubit state $|00\rangle$, and apply an $R_x$ rotation with an ...
DaftWullie's user avatar
7 votes
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What does $M_m |\psi_i\rangle$ mean in the equation $p(m|i)=\langle\psi_i|M_m^\dagger M_m|\psi_i\rangle$?

Measurement postulate The statement you are asking about is a postulate of quantum mechanics, so it cannot be mathematically derived from other facts in the theory. Instead, it is justified by its ...
Adam Zalcman's user avatar
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7 votes
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Can quantum circuits/operations have truth tables?

Quoting Wikipedia, a truth table is a table "which sets out the functional values of logical expressions on each of their functional arguments". Thus, a quantum operation cannot have a ...
Mariia Mykhailova's user avatar
7 votes

How to describe a known quantum state using classical information?

Generally speaking, in order to describe elements of a set $A$ using classical information we need two ingredients: a non-empty finite alphabet $\Sigma$ and an encoding $E: A\to\Sigma^\omega$ which ...
Adam Zalcman's user avatar
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6 votes
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Why isn't the circuit performing a measurement in the Bell basis?

You're correct that unitary gates like the Hadamard and CNOT do not perform any measurement. In fact, measurement isn't a unitary operation at all! There are separate measurements gates to measure the ...
Sanchayan Dutta's user avatar
6 votes
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Exercise 10.34 of Nielsen and Chuang : $-I$ not element of stabilizer group iff $g_j^2=I$ and $g_j \neq I$ where $g_j$ generators

Consulting my local copy of Nielsen & Chuang (10th anniversary edition, p. 457), the complete statement of the exercise is pretty much exactly as you have given it: Exercise 10.34. Let $\...
Niel de Beaudrap's user avatar
6 votes
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What kind of errors does the master equation in the Lindblad form describe, continuous errors or discrete errors?

The errors that are described by the Master equation are continuous errors. The action of error correction is to discretize those errors. For example, dephasing noise can be described by the Master ...
DaftWullie's user avatar
6 votes
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Understanding Steps in Deutsch's Algorithm

To answer your first question, the quantum oracles are defined by their effect on the basis states $|0\rangle$ and $|1\rangle$, and if the oracle has to be computed on a superposition of basis states, ...
Mariia Mykhailova's user avatar
6 votes
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Is the quantum state fidelity defined as $F(\rho, \sigma)=\text{tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ or its square?

Both definitions are used and authors usually make it clear which one they mean. Wikipedia also points this out under the Alternative Defintion section.
user1936752's user avatar
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6 votes
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Question Regarding Simulating Hamiltonian With Quantum Circuit

I think the problem is in this assumption: $$e^{-i Z \otimes Z \otimes Z t} |110\rangle = e^{Z}|1\rangle e^{Z}|1\rangle e^{-iZt}|0\rangle$$ It shouldn't be right, because, at least, $e^{Z}$ is not ...
Davit Khachatryan's user avatar

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