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Accepted

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. ...
• 26.1k
Accepted

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 $n$-...
• 12.2k
Accepted

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 ...
• 6,177
Accepted

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 ...
• 5,589
Accepted

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....
• 40.1k
Accepted

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 ...
• 5,589
Accepted

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 ...
• 12.2k
Accepted

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 ...
• 26.1k
Accepted

• 4,391

How to find the operator sum representation of the depolarizing channel?

While the procedure in the existing answer, based on channel-state duality, applies to general channels, there's a more direct way to obtain Kraus operators for this particular case of the ...
• 201

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 ...
Accepted

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{\...
• 26.1k
Accepted

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 ...
• 60.2k
Accepted

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 ...
• 23.5k

How to visualize Hadamard gate as $X$-$Z$-$X$ decomposition?

As mentioned in the other answer, the Hadamard gate is a pi rotation (180 degree) around the $X + Z$ axis. That is, it is a 180 degree rotation around the purple axis indicated in the below figure: ...
• 13.9k
Accepted

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 ...
• 9,175

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 ...
• 23.5k
Accepted

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 ...
• 17.7k