Questions tagged [unitarity]

In quantum mechanics, a unitary operator satisfies U<sup>†</sup> U = UU<sup>†</sup> = I, where † denotes Hermitean conjugation; such operators then specify Hilbert space automorphisms and preserve state norms, so then probability amplitudes and hence probabilities. Use for conservation of probability questions under unitary state transformations.

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3answers
271 views

Ground state of local parent Hamiltonians and invariance under local unitaries

Assume that a finite-dimensional pure state $|\psi\rangle\in \mathcal{H}\simeq \mathbb{C}^m$, $m<\infty$, is the (unique) frustration-free ground state of a local parent Hamiltonian and suppose ...
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325 views

Conservation of probablities with non-unitary matrices?

In quantum mechanics, in the context of symmetry transformations, it is often said that for a transformation $T$ to conserve probabilities it must be unitary. But by performing any (even non-unitary)...
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Why do quantum gates have to be reversible?

One possible reason I have come up with is that we are modeling quantum gates by unitary matrices. And since unitary operations are reversible we have to be able reverse the operation in the physical ...
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1answer
173 views

Norm preserving Unitary operators in Rigged Hilbert space

If we take the free particle Hamiltonian, the eigenvectors (or eigenfunctions), say in position representation, are like $e^{ikx}$. Now these eigenfunctions are non-normalisable,so they don't belong ...
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Applying controlled Hadamard gate

I am unable to explain the output of a controlled Hadamard gate. If U is a single qubit gate $$U= \begin{pmatrix}u11 & u12\\ u21 & u22\end{pmatrix},$$ then the controlled gate is $$\mathrm{...
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Do the Cutkosky rules imply (perturbative) unitarity?

In most standard textbooks on relativistic QFT, the Cutksoky rules are presented as a consequence of unitarity of the S-Matrix. However, at least for scalar field theories, it appears that the ...
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1answer
457 views

Find unitary for given rotations on Bloch sphere

I want to characterize a unitary by given rotations on the Bloch sphere. I know, that when I send in the State $|\Psi\rangle =\begin{pmatrix}1\\0 \end{pmatrix}$, I get the state $U|\Psi\rangle=\...
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2answers
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What exactly does No cloning mean, in the context of Quantum Computing?

I am trying to get an intuitive idea of how the No-Cloning theorem affects Quantum computation. My understanding is that given a qubit $Q$ in superposition $Q_0 \left| 0 \right> + Q_1 \left| 1 \...
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1answer
796 views

Going to the interaction picture in the Jaynes–Cummings model [closed]

In the Jaynes–Cummings model for a two level atom, the Hamiltonian for the atom is defined as (I let $\bar{h}=1$) $$H_a=\omega_a\frac{\sigma_z}{2}$$ and the field Hamiltonian is $$H_f=\omega_ca^{\...
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How do derivative couplings affect canonical quantization?

Consider a Lagrangian for a scalar field $\phi$ with an interaction term $$\mathcal{L}_{int} = (\partial^2 \phi)^2 \phi.$$ Here I'm suppressing all indices for brevity. Now, this is just a three-...
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106 views

Unitarity of quantum evolution

In this paper by Charles Bennett, he says on page 25, I understand why U(XOR) gives the result it does but why is that a consequence of its unitary property? Thanks
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Path Integral Quantization in General Relativity

In Ref. 1 I have seen that the action must contain only the first derivative of the metric as required by the path integral approach. I don't understand why. I mean why the path integral approach of ...
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1answer
734 views

Griffith's Proof that a wave function will stay normalized is incorrect?

In Griffith's book, Introduction to Quantum Mechanics, in the equation: $$ \frac{\partial}{\partial t} \left| \Psi\right|^2 = \frac{i\hbar}{2m} \left( \Psi^* \frac{\partial^2 \Psi}{\partial x^2} - \...
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time evolution of the state $|a'\rangle$ with Hamiltonian $H=|a'\rangle\delta\langle{a}''|+|a''\rangle\delta\langle{a}'|$ [closed]

Reference to Chapter2, Problem8.b, Modern Quantum Mechanics, Sakurai: $|a'\rangle$,$|a''\rangle$: eigenket of the hermitian operator $A$ and the Hamiltonian, $$ H=|a'\rangle\delta\langle{a}''|+|a''\...
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What does the continuity equation for probability in quantum mechanics mean?

In quantum mechanics, the continuity equation $-{d\rho}/{dt}=\nabla\cdot{J}$ holds for a probability density $\rho$ and probability current $J$. But what does it mean, from a physical point of view? ...
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Where does the $i$ come from in the Schrödinger equation?

I am currently trying to follow Leonard Susskind's "Theoretical Minimum" lecture series on quantum mechanics. (I know a bit of linear algebra and calculus, so far it seems definitely enough to follow ...
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1answer
427 views

Expectation Value of Unitary Time Evolution Operator in Quantum Mechanics

Does the expression $\langle \Psi_i|U(t)|\Psi_i\rangle$ have a specific meaning, where $U(T)$ is the unitary time evolution operator of $\Psi$, and $\Psi_i$ is the initial state of $\Psi$? If so, ...
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Prove time-dependent hamiltonian is hermitian from unitarity of time-evolution operator

When we solve the Schrodinger equation for the time-evolution operator: \begin{equation} i\hbar\frac{\partial}{\partial t}U(t,t_{0})=HU(t,t_{0}), \end{equation} We have three cases to be treated ...
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How is the Born rule consistent with unitary evolution?

Consider a system $|\Psi_T \rangle_{t = 0} = |\Psi_E \rangle \otimes |\Psi_S \rangle$ where $|\Psi_S \rangle$ is a system that collapses into an eigenstate upon measurement. $|\Psi_E \rangle$ is the ...
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1answer
260 views

Unitary Bose gas

A unitary Bose gas (more about it [here]) is defined to occur when the scattering length diverges. What I don't understand, however, is which quantity/matrix is actually unitary? I mean, they could ...
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Unitarity of the S-matrix and Feynman Diagrams

There are several questions on the unitarity of the S matrix, but unfortunately non of them answers directly the following question. The S matrix is unitary and that can be proven by the fact that ...
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2answers
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Why is wavefunction collapse always non unitary?

The 'wavefunction collapse' upon measurement is usually referred to as being a non-unitary transformation, since it does not preserve the norm of the state vector. Indeed, if a linear superposition ...
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1answer
464 views

What is the time evolution operator in quantum mechanics [duplicate]

I'm curious about what happen to a system when the configuration of the system changes. If we have a system in a state $|\psi_{\textrm{in}}\rangle$ and we change the configuration of the system, the ...
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1answer
944 views

A wave function that is normalized initially remains normalized

Suppose that $\Psi(x,t)$ is normalized at time $t=0$. Show that this implies that $\Psi(x,t)$ is normalized at all other times. I know that this makes intuitive sense, and we'd certainly want our ...
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1answer
227 views

Deriving the form of generators of transformations

I'm struggling to understand a bit of quantum mechanics relating to the transformation generators. This specific bit contains quite a few guesses and assumtions which probably do make sense in ...
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How can I make this toy quantum random walk model unitary?

Take a toy $(1+1)$-dimensional lattice model of the universe. A particle begins at $x=0$ at $t=0$. It has an amplitude ${1}/{\sqrt{2}}$ to move one step to the left and amplitude ${1}/{\sqrt{2}}$ to ...
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Peskin-Schroeder, Unitarity of the S matrix, eq 9.61

I have a question regarding a derivation in Peskin and Schroeder's QFT book. On page 298, he is discussing a method for defining a gauge invariant S matrix. He does this by defining projection ...
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567 views

What experiment supports the axiom that quantum operations are reversible?

Among the axioms of quantum mechanics there is one axiom that says transformations of a quantum state need to be continuous, linear, and reversible (and this together with the other axioms results in ...
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The black hole paradox

I recently read in the news that Stephen Hawking claims to have solved the Black Hole Information paradox. I researched a bit about the paradox and the research that Stephen Hawking did to solve it. ...
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3answers
937 views

Why does Hamiltonian follow the property $H^*_{ij} = H_{ji} $?

I was reading Feynman's Lectures III's Hamiltonian Matrix. There I found this property of Hamiltonian Matrix: The Hamiltonian has one property that can be deduced right away, namely, that $$H^*_{...
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496 views

Can the sign of metric change physics?

Consider the Lagrangian of a massless real scalar (classical field) in $\phi(\textbf{x},t)$: $$\mathcal{L}=\frac{1}{2}\partial^\mu\phi\partial_\mu\phi$$ The Hamiltonian density in two different ...
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1answer
973 views

QED and anomaly

I've just started to learn anomalies in quantum field theories. I have a question. How to show that QED is free from vector current anomaly and what would happen if it were not? In other words, how ...
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Locality, unitarity & vacuum energy

I've read in a set of lecture notes that the requirement of locality and unitarity in QFT imply that the vacuum must have a non-zero energy associated with it (http://arxiv.org/abs/1502.05296 , top of ...
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1answer
769 views

Probability conservation in WKB tunneling

Suppose we have quantum mechanical plane waves of energy $E$ incident upon a one-dimensional potential barrier $V(x)$ with sloping sides. One can compare the WKB solutions in the three relevant ...
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2answers
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Violation of unitarity: meaning and consequences

What is meant by unitarity and violation of unitarity of a QFT? For example, Fermi theory of beta decay is said to violate unitarity. How does violation of unitarity make a theory sick?
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What do we mean by Unitary Dynamics in Quantum Computing?

In the afterword to the Tenth Anniversary Edition of the book Quantum Computation and Quantum Information the authors say: For many years, the conventional wisdom was that coherent superposition-...
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1answer
436 views

Ghost in the quantization of relativistic particle

It is well known that in the quantization of certain relativistic theories such electromagnetism or relativistic string negative norm states could arise when quantizing covariantly. Acting with ...
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4answers
786 views

Importance of local conservation of probability

In almost every textbook of quantum mechanics we can find the derivation of the local conservation of probability. $$\nabla\cdot\vec{J}+\partial_t (\psi^*\psi)=0$$ where $\vec{J}$ is probabilty ...
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Ballentine's proof of (one half of) Stone's theorem

Reading Ballentine's "Quantum Mechanics; A Modern Development" I got stuck on his really short proof of what I think is Stone's theorem. On page 65 (paperback, reprint of 2008) he writes about about a ...
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1answer
275 views

What does it mean “Hawking radiation is in a pure state”?

I'm trying to understand black hole paradox but I'm not sure if I understand what does it mean "Hawking radiation is in a pure state". Does it mean if Hawking radiation is in a mixed state then ...
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1answer
130 views

What sort of operations can be applied on a Hilbert spaces?

I was reading the paper No Universal Flipper for Quantum States. In this paper they have tried to prove by contradiction that a universal flipping machine cannot exist. By flipping I mean if I have a ...
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1answer
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Prove that this operator is unitary

$$\hat{O}\equiv(1/\sqrt{2\pi})\int e^{-iNz}dz$$ $$\hat{O}^\dagger\equiv(1/\sqrt{2\pi})\int e^{iN'x}dx$$ We have the operator $\hat{O}$ and its Hermitian adjoint $\hat{O}^\dagger$, in the one ...
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Why is time evolution unitary?

Is the reason why the time evolution operator is unitary based on purely physical arguments, i.e. that the physical processes that an isolated system undergoes shouldn't depend on any particular ...
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Non-Hermitian Lagrangian in Quantum Field Theory

I have seen more than once non-Hermitian Lagrangian densities being used in effective field theories. Usually the problem of unitarity is explained away with decays into some degree of freedom not ...
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1answer
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Embedding of particles into fields

For the classification of particles (Wigner 1939), we look for unitary representations of the Poincaré/Lorentz group. There are only infinite-dimensional (non-trivial) unitary representations! To ...
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1answer
79 views

What does it mean to have a degenerate $S$-matrix?

The Coleman-Mandula theorem $D>2$ assumes that the quantum field theory may not have a degenerate $S$-matrix. But what does it mean to have a degenerate $S$-matrix? The $S$-matrix if I got it ...
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351 views

How can an inverted anharmonic potential $V(x)=-x^4$ have discrete bound states?

I've been watching the lectures on mathematical physics by Carl Bender on youtube where he uses the non-Hermitian Hamiltonian methods to prove that the inverted anharmonic potential $V(x)=-x^4$ has a ...
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1answer
979 views

When is a unitary operator a quantum gate?

Quantum gates we use like X, Y, Z, H, CNOT, etc. are all unitary. When can an arbitrary unitary operator be considered as a quantum gate?
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1answer
264 views

Reality of the action in QFT [duplicate]

Following Ramond, 1.5 Field Theory, it is mentioned that the classical Lagrangian density in (workable for HEP) QFT theories has to be Real, otherwise total probability is not conserved. Can someone ...
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1answer
286 views

Is the process on initialization of qubits unitary?

It is said in some texts, that quantum computer undergoes only 2 types of transofrmations: 1) unitary evolution while computing and 2) non-unitary transformation ...