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The preparation of states in quantum mechanics

### Abstract

The important problem of how to prepare a quantum mechanical system, S, in a specific initial state of interest—e.g., for the purposes of some experiment—is addressed. Three distinct methods of state preparation are described. One of these methods has the attractive feature that it enables one to prepare S in a preassigned initial state with certainty, i.e., the probability of success in preparing S in a given state is unity. This method relies on coupling S to an open quantum-mechanical environment, E, in such a way that the dynamics of S∨E pulls the state of S towards an “attractor,” which is the desired initial state of S. This method is analyzed in detail.

© 2016 AIP Publishing LLC

Received 06 January 2015
Accepted 04 January 2016
Published online 01 April 2016

Article outline:

I. AIM OF THE PAPER, MODELS AND SUMMARY OF RESULTS
A. Three different methods for state preparation in quantum mechanics
1. Quantum state preparation via weak interaction with a dispersive environment
2. Preparation of states via adiabatic evolution
3. State preparation via duplication of systems and state selection
4. Plan of the paper
B. The model
1. The Hamiltonian of the system
2. Initial states and “observables”
3. Basic assumptions
C. Main result
D. Outline of the proof
1. Step 1. Analysis of the reduced dynamics on the van Hove time scale
2. Step 2. Reduced dynamics at arbitrarily large times: The cluster expansion
3. Step 3. The limit *t* → ∞
4. Generalization of Theorem 1.3.1 to initial states with finitely many “photons” and to thermal equilibrium states
II. ANALYSIS OF FOR *t* − *s* ∝ *λ*^{−2}(*s*)
A. Notations
1. Inner products and norms
2. Dyson expansion
B. Dyson expansions for and
C. Comparison of the effective propagator with the semigroup generated by a Lindbladian
D. Properties of
1. Spectrum of
III. REWRITING AS A SUM OF TERMS LABELLED BY GRAPHS
A. Discretization of time and Feynman rules
B. Resummation
1. Re-summing the Dyson expansion inside isolated double intervals
C. The cluster expansion
1. Construction of the set ℙ_{N}
2. The cluster expansion
3. The exponentiated form of the cluster expansion
IV. PROOF OF THE KOTECKỲ-PREISS CRITERION AND CONVERGENCE OF THE CLUSTER EXPANSION WHEN *N* → ∞
A. The Koteckỳ-Preiss criterion for
1. Upper bound on and summability of weights
2. Proof of Lemma 4.1.1
3. Proof of the Koteckỳ-Preiss criterion
B. Convergence of the cluster expansion as *N* → ∞
1. Convergence to the ground state
2. Proof of Lemma 4.2.1
V. EXTENSIONS OF THEOREM 1.3.1
A. Extension to initial field states with a finite number of photons
B. Thermalization at positive temperature

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2016-04-01

2016-09-27

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