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## Hot answers tagged mathematics

39 votes
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### How to understand the Haar measure from a quantum information perspective?

Computations in quantum information processing are implemented by means of unitary operations. Sometimes, we need to think not about a specific unitary operation required to execute a specific ...
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35 votes
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### How to derive the CNOT matrix for a 3-qubit system where the control & target qubits are not adjacent?

For a presentation from first principles, I like Ryan O'Donnell's answer. But for a slightly higher-level algebraic treatment, here's how I would do it. The main feature of a controlled-$U$ operation,...
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23 votes
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13 votes
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### How to check if a quantum circuit can be constructed for a given matrix representation?

Correct, unitarity is a sufficient and necessary condition. From Nielson and Chuang page 18: Amazingly, this unitary constraint is the only constraint on quantum gates. Any unitary matrix specifies a ...
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13 votes
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### Minimum number of 2 qubit gates to build any unitary

Theoretical lower bound In contrast to the answer by Bertrand, I will assume that along with a $CNOT$ gate we have arbitrary single-qubit unitaries on our disposal. In this case, one can derive the ...
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11 votes
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### How to show a density matrix is in a pure/mixed state?

First, the example that you give is not a density matrix (they must have trace 1). Second, you’re asking how to go from the matrix into an operator representation that is not unique. So, there are ...
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11 votes
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### How is the ground state of a Hamiltonian defined?

A couple of points: The ground state is by definition the eigenvector associated with the minimum valued eigenvalue. Lets consider the Pauli Z matrix as you have. First, \begin{align*} Z = \begin{...
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10 votes
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### Quantum states are unit vectors... with respect to which norm?

Born's rule states that $|\psi(x)|^2 = P(x)$ which is the probability of finding the quantum system in the state $|x\rangle$ after a measurement. We need the sum (or integral!) over all $x$ to be 1: \...
10 votes
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### What are the possible non-entangling two-qubit gates?

There are no other non-entangling gates in $SU(d^2)$ in any dimension $d=2,3,\dots$. Note that the global phase is irrelevant to the problem, so we lose no generality by considering non-entangling ...
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9 votes

### Quantum states are unit vectors... with respect to which norm?

Some terminology seems a little bit jumbled here. Quantum states are represented (within a finite dimensional Hilbert space) by complex vectors of length 1, where length is measured by the Euclidean ...
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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 ...
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9 votes

### When can a matrix be "extended" into a unitary?

A necessary and sufficient condition is that, given an $n\times n$ matrix $M$, you can construct a $2n\times 2n$ unitary matrix $U$ provided the singular values of $M$ are all upper bounded by 1. ...
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9 votes

• 60.3k
9 votes
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### How do I prove that the Hadamard satisfies $H\equiv e^{i\pi H/2}$?

First of all, note that the statement, as written, is wrong (or rather, it is correct only as long as the "$\equiv$" symbol is taken to mean "equal up to a phase"). An easy way to see it is by ...
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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 ...
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9 votes
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9 votes

### If two unitary operators commute, do their roots also commute?

It's always possible to find some square roots that commute. But it's not guaranteed that all square roots commute. If two operations commute, then you can find square roots that commute by ...
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8 votes

### Is acting with a positive map on a state not part of a larger system allowed?

Any map which is not Completely Positive, Trace Preserving (CPTP), is not possible as an "allowed operation" (a more-or-less complete account of how some system transforms) in quantum mechanics, ...
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