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Continuous Time Limit of Repeated Quantum Observations

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Continuous Time Limit of Repeated Quantum Observations Continuous time limit of repeated quantum observations Von der QUEST-Leibniz-Forschungsschule der der Gottfried Wilhelm Leibniz Universit¨at Hannover zur Erlangung des Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation von M.Sc. Bernhard Neukirchen 2016 Referent: Prof. Dr. Reinhard F. Werner, ITP Leibniz Universit¨at Hannover Korreferent: Prof. Dr. David Gross, THP Universit¨at K¨oln Tag der Promotion: 18.12.2015 Abstract In this thesis we study the problem of continuous measurement on an open quantum system. The starting point of our analysis is the time-evolution of an open quantum system where the evolution of the environment is not explicitly described, i.e. traced out. More explicitly we restrict to time-evolutions having the structure of a semigroup or evolution system. That is, for time-points 0 r s t there exists a two-parameter family of ≤ ≤ ≤ completely-positive maps E (t,s), such that E (t,s)E (s,r) = E (t,r) and the systems ∗ ∗ ∗ ∗ state evolved according to ρ(t)= E (t,s)(ρ(s)). ∗ Physically this means that in our Ansatz the interaction between system and environ- ment is already fixed and can only be weak. The main goal of this thesis is now to describe, mathematically rigorous, all continuous measurements compatible with the given evolu- tion. A measurement is compatible with the given time-evolution, if the evolution does not change when measurement results are ignored. In other words the measurement is indirect and does not further disturb the systems evolution. Our results can be interpreted as a rigorous construction of continuous matrix product states, and are related to the Hudson-Parthasarathy quantum stochastic calculus. The analysis in this thesis separates into two parts. In the first we study the class of quantum systems having an evolution system structure as described above. Precisely we study the theory of minimal solutions to a Cauchy equation with a generator of Gorini- Kossakowski-Sudarshan-Lindblad type. We do this for the case of time-dependent and unbounded generators. This analysis culminates in the description of a counterexample due to Holevo of a semigroup which is not the minimal solution to such a standard Lindblad equation. In the second part we construct a description of all measurements compatible with a given evolution and thus achieve our main goal. The analysis starts with the discrete-time version of the problem, i.e. where the time-points r,s,t in the above description of the evolution are chosen from a finite set. This discrete-time problem is completely solved by Stinespring dilation theory. We then construct our solution of the continuous-time case as a refinement limit over arbitrary discretizations of the evolution. Hence, our construction gives a continuous-time analogue of the Stinespring dilation for semigroups and evolution systems. It can be interpreted as a complete quantum description of the information the system radiates into the environment, i.e. it yields a quantum state of the environment. We complement the description of the state with a description of the observables for continuous measurements and the calculation expectation values. Keywords: open quantum systems, delayed-choice quantum measurement, lindblad gen- erators iii Zusammenfassung In der vorliegenden Arbeit untersuchen wir das Problem der Beschreibung von kontinu- ierlichen Messungen an einem offenen Quantensystem. Der Startpunkt unserer Analyse ist die Zeitentwicklung eines offenen Quantensystems, bei welcher die Zeitentwicklung der Umgebung nicht Teil der Beschreibung, also ausge- spurt, ist. Genauer gesagt, beschr¨anken wir die Klasse an m¨oglichen Zeitentwicklungen auf diejenigen, welche die Stuktur einer Halbgruppe oder eines Propagators haben. Zu Zeit- punkten 0 r s t existiert also eine zwei-Parameter-Familie von vollst¨andig positiven ≤ ≤ ≤ Abbildungen, E (t,s), mit der Eigenschaft E (t,s)E (s,r)= E (t,r). Die Zeitentwicklung ∗ ∗ ∗ ∗ des Systems ist gegeben durch die Gleichung ρ(t)= E (t,s)(ρ(s)). ∗ Auf physikalischer Ebene bedeutet dieser Ansatz, dass die Wechselwirkung zwischen System und Umgebung von vorne herein festgelegt ist und nur schwach sein kann. Das Prim¨arziel dieser Arbeit ist die mathematisch strenge Beschreibung aller Messungen in kontinuierlicher Zeit die mit der gegebenen Zeitentwicklung vertr¨aglich sind. Vertr¨aglich- keit von Messung und Zeitentwicklung bedeutet hierbei, dass die Zeitentwicklung un- ver¨andert bleibt falls die Messergebnisse nicht beachtet werden. Die Messung verl¨auft also indirekt und st¨ort das System nicht zus¨atzlich. Unsere Ergebnisse k¨onnen als eine kontinuierliche Stinespring Dilatation verstanden werde. Es besteht ein Zusammenhang mit kontinuierlichen Matrixproduktzust¨anden und dem quantenstochastischen Calculus von Hudson und Parthasarathy. Diese Arbeit gliedert sich in zwei Abschnitte. Im ersten Abschnitt werden Zeitentwick- lungen mit der oben beschriebenen Propagator-Struktur n¨aher beleuchtet. Genauer gesagt, untersuchen wir minimale L¨osungen von Cauchy Gleichungen mit einem infinitesimalem Erzeuger von Gorini-Kossakowski-Sudarshan-Lindblad Form. Wir behandeln hier den Fall von zeitabh¨angigen und unbeschr¨ankten Erzeugern. Dieser Abschnitt der Arbeit endet mit der Beschreibung eines Gegenbeispiels von Holevo, einer Halbgruppe die nicht minimale L¨osung einer Lindblad Gleichung ist. Im zweiten Abschnitt der Arbeit konstruieren wir die Beschreibung aller mit einer vor- gegebenen Zeitentwicklung vertr¨aglichen Messungen und erfullen¨ damit unser Prim¨arziel. Unser L¨osungsansatz basiert auf der Diskretisierung der Zeitentwicklung, das heißt die m¨oglichen Zeitpunkte r,s,t in der obigen Beschreibung beschr¨anken sich auf eine endliche Menge. Diese diskretisierte Variante des Problems wird vollst¨andig durch die Stinespring Dilatation gel¨ost. Der Kontinuumsfall ergibt sich daraus als ein Verfeinerungslimes uber¨ beliebige Diskretisierungen. Die Konstruktion kann also als ein kontinuierliches Analogon zur Stinespring Dilatation fur¨ Halbgruppen und Propagatoren betrachtet werden. Das Ergebnis ist eine vollst¨andige Beschreibung der Quanteninformation die das System an die Umgebung abgibt. Wir erg¨anzen diese Beschreibung der Zust¨ande mit einer Theorie der dazu geh¨origen Observa- blen und der Berechnung der Erwartungswerte. Schlagw¨orter: offene Quantensysteme,verz¨ogerte Quantenmessung, Lindblad Erzeuger v Contents Abstract iii Zusammenfassung v List of Figures xi List of Notations xiii 1. Outline 1 1.1. Measurements compatible with an evolution . 1 1.2. Thesisoutline................................... 2 1.3. Central ideas of part I . 3 1.4. Central ideas of part II . 4 2. Mathematical and physical basics 7 2.1. Inductivelimits.................................. 7 2.1.1. Directed sets and interval decompositions . 7 2.1.2. Nets . 9 2.1.3. Inductivelimits.............................. 10 2.1.4. Projectivelimits ............................. 11 2.1.5. Operators between inductive limit spaces . 12 2.1.6. Generalizedinductivelimits . 15 2.2. Evolution of open quantum systems . 16 2.2.1. The Stinespring dilation . 17 2.2.2. Interpretation of the Stinespring dilation . 20 2.2.3. The Heisenberg microscope . 22 2.2.4. Cavity quantum electrodynamics . 24 2.2.5. Beams and rates . 24 2.3. Semigroups and evolution systems . 26 2.3.1. Some physical considerations . 26 2.3.2. Basic properties . 28 2.3.3. The Markov property . 36 2.3.4. Evolution systems . 37 2.3.5. -valued evolution systems . 40 D 2.3.6. Evolution systems and non autonomous Cauchy equations . 41 vii Contents I. The Lindblad equation 43 3. Lindblad generators 45 3.1. Norm continuous semigroups . 45 3.1.1. The Lindblad theorem . 45 3.1.2. Interpretation of Lindblad generators . 46 3.2. Strongly continuous semigroups . 47 3.2.1. Minimal solutions . 48 3.2.2. Unitality of minimal solutions . 51 3.3. Gauge invariance . 52 3.4. Stinespringcontinuity . 54 4. Measurement on arrival 57 4.1. Arrival-time measurements . 58 4.1.1. Definition . 59 4.1.2. Outline . 61 4.1.3. Infinitesimal description of arrival-time measures . 61 4.1.4. dom( ) as an operator space . 65 Z∗ 4.1.5. Proof of the characterization theorems . 66 4.2. Exitspaces .................................... 67 4.2.1. General remarks . 67 4.2.2. Definition . 68 4.2.3. Relation to arrival-time measures . 69 4.2.4. Re-insertions . 70 4.2.5. Re-insertions and Lindblad generators . 71 4.2.6. Measurements on arrival . 72 4.2.7. Exit spaces for evolution systems . 72 4.2.8. Exit spaces for general contractive semigroups . 73 4.3. Completely-positive perturbations of semigroups . 74 4.3.1. Integration against arrival-time measures . 74 4.3.2. Repeated integration and perturbation of semigroups . 75 4.3.3. Resolvents for perturbed semigroups . 77 4.3.4. Delayed choice measurements . 79 5. Holevo’s counterexample 83 5.1. Outline ...................................... 83 5.1.1. Setup . 83 5.2. Construction of the counterexample . 84 5.2.1. The diffusion on R+ ........................... 84 5.2.2. Two types of diffusion on R+ R+ . 87 × 5.3. A non-standard semigroup . 92 5.3.1. Definition . 92 viii Contents II. Continuous time limit of repeated observations 97 6. Delayed choice measurement 99 6.1. Iterated measurements . 99 6.2. Main results . 101 6.3. Finitely correlated states . 102 6.4. Limit space . 104 6.4.1. Hilbert space of cMPS . 105 6.4.2. Exponential vectors . 111 6.5. Other dilation spaces . 112 6.5.1. A projective limit construction . 112 6.5.2. A minimal dilation space . 113 7. Limit of repeated quantum observations 115 7.1. Main results . 115 7.1.1. Results for unbounded generators . 115 7.1.2. Results for bounded generators . 117 7.1.3. Discussion of the assumptions . 118 7.1.4. Discussion of the dilation . 119 7.2. cMPS . 119 7.2.1. Basic properties . 120 7.2.2. cMPS over R ...............................120 7.2.3. cMPS as a variational class . 121 7.3. Comparison with the literature . 121 7.3.1. Quantum stochastic calculus . 122 7.3.2. Others . 124 7.4. Proof: unbounded case . 125 7.4.1. Technical preliminaries .
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