# 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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### The unitary operation of the Schrödinger cat example

If the initial state of the system is: $|\psi(0)\rangle$ = $|0_{cat}0_{gas}\rangle$ Where $|00\rangle$ is the state where the gas has yet to decay and the cat is still alive. Then say the state ...
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### Positive Definiteness of Killing Form in Gauge Theory

This question is related to requirement that the gauge group of a gauge theory be a direct product of compact simple groups and $U(1)$ factors but is not the same as, for example, this question (...
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### Help proving bound on POVM measurement probabilities

I am trying to follow Nielsen and Chuang's 1 proof that the difference in measurement probabilities is bounded by the difference between two unitary operators applied to a given state. Can someone ...
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### How do I find the state and the energy exchange of a system after time $t$?

Assuming that the Hamiltonian of my system is of the form: $$H=k\left( \left| 0\right> \left<1 \right| + \left| 1\right> \left< 0\right| \right)$$ and that my initial state can be ...
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When studying unitary transformations in QM, most of the textbooks I used defined the unitary transformation operator as $$\hat{U}(\alpha) = e^{-i \alpha \hat{G}} \, \, ,$$ where $\alpha$ is the ...
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### Determining if a process is information preserving in quantum mechanics

I was wondering if one is provided with two quantum states such that one is the evolution of another through some unknown process, is it possible to determine if the aforementioned process is ...
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### Conservation of Distinctions in Quantum Mechanics

Recently I have been reading Quantum Mechanics The Theoretical Minimum by Leonard Susskind. In the book he mentions the law of conservation of distinctions, i.e. the conservation of information. He ...
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### Unitarity implies branch cuts in $s > 4m^2$ in the $S$-matrix

Why does unitarity imply a branch cut in the $S$-matrix after $s > 4m^2$ where $s$ is the Mandlestam variable and $m$ is the mass of the particle? Assume identical particle scattering.
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### Proof that $U$ is an unitary operator [closed]

I have a function $f$ mapping a bit onto another bit, i.e. $f : \{0, 1\} \rightarrow \{0, 1\}$. The function f is either constant, so f(0) = f(1) or balanced, so f(0) $\neq$ $f$(1). The quantum gate ...
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### Are there finite-dimensional unitary irreducible representations in Euclidean space?

One can show that there are no unitary finite-dimensional irreducible representations of the Poincaré group. The reason is that the generators for boosts are antihermitian. Since boosts are basically ...
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### Analytic Continuation of the $S$-Matrix and Analytic Continuation of Spinors

Broadly speaking, what is actually being done when one analytically continues the external particles' momenta in scattering amplitudes in QFT? Some associated more detailed questions: When thinking of ...
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### Positive mass but negative/positive/neutral charge

I have heard that the mass of a particle can't be negative because the hamiltonian should be bounded from below. What is the formal argument regarding this and why the same argument doesn't follow ...
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### Can be a Schmidt Decomposition Written in this Form?

The Schmidt Decomposition of a state $|\Psi_{AB}\rangle = \sum C_{ij}|i_A\rangle |j_B\rangle$ can be written in this form $|\Psi_{AB}\rangle = \sum \lambda_k|k_A\rangle |k_B\rangle$ where $\lambda_k$ ...
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### Weinberg derivation of Lie Algebra [duplicate]

In the derivation of the Lie algebra in the first volume of Quantum Theory of Fileds by Weinberg, it is assumed that the operator $U(T(\theta)))$ in equation (2.2.17) is unitary, and the rhs of the ...
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### Isometric equivalence of purifications of quantum states

Following the notes here (Quantum Information Theory Tips 5 at ETH), we state the following result. For any quantum state $\rho_A$ and purifications $\vert\psi\rangle_{AB}$ and $\vert\phi\rangle_{AC}$,...
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### Connection between isometries and projectors in QM

I realize this question is technically a mathematical one but I think it is seen often enough in quantum information so I ask it here. The following is the definition of an isometry in Mark Wilde's ...
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### Under what assumptions does a state following the TDSE converge to its ground state?

Until $t=0$ a system is in an eigenstate $\psi_0(x)$ of the Hamiltonian $\hat{H}_0$. The time-evolution is the trivial phase factor. Now at $t=0$ the system changes to $\hat{H}$ (you can assume it is ...
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### Physical interpretation of Lorentz group non-compactness in the case of Weyl spinors

If we consider the generators of Lorentz group $J$ and $K$, it is possible to indroduce the operators $J^{\pm}=\frac{J\pm iK}{2}$ which shows the $SU(2)\times SU(2)$ structure of the Lorentz group. ...
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### Must all operators be Unitary? [closed]

Do all operators on a quantum state have to be unitary? Does this mean that there are non-unitary operators that act on quantum states in real life? Context: Let's say we have a composite quantum ...
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### Is it possible to build a non-unitary quantum circuit?

Is it possible to build a non-unitary circuit of quantum Boolean gates, apart from using measurement to achieve this?
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### Deriving Unitarity of $S$-matrix in 1D Quantum Mechanics

So, I was studying about scattering across a one-dimensional unknown potential ( pretty elementary Quantum Mechanics) and how if we know the $S$-matrix of such a system, we can deduce an awful lot of ...
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### Do all unitary-preserving regulators necessarily turn real loop integrals into pure imaginary numbers?

The optical theorem, which results from the unitarity of the $S$-matrix, relates the imaginary part of the forward scattering amplitude to the total cross section. When using this theorem in practice, ...
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### Perturbative proof of unitarity of $S$-matrix in QED

In any standard textbook on QFT I know it is claimed that the $S$-matrix in QED is a unitary operator. I have never seen any proof of it. This should be compared with the analogous property of $S$-...
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### Stone's theorem on one-parameter unitary groups and new self-adjoint operators

I have been following the proof of the Stone's theorem on one-parameter unitary groups. The question is if the current list of self-adjoint operators used in quantum mechanics, including position, ...
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### What are the theoretical / mathematical problems in discarding negative solutions of Dirac equation?

I read some Q&A about it, but my question is why Dirac was so sure that he could not discard negative energy solutions. It seems so natural that energy must be positive, that I suppose that if ...
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### Proof of normalizing the wave function

So suppose we have the wave function $\phi(x,t)$ in the context of quantum mechanics, that satisfies the Schrodinger's equation. We want to see that if we normalize this function at $t=0$ then it will ...
I'm trying to prove that $\{\tilde a_i,\tilde a_j^{\dagger} \}=\delta_{ij}$, by defining $\tilde a_i=\sum_j \bar U_{ji}a_j$. U is an unitary matrix and $a_i$ refers to an element of the operator $a$. ...
### Prove that there exists a $d \times d$ unitary matrix $U$ which cannot be decomposed as a product of fewer than $d-1$ two-level unitary matrices
This is exercise 4.38 from Nielsen and Chuang: Prove that there exists a $d \times d$ unitary matrix $U$ which cannot be decomposed as a product of fewer than $d-1$ two-level unitary matrices. If ...