Recent content by Truecrimson

Would there be a problem with products of non-commuting operators? Classically, ##f(x,p) = x^2p^2 = q xp^2 x + (1-q) px^2p## for any ##0\le q \le 1## but different choices of ##q## lead to different observables.

This problem in general is known as quantum gate/circuit synthesis, or quantum compiling. This paper from 2004 that I found has been cited almost three hundred times. As for your questions, 1) Isn't the Fredkin gate already a controlled SWAP? What are the sets of gates allowed for your...

To say it in another way, commutativity is not transitive. ##[A,B] = 0## and ##[B,C]=0## does not imply that ##[A,C]=0##.

I didn't realize Gribbin wrote a book with the same title a long time ago (1998?). Anyway, DarMM's book is from 2017. http://qisforquantum.org/

Yes, you can. You have made a few threads to ask basically the same question. The answer will still be the same: a choice of bases for a vector space is arbitrary and has no physical meaning, and as such cannot effect physically relevant notions such as the locality of a Hamiltonian or the...

They are different. Wigner proved that the most general transformation that preserves the total probability has to be linear and unitary or anti-unitary. On the other hand, such ##U## is a symmetry of ##H## (i.e. of a system with Hamiltonian ##H##) if ##U^{\dagger}HU = H##.

You either multiplying out ##|\psi^+ \rangle \langle \psi^+|## wrong or writing + in the result where there shouldn't be. (Each term in ##\rho_A## should be a ket-bra ##|\dots \rangle \langle \dots |##) Other than that, use the orthonormality ##\langle x|y \rangle = \delta_{xy}## of subsystem B...

I didn't check the calculation, but having ##i## in the off-diagonal elements is fine as long as your density matrix is Hermitian.

I'm not sure I understand your question, but not every state in ##U \otimes V## is of the form ##|u\rangle \otimes |v\rangle, |u\rangle \in U, |v\rangle \in V##. Only your basis vectors are, but the choice of bases is arbitrary and has no physical meaning.

Yes, the last two paragraphs of my reply were all about pure states.

What you did was akin to linear inversion, one of the methods for doing quantum tomography which is what you need in general. The problem of determining pure states ("wave functions" as opposed to mixtures) came to be known as the Pauli problem. Pauli originally asked whether the measurements...

Schroeder (thermal physics) and Griffiths (QM) are popular undergrad textbooks. I myself have never used Schroeder. Kittel (1st edition, not Kittel and Kroemer) is a great introduction to thermal physics, starting right from the non-mysterious notion of entropy by counting microstates. However...

Late correction: I meant to write ## (|u\rangle + |v\rangle) (\langle u| + \langle v|) ##, not ## |u+v\rangle \langle u+v| ## in the last reply.

Hi @Joao . First, I have a nitpick: you should keep ##1/\sqrt{2}## instead of 0.7 (and not 0,7). Otherwise, the state will not be normalized to exactly 1. Now, the reason for tensoring the gate is exactly the same reason that you tensor the two qubits. If you apply gates independently to the...

## \langle p|q \rangle ## is a number. Also ## |v\rangle \langle u| ## cannot be a state (density matrix) unless ##u = v##. (A density matrix must be positive and therefore Hermitian.) So let's say that you have ## |u\rangle \langle u| ## where ##|u\rangle## is normalized. What's the point of...