Quantum Evolution and Anticipation
Quantum Evolution and Anticipation
by Hans-Rudolf Thomann March 2010
Author address
Im Altried 1h, 8051 ZΓΌrich,
[email protected]
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Quantum Evolution and Anticipation
CONTENTS 0
Introduction ................................................................................................................. 4
1
Quantum Evolution Systematic ................................................................................. 5 1.1 Basic Terminology and Definitions ................................................................................ 5 Definition 1: Evolution Scenario ...................................................................................... 5 Definition 2: Spectral measures and quantities ............................................................... 6 Proposition 1: Spectral Measures ................................................................................... 7 Theorem 1: Embedded Orthogonal Evolution ................................................................. 8 1.2 Characterization of Finite-dimensional Evolution......................................................... 8 1.2.1 Amplitudes, Characteristic Polynomial, Eigenvalues and Spectra .................... 8 1.2.2 Domain of Positivity .......................................................................................... 9 1.2.3 Summary.........................................................................................................10 Theorem 2: Characterization of Finite-dimensional Evolution ........................................10 1.3 Characterization of Infinite-dimensional Evolution .................................................... 11 Theorem 3: Characterization of Infinite-dimensional Evolution ......................................11
2
Anticipation in Quantum Evolution ......................................................................... 12 2.1 Anticipation and Retrospection ................................................................................... 12 Definition 3: Anticipation Amplitudes and Probabilities ..................................................12 2.2 Anticipation Amplitudes and the Spectral Difference ................................................ 12 Definition 4: Spectral Difference ....................................................................................13 Lemma 1: Well-behaved measures ...............................................................................13 Proposition 2: Representation of Anticipation Amplitudes ..............................................13 2.3 Assessment Methodology ............................................................................................ 14 Definition 5: ...................................................................................................................14 2.4 Anticipation Strength .................................................................................................... 14 Theorem 4: Anticipation Strength ..................................................................................14 Lemma 2 .......................................................................................................................15
3
Conclusions .............................................................................................................. 16
4
Bibliography .............................................................................................................. 17
Appendix A: Spectral-analytic Lemmata ....................................................................... 18 A.1 Inversion of the Discrete Fourier Transform ππ^π§ π§ β β€............................................. 18 Lemma A.1: Inversion of the Discrete Fourier Transform ππ^π§ π§ β β€ ............................18 Examples ......................................................................................................................20 A.2 Representation of the Dirac ο€-Functional .................................................................... 20 Lemma A.2: Representation of the Dirac ο€-Functional ...................................................21 Corollary A.2 .................................................................................................................21 A.3 Time-averages of π
π^ππ .................................................................................................... 22
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Quantum Evolution and Anticipation
Abstract A state is said to exhibit orthogonal evolution of order πΏ and step size π iff it evolves into orπ thogonal states at times 0 < π π β€ πΏ. Measurement at time 2 yields states from this orthogonal set with certain probabilities. This effect, which we name anticipation, has been described in a previous paper in a special setting. Here we extend our analyzes to general type quantum evolutions and spectral measures and prove that quantum evolutions possessing an embedded orthogonal evolution are characterized by positive joint spectral measure. Furthermore, we categorize quantum evolution, assess anticipation strength and provide a framework of analytic tools and results, thus preparing for further investigation and experimental verification of anticipation in concrete physical situations such as the H-atom, which we have found to exhibit anticipation.
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Quantum Evolution and Anticipation
0
Introduction
This paper generalizes and extends the findings of (Thomann, 2008), where we studied the special case of orthogonal evolutions of maximal order with pure point and absolutely continuous measures, to arbitrary types of spectral measures and evolutions. ππ»
Under a Hamiltonian π», a quantum state π evolves into an orbit ππ‘ = ππ‘ π, where ππ‘ = π β β π‘ is defined on the closure of π· π» . By spectral theory (Reed, 1980), the spectral measure ππ π of π and π» is uniquely defined by π, π π» π = β« π π» πππ π for any analytic function π. For fixed step size π > 0, π0 = π and π β β€, an evolution of order πΏ β₯ 0 is given by the primary sequence ππ = πππ π0 and the dual sequence, ππ = πππ π0 , such that ππ , ππ = πΏππ π, π β β€ , π β π β€ πΏ (see definition 1). The amplitudes ππ‘ , π0 are determined by the spectral measure ππππ (see definition 2). Positive evolutions are defined as those with positive π. Theorem 1 states that every positive evolution contains an embedded orthogonal evolution, driving the component π 0 of π0 of size π through πΏ + 1 mutually orthogonal states. Theorems 2 and 3 prove the existence of (not necessarily positive) evolutions of any order for point, singular-continuous and absolutely continuous spectrum, respectively. Quantum-Mechanical anticipation (definition 3) is defined for the embedded orthogonal evolution in terms of the anticipation amplitudes πΌπ = π π , ππ 2 π 0 , and the anticipation probabilities ππ = πΌπ 2 , expressing the result of a measurement of π 0 at time π 2, where π β€ πΏ. Quantum-mechanical retrospection is the time-reverse of anticipation. Measurements anticipate future (or recall past) states π π with probability ππ , which is the reason for the naming. Theorem 4 assesses the anticipation strength in various situations. This paper provides analyses and tools facilitating further research about anticipation. In chapter 1 we introduce the basic terminology and definitions, characterize finite-dimensional and infinite-dimensional evolutions and their spectral measures, and explore the topology of the manifold of positive evolutions. Chapter 2 explains and defines anticipation, provides a representation of anticipation amplitudes and analyzes anticipation strength. To cover not only point and absolutely continuous but also singular-continuous measures, we use some formalism from spectral theory an provide mathematical theorems in appendix A. Theorem A.1 establishes explicit formulae for the Fourier coefficients of continuous tions πΉ supported on πΎ whose second distributional derivatives yield a given measure π, inverting the discrete Fourier transform ππ ^ π . Theorem A.2 proves a very general representation of Dirac's πΏ functional. In a future paper we will present results from numerical simulations of the point spectrum of the H-atom as well as equidistant and random point spectra, demonstrating that anticipation and retrospection occur in common situations. The quantum anticipation simulator software running und Microsoft Windows is already downloadable from the author's homepage http://www.thomannconsulting.ch/public/aboutus/aboutus-en.htm.
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Quantum Evolution and Anticipation
1
Quantum Evolution Systematic
In this chapter we introduce the basic definitions and terminology, cast quantum evolutions into a set of categories and derive the characteristic spectral properties of each category. 1.1 Basic Terminology and Definitions Let π» be an essentially self-adjoint Hamiltonian. The set of states we are considering is the Hilbert space β which equals the closure of π· π» . Throughout the paper all states will be assumed to have unit norm. New technical terms are introduced in italic type. For any state π β β the finite spectral measure ππ π is uniquely defined by the inner product π, π π» π = β« π π» πππ π for any analytic function π (see e.g. (Reed, 1980)) This relation induces a unique unitary mapping βπ : β β β 2 πππ such that the image of π is the constant function 1. We will sometimes identify states with their images under this mapping. ππ»
The evolution operator ππ‘ = π β β π‘ is a unitary isomorphism of β, under which π evolves into an orbit ππ‘ = ππ‘ π. The amplitude π0 , ππ equals the Fourier transform ^ βπππ β πππ π = β« π πππ π . By the unitarity of ππ‘ , π0 , ππ = ππ‘ , ππ‘+π is independent from π‘. Our subject is the evolution of states at equidistant time steps. To analyze them we define evolution scenarios comprising a primary and an associated dual sequence of states. Definition 1: Evolution Scenario An evolution scenario (or short: evolution) is characterized by the following elements: A Hamiltonian π». The step time π > 0. The primary base state π0 . The doubly-infinite primary sequence ππ = πππ π0 π β β€ . The primary amplitudes π½π = π0 , ππ satisfying π½βπ = π½π . The cyclic space πΆ = Span ππ π β β€ , which is invariant under ππ . The dimension π = dim πΆ , 0 < π β€ β. The integer order πΏ, π > πΏ β₯ 0. The subspace πΆπΏ = Span ππ 0 β€ π β€ πΏ of πΆ. The dual base state π0 β πΆπΏ of unit norm meeting π0 , ππ = πΏπ π β β€ , 0 β€ π β€ πΏ and ππ , ππ = ππ , ππ π, π β β€ . k) The doubly-infinite dual sequence ππ = πππ π0 π β β€ in πΆ. a) b) c) d) e) f) g) h) i) j)
An evolution scenario is said to exhibit orthogonal evolution of order πΏ iff ππ = ππ . Evolution scenarios exist for any ππ , π0 and 0 < πΏ β€ theorems 2 and 3. For πΏ = β see theorem 2e).
β
π β 1 2 < β, as we will prove in
Provided its existence π0 is that vector in the orthogonal complement of the hyperplane Span ππ 0 < π β€ πΏ relative to πΆπΏ uniquely defined by the requirement that π0 , π0 = 1, and β > π0 2 β₯ 1.
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Quantum Evolution and Anticipation
As, by the unitarity of ππ‘ , π0 , ππ = ππ , ππ+π is independent from π, our construction ensures that ππ+π , ππ = πΏπ π, π β β€ , 0 β€ π β€ πΏ . Thus, within the order, the dual sequence is orthogonal to the primary sequence. In case of orthogonal evolution, the primary and dual sequence coincide, thus π0 , π0 = π0 22 = 1. State evolution is completely determined by spectral measures and quantities, which we introduce now. Definition 2: Spectral measures and quantities Until the end of this chapter β
π
denotes the β π πππ norm. The Lebesgue norm is β
π,ππ .
a) The spectral measure of the primary and dual sequence are denoted by ππ π and ππ π , respectively. π b) With π = βπ π0 β β 2 πππ the joint spectral measure πs π is πππ = π πππ . 1
c) Positive evolution is characterized by π being a non-negative function. π
0β€π
1. π=
π π 1 1
= π 0
1
π
1
= β« π πππ = π0 , π0 = 1, πππ = ππππ and βπ π 0 = π1 2 ,
is a positive quantity. 0 < π β€ 1, as it is the inner product of two unit
vectors. In case of positive evolution π = β« π1
2
πππ .
The definition of 1β² requires that π be real-valued. This holds because π0 , ππ + πβπ = 2π cos πππ β, π0 , πβπ β ππ = 2 β« π sin πππ β πππ and ππ Β± πβπ , π0 = 0 0 < π < πΏ . As β« π πππ = 1 > 0, π is the unique solution of a system of purely real (integral) equations. All rights reserved 6/22
Quantum Evolution and Anticipation
πππ =
π πππ π 1
is a positive measure, π0 , ππ = π
positive evolution 1β² = 1 a.e., π
1
1 β«1
β²
π βππππ
β
πππ = π 0 , π πβ² . In case of
= 1, π πβ² = π π and π π has orthogonal evolution.
π is periodic with period 2π β π, as it has a representation π = π β€πΏ ππ π βππππ β for certain coefficients ππ , which satisfy ππ = πβπ because π is real-valued. The same follows for 1β² and βπ π π . The unitary mappings βπ₯ : β β β 2 πππ₯ , existing for π₯ = π, π , induce unitary mappings ππ₯ : πΆ β β 2 πππ₯ between the cyclic space πΆ and the Lebesgue spaces of the reduced spectral measures. Due to the periodicity of π, 1β² and βπ π 0 there holds ππ ππ = ππ βππππ β in β 2 πππ , πππ =
π πππ π 1
ππ π0 = 1β²
1
π
and ππ π π = π βππππ
π , as well as π
π
= π
β
in β 2 πππ , furthermore ππ π0 =
π 1,π π π π
and
π,ππ π .
The reduced spectral measures as well determine the state evolution at integer times: π0 , ππ = πππ^ ππ β and π0 , ππ = π 1 πππ ^ ππ β . The proof is given in the following proposition which summarizes the above findings. Until stated otherwise, we assume below π β = 1 and omit this factor from all formulae. Proposition 1: Spectral Measures a) The measures ππ₯ π₯ = π, π are positive and have total measure ππ₯ β = 1. πππ = π 2 πππ , βπ π0 = π β1 2 π 11 2 and βπ π0 = 1β² π 1 2 π 1β1 2 . b) π, 1β² and βπ π 0 = π 1 2 π 1β1 2 are real, periodic functions. c) The measures ππ₯ π₯ = π, π are positive and have total measure ππ₯ πΎ = 1. πππ = π 1β1 π πππ , ππ π0 = π β1 2 π 11 2 and ππ π0 = 1β² π 1 2 π 1β1 2 . d) The state evolution at integer times is determined by the reduced measures: π0 , ππ = πππ^ π and π 0 , π πβ² = π 1 1β²πππ ^ π . e) In case of positive evolution π π has orthogonal evolution, π 1 = 1, βπ π 0 = βπ π 0β² = π1 2 and π = β« π1 2 πππ . f) ππ has orthogonal evolution of order πΏ β₯ 0 iff π = 1 iff ππ = ππ = ππ . Proof: For parts a-c) and e-g) see the above derivations. Part d) follows from 2π π +1
π0 , ππ =
π
βπππ
πππ π =
= πππ^ π
π
π
βπππ
π βππ
π
πππ π =
π ββ€ π=2ππ
π
=βπ
πππ π π
=Shift π mod 2π
β
From this proposition there follows a simple, but very important observation, which we state in our first theorem.
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Quantum Evolution and Anticipation
Theorem 1: Embedded Orthogonal Evolution Let π0 have positive evolution of order πΏ. Then there is a component of π0 which has orthogonal evolution of order πΏ. This component is ππ 0 and has norm 0 < π β€ 1. Furthermore, Supp ππ β Supp ππ . The evolution of π 0 is the embedded orthogonal evolution of π0 , and π is its size. Proof: By proposition 1, π 0 has positive evolution. The inclusion of the supports is proper, as e.g. a spectral measure of dimension π whose support contains an equidistant point spectrum of dimension πΏ + 1 β€ π has an embedded orthogonal evolution of order πΏ. β Until the end of this chapter we will, based on proposition 1d, confine to the reduced spectral measures. 1.2 Characterization of Finite-dimensional Evolution In the finite-dimensional case the cyclic space and the Lebesgue spaces have finite dimension. The spectrum is pure point, and the reduced spectral measure has support on π points. This occurs if either the underlying state space β is finite-dimensional, or if the support of ππ has a regular structure such that the reduction yields only finitely many points. An example is the reduction of the spectrum of the SchrΓΆdinger operator ββ + π 2 modulo a rational number. 1.2.1 Amplitudes, Characteristic Polynomial, Eigenvalues and Spectra In this section we explore the relationships of amplitudes, characteristic polynomial, eigenvalues and spectra. Let π
π 0 β€ π < π be the ordered sequence of the reduced spectrum, ππ = π βππ
π the eigenvalues of ππ πΆ, and π€π = ππ π
π the weights. Finiteness of the dimension implies ππ = 0β€π πΏ > π β 1 2 , then ππ,πΏ is an open manifold of dimension 2 π β πΏ β 1, because in this case, as we have seen in the preceding section, some ππ = 0, and each of these complex equations constrains two real degrees of freedom. The boundary of ππ,πΏ consists on the one hand of the manifold ππ,πΏ of eigenvalues admitting at least one non-negative but no positive solution of order πΏ, and on the other hand of degenerate elements with one or more degenerate pairs π
π = π
π . By continuity, the ππ 's either converge to positive limits as elements of ππ,πΏ are degenerating, or the degenerate elements are lying on ππ,πΏ . In the former case we identify the degenerate element with an element of ππ,πΏ , where πΏ < π < π, as ππ,πΏ = β
for π β€ πΏ. Each element of ππ,πΏ corresponds to π π β π degenerate elements of ππ,πΏ . Limit points of ππ,πΏ (degenerate and non-degenerate) with some zero ππ 's are lying on ππ,πΏ . Each element of ππ,πΏ can be constructed as an element of ππ,πΏ combined with π β π points in 0,2π of zero weight, where πΏ < π < π. Let ππ,πΏ = ππ,0 for πΏ β€ π β 1 2 , and the sub-set of ππ,0 satisfying the special constraints mentioned above in case of πΏ > π β 1 2 . Then ππ,πΏ is open in the topology of ππ,πΏ , and ππ,πΏ βͺ ππ,πΏ β ππ,πΏ . ππ,πΏ β ππ,πΏβ1 , as ππ,πΏ satisfies all constraints that ππ,πΏβ1 does. ππ,πΏ partitions ππ,πΏβ1 β© ππ,πΏ into ππ,πΏ and ππ,πΏβ1 β© ππ,πΏ β ππ,πΏ . 1.2.3 Summary The following theorem summarizes in parts a-c and e-f the results of the preceding two sections and completes them in parts d and g. It does however not reproduce the full details and all finesses of the above discussion. Theorem 2: Characterization of Finite-dimensional Evolution If π < β, then the reduced spectrum is pure point and a) ππ is completely determined by π½π 0 β€ π β€ π 2 . b) Evolution scenarios exist for any ππ and 0 < πΏ β€ π β 1 2 , as well as for π β 1 2 < πΏ β€ π provided that ππ = 0 π β 1 2 < π < πΏ . c) The solutions of the system (4.1-2) form the π β 1 β 2πΏ-dimensional linear space π·π,πΏ . Each solution completely determines ππ , ππ and π. d) If π > 2, then for any πs and 0 < π < 1 there are several solutions ππ , ππ . e) πΏ = π β 1 iff the motion is essentially periodic with π = π iff ππ = πβ1 . f) The domain of positivity, ππ,πΏ , is an open manifold of dimension π π > 2πΏ β₯ 0 and 2 π β πΏ β 1 2πΏ β₯ π > πΏ > 0 , respectively, nested into ππ,πΏβ1 . Its boundary is constructible from ππ,πΏ , where πΏ < π < π. g) Necessary and sufficient for any measure ππ to admit non-negative solutions of order πΏ = 1 is that sup Supp ππ β inf Supp ππ > π. Proof: For parts a-c and e-f) see the derivations above. Part d) We prove the proposition for 1 < π β€ β and arbitrary spectrum. Let the support of the reduced measures be partitioned into two sets π΄, π΅ with non-zero measure π, π with respect All rights reserved 10/22
Quantum Evolution and Anticipation
to πππ , respectively. Then there is exactly one solution such that βπ π0 has constant value 0 < π₯ β€ 1, π¦ β₯ 1 on π΄, π΅, respectively, because proposition 1 implies ππ₯ + ππ¦ = π and ππ₯ 2 + ππ¦ 2 = 1, which yields a quadratic equation with non-negative discriminant, such that there is a unique solution meeting the inequalities. The claim follows from the fact that in the case of pure point spectrum there are 2π β 2 partitions satisfying the premises. Part g) One verifies this criterion easily in case of point spectrum by noticing that it guarantees opposite sign of the minimum and maximum terms. The extension to continuous spectrum is straight-forward. β
1.3 Characterization of Infinite-dimensional Evolution In the infinite-dimensional case the cyclic space and the Lebesgue spaces have infinite dimension. The spectrum is a mixture of absolutely continuous, singular-continuous and discontinuous spectrum. Infinite-dimensional reduced point spectrum has isolated or even dense points of accumulation. An example for the latter is the reduction of the spectrum of π 2 modulo an irrational number. However, the formulae of theorem 2 below hold even in this case. Relationships similar to those in the finite-dimensional case hold between the various quantities, with slight differences. Theorem 3: Characterization of Infinite-dimensional Evolution If π = β, the reduced spectrum is a mixture of absolutely continuous, singular-continuous and discontinuous spectrum. The motion is aperiodic. a) ππ is completely determined by π½π π β β€, π β₯ 0 , πππ = π· 2 πΉ β π½0 πΏπ β πΎ0 π·πΏπ for the continuous function πΉ and constant πΎ0 given by Lemma A.1 below. b) If ππ has dense support in πΌ = π
0 , π
β , then for any 0 < πΏ < β there is a solution π. c) For any real measure πs , 0 < π < 1 and πΏ < β there are infinitely many solutions ππ . d) π = 1 implies orthogonal evolution ππ = ππ = ππ of length πΏ β€ β. 1
e) πΏ = β implies absolutely continuous spectrum πππ π
= 2π ππ
on πΎ, while 0 < π β€ 1. f) Let ππ have dense support in π
π β π, π
π + π , where the π-tuple of π
π 's is an element of ππ,πΏ . Then positive evolution occurs for sufficiently small 0 < π βͺ πΏβ1 . Proof: Part a) From π½βπ = π½π and Lemma A.1 below. Part b) Partition the support of ππ into 2πΏ + 1 intervals πΌπ , define πππ = β«πΌ π βπππ
πππ π
and π solve π ππ πππ = πΏπ , ππ 2 = 1. Then π π
= ππ π
β πΌπ is a solution. Part c) The proof of theorem 1b) applies and yields the claim. Part d) By proposition 1f. Part e) By Lemma A.1, the given solution (see example A.1) is unique. Part f) Partition the support into π intervals and determine π as in part b). As π β β, the solutions approximate those of the finite-dimensional problem, and the openness of ππ,πΏ guarantees positivity. β All rights reserved 11/22
Quantum Evolution and Anticipation
2
Anticipation in Quantum Evolution
In this chapter we formally define anticipation for positive evolutions, establish a representation of the anticipation amplitudes in terms of the spectral difference and set up the statistical model for the assessment of anticipation in the remaining chapters. 2.1
Anticipation and Retrospection
Definition 3: Anticipation Amplitudes and Probabilities Consider a positive evolution of order πΏ. Then π 0 has orthogonal evolution of order πΏ, and for 0 β€ π β€ πΏ and 0 < π β€ 1 π
1
a) πΌπ = π πππ π 0 , πβπ 2 π 0 = ππππ π + 2 = π β«βπ π amplitude b) ππ = πΌπ 2 is the πth anticipation probability.
1 2
βπ π+ π
πππ π
is the πth anticipation
β
Let π΄ be an observable defined on Span π π 0 β€ π β€ πΏ (whose dimension lies in the range πΏ + 1, π ), such that π π 0 β€ π β€ πΏ is an eigenfunction of π΄. Then ππ is the probability to obπ π tain π π , when measuring π΄ at time β 2 . Thus ππ is the probability to observe at time β 2 that component of the embedded orthogonal evolution appearing in the sequence of discrete evoπ lution steps not before the πth step: With probability ππ , measurement at time β 2 anticipates the state at time ππ, and projects the target into state π π . This is the reason to name the effect anticipation. Due to anticipation, measurement interactions can speed-up processes and de-stabilize states. Let's define "computation" as "controlled state preparation by quantum evolution". Then due to anticipation measurements at π time β 2 may provide a rapid advance of computation results. π
If π΅ is an observable with eigenfunctions π π βπΏ β€ π β€ 0 , then measurement of π΅ at time + 2 will project the state back to a previous state. This effect, which we name retrospection, can slow-down processes and stabilize states. Retrospection amplitudes and probabilities can be defined analogous to definition 3 above, and all of the following analyzes of anticipation hold as well for retrospection. In case of non-positive evolution measurements are less likely to project states forward or backwards on their trajectory, as there is no embedded orthogonal evolution. Therefore we confine our definition and analysis of anticipation and retrospection to positive evolutions. The occurrence of positive evolutions will be investigated in a future paper. 2.2
Anticipation Amplitudes and the Spectral Difference
In chapter 1 we have seen that the evolution is determined by the reduced spectra. The anticipation amplitudes being defined at half-step time thus can be represented by the reduced spectra of the evolution with step size 1 2. In definition a) below the modular factor selects from the real axis the odd and even numbered intervals of length 2π, respectively. Recall from (Last, 1996) that the measure π is absolutely continuous with respect to the measure π, iff ππ = πππ, for some Borel function π. All rights reserved 12/22
Quantum Evolution and Anticipation
Definition 4: Spectral Difference a) b) c) d)
ππ₯π ,π Shift π
= πππ ,4π π
+ 2ππ π = 0,1 0 β€ π
β€ 2π, π = 0,1 . ππ₯π = ππ₯π ,0 β ππ₯π ,1 is the spectral difference measure. π is well-behaved iff ππ₯π ,π = ππ ππ π. π. for some measure π and Borel functions ππ . π βπ Let π be well-behaved. Then π¦ = π0 +π1 is the spectral difference function. 0
1
See definition 2d-e) for the notation used in a). β In the sequel, we require π to be well-behaved, where π is admitted to be a spectral measure of arbitrarily general type. The following lemma shows that this is a weak condition. Lemma 1: Well-behaved measures a) If π is pure point, then it is well-behaved. b) If π is absolutely continuous, then it is well-behaved. c) If the support of ππ₯π can be partitioned into measurable sets, such that on each set the restrictions of ππ₯π ,0 and ππ₯π ,1 to this set are normally related, then π is wellbehaved. Two measures, π and π, supported on a set π, are said to be normally related if either π is absolutely continuous w.r.t. π, or vice versa, or π has zero measure under one of them. Proof: a) ππ₯π ,0 and ππ₯π ,1 are pure point and meet the premises of c). b) ππ₯π ,0 and ππ₯π ,1 are absolutely continuous and meet the premises of c). c) Consider a set π and the restriction of the measures to this set. If ππ₯π ,0 is absolutely continuous w.r.t. ππ₯π ,1 , then set ππ = ππ₯π ,1 , π1 = 1, π0 =
ππ₯ π ,0 ππ₯ π ,1
. If ππ₯π ,0 is singular w.r.t.
ππ₯π ,1 , then set ππ = ππ₯π ,0 , π0 = 1, π1 = 0. The reversed cases are analogous.
In the well-behaved case the following representations hold: Proposition 2: Representation of Anticipation Amplitudes a) ππ₯π ,0 + ππ₯π ,1 = πππ , ππ₯π = π¦πππ . 1
π
b) πΌπ = ππ₯π ^ π + 2 = β«βπ π
1 2
βπ π+ π
ππ₯π π
.
2
c) π¦ is a real function in β πππ , π¦ β€ 1. 1 d) πΌπ = π π¦πππ ^ π + 2 . Proof: a) By definition 2 and 4) and the well-behavior of π. b) π
πΌπ = π
π βπ
1 βπ π+ π
2 ππ₯π ,0
+π
1 βπ π+ 2π+π
2
ππ₯π ,1
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β
Quantum Evolution and Anticipation
π
=π
π
1 βπ π+ π
2
βπ
ππ₯π ,0 β ππ₯π ,1 .
c) By definition 4d). d) Straight-forward from part a-b) and definition 2. β
2.3 Assessment Methodology To assess anticipation strength, we consider laboratory and field situations where the primary base state is sampled from a certain distribution and its evolution at a certain step time is observed. This distribution induces a distribution of πππ defined on a probability space of spectral measures. In our analyses we focus on distributions conditional under ππ₯π,0 + ππ₯π,1 = πππ , from which any other distributions are obtained as products with the distribution of πππ . In this conditional model only the spectral difference measure ππ₯π is stochastic, whereas πππ is pre-scribed. In contrast to (Thomann, 2008), where we asserted, that technologies for the preparation of orthogonal evolutions of a given step size were likely to exhibit orthogonal anticipation of significant strength, we do not make here any specific assumptions on the distribution of the spectral difference, but will make reference to its mean π¦1 and variance π¦2 = πΈ π¦ β π¦1 2 under the given distribution, which by the boundedness of π¦ always exist. The following definitions are used below: Definition 5: a) ππ = 0β€π 1 . Ξ΅π 3. ππΏ = 4Ξ΅β2 Οβ2 π 2 sin2 2 1 + O πΏβ1
β π 2 π β 0, πΏ β β .
Proof: Part a.1) By theorem 2, π 0 is the embedded orthogonal evolution. As π 0 , β¦ , π πΏ are mutually orthogonal, ππΏ = π§ 22,ππ π , where π§ equals the projection of π¦ on Span π 0 , β¦ , π πΏ . Part a.2) By (Last, 1996) ππ can be decomposed in a continuous and a pure point part, which are mutually singular. Therefore the second integral in the decomposition 1
βπ π+
π
βπ
β²
2 πΈ ππΏ = β¬π
β π
β² π π¦1 π
π¦1 π
β² πππ π
πππ π
β² + β« πΈ π¦ 2 π
πππ π
πππ π
only 2 depends on πππ ,ππ . The result follows from the identity π¦2 = πΈ π¦ 2 β π¦12 . Notice that, if the evolution is of order πΏ, then πππ ,ππ is either zero or at least of dimen-
2
2 sion πΏ + 1, thus πΏ + 1 β« πππ ,ππ β€ β« πππ ,ππ . Part b.1) As π β β the anticipation amplitudes approximate π 2
1
β2ππ πβ
π+π
π 2
πΏ ππ
1 β2ππ πβ π
πΏ ππ
2 2π
2 πΌπ = πΏβ1 = π β«βπ 2 π π =0 β«βπ 2 ππ 2π The maxima, minima and asymptotics are elementary. Part b.2) From b.1) by the Euler summation formula. Part b.3) By Lemma 2 below. Part c.1) In the same way as part b.1).
Part c.2) From c.1) and c.3), as ππ β₯ Part c.3) By Lemma 1 below.
πΏ π 4
1
πΏβ1 β2ππ πβ2 π πΏ . π =0 π
ππΏ . β
Notice in parts b-c that π₯π approximates an evolution of order πΏ, but this πΏ is unrelated to the order of the evolution under ππ . The following lemma is obtained by direct summation, adapting theorem 4.9a of (P.Henrici, 1988) to periodic functions. Lemma 2 πβ1 β2 sinβ2 π=0 π
π πβ
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1 2
π = 1.
Quantum Evolution and Anticipation
3
Conclusions
We have classified and analyzed quantum evolutions with arbitrary spectrum and shown in theorem 1 that every positive evolution (see definition 2.c) possesses an embedded orthogonal evolution. Our findings on eigenvalues, spectra and domains of positivity have been summarized in theorem 2 for the finite-dimensional and in theorem 3 for the infinitdimensional case. Anticipation and retrospection have been introduced (see definition 3) for positive evolutions, and their strength has been assessed in theorem 4 in the general setting of well-behaved measures (see definition 4.c). Our analyzes show that anticipation and retrospection occur in a wide class of quantum evolutions, generalizing and extending (Thomann, 2008) where only equally distributed measures were considered. An interesting open question is how systems exhibiting anticipation and retrospection interact with other systems, particularly with the environment. In the presence of einselection (Zurek, 2003), interesting phenomena may occur. E.g., if decoherence sets in randomly in the interπΏβ1 val 0, πΏπ , then theorem 4.a.1. implies π πΏ β€ 2 π 2 π¦ 22,ππ π . Given suitable timing, a Zenolike effect may occur, but due to anticipation and retrospection with rapid evolution forward or even backward in time. While the domain of positivity has been analyzed in our study, the requirement of positive evolution in the definition of anticipation and retrospection raises the demand for the demonstration of the presence of positive evolution in physical model systems. This will be the subject of a future paper, presenting numerical simulations showing that anticipation and retrospection occur in a variety of spectra including that of the H-atom. The quantum anticipation simulator software running und Microsoft Windows is already downloadable from the author's homepage http://www.thomannconsulting.ch/public/aboutus/aboutus-en.htm.
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Quantum Evolution and Anticipation
4
Bibliography
Aitken, A. (1956). Determinants and Matrices. Oliver and Boyd. Killip, R. ,. (2001). Dynamical Upper Bounds on Wavepacket Spreading. arXiv:math.SP/0112078v2. Kiselev, A. L. (1999). Solutions, Spectrum and Dynamics for SchrΓΆdinger Operators on Infinite Domains. math.SP/9906021. Last, Y. (1996). Quantum Dynamics and Decompositions of Singular-Continuous Spectra. J. Funct. Anal. 142, pp.406-445. Nielsen, O. (1997). An introduction to integration and measure theory. New York: Wiley. P.Henrici. (1988). Applied Computational and Complex Analysis", Vol.I . Wiley. Reed, M. S. (1980). Methods of Mathematical Physics, Vol.1-4. Academic Press. Rudin, W. (1973). Functional Analysis. Mc Graw Hill. Simon, B. (1990). Absence of Ballistic Motion. Commun. Math. Phys. 134, pp.209-212. Strichartz, R. (1990). Fourier asymptotics of fractal measures. J.Funct.Anal. 89 , pp.154-187. Thomann, H. (2006). Instant Computing β a new computing paradigm. arXiv:cs/0610114v3. Thomann, H. (2008). Orthogonal Evolution and Anticipation. arXiv:0810.1183v1. Thomann, H. (July 2009). Quantum anticipation simulator. http://www.thomannconsulting.ch ("About us" page) . Zurek, W. (July 2003). Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys., Vol 75 .
All rights reserved 17/22
Quantum Evolution and Anticipation
Appendix A: Spectral-analytic Lemmata This appendix contains some useful lemmas from spectral analysis which are referenced in the main body of this paper. Our proofs are original, and we do not know of any references in the literature for them. A.1 Inversion of the Discrete Fourier Transform ππ^ π§ π§ β β€ Integrating any finite measure π yields a continuous function π. As the reduced measure ππ has support in the compact set πΎ, it depends only on πΉ = π πΎ, which in turn can be represented by a Fourier series. Explicit formulae for the coefficients of this series in terms of the discrete Fourier transform ππ ^ π are given below. Lemma A.1: Inversion of the Discrete Fourier Transform ππ^ π§
π§ββ€
Let π be a finite complex measure supported in the half-open interval πΎ = βπ,π , and π½π = ππ ^ π π β β€ be given, where ππ ^ π‘ = β« π βππ
π‘ ππ π . Then a) There is an absolutely continuous function πΉ supported on πΎ such that ππ = π· 2 πΉ β π½0 πΏπ β πΎ0 π·πΏπ , π where the derivative π· = 2ππ is taken in the distributional sense. ππ
2π 2 πΉ = π΅ π
+ π β 2πππΎ0 π
β π½0
b)
πΎ0 =
π ππ
π
2 2
π
β€ π , where π΅ π
= π 4
π½π π β 0 π 2
π βπππ
,
π΅ βπ + π½0 π and π = βπ΅ βπ β ππΎ0 π + π½0 .
c) For any test function π π½π βπππ
2πππ β π = π β π + ππ½0 π
β 2ππΎ0 β π π πβ 0
ππ β π =
π½π π βπππ
β π
Proof: All references qualified by an "R" in the proof below address (Rudin, 1973). Notice that, other than (Rudin, 1973), we are scaling π· by the factor 2π, to let πΎ have unit Lebesgue measure. W.l.o.g. π β«βπ
π
π
π π
π½π β€ 1. Define π π
= π ββ,π
. Then π βπ = 0 and π π = π½0 . Let πΎπ = ππ
. 2π
First notice that Ξπ = β« πππ, defined for test functions π, is a functional of zero order in the sense of the note following theorem R6.8, as Ξπ β€ maxπ
βπΎ π . By theorem R6.27 and exercise R6.17 there is a continuous function π β πΆ β such that ππ = π· 2 π in the distributional sense, i.e. π·Ξπ = βΞπ·π (see R6.12). To adjust for the factor 2ππ, π·π = βπ 2π β1 π. W.l.o.g. π π
= 0 π
β€ βπ .
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Quantum Evolution and Anticipation
We represent πΉ = π πΎ as a function on β by defining πΉ = π 1 β π»π , using the Heaviside π ππ
function π»π = π
β₯ π . With π·π»π₯ = 2πππΏπ₯ , ππΏπ₯ = π π₯ πΏπ₯ and π π = β«βπ βπ π π
= 2π 2 βπ 2π
β1
πΎ0 we find
πΉ = π 1 β π»π π·πΉ = βπ 2π β1 π 1 β π»π β 2ππππΏπ = βπ 2π β1 π 1 β π»π β πΎ0 πΏπ π π· 2 πΉ = 1 β π»π ππ β ππΏπ β πΎ0 π·πΏπ = ππ β π½0 + 2πππΎ0 ππ
πΏπ
(1.1) (1.2) (1.3)
The products π 1 β π»π and π 1 β π»π are functionals, have been differentiated using the product rule, which is not generally applicable (see R6.14 and R6.15). However, integration π by parts of the r.h.s. of π·π 1 β π»π β π = βπ·π 1 β π»π β π β² = β β«βπ ππ β² yields the r.h.s. of 1.2, and π· βπ 2π
β1
β
β
β π = βπ 1 β π»π β π β² = β β«ββ ππ β² + β«π ππ β² = ππ β ππΏπ β π.
π 1 β π»π
This proves part a). Now by theorem 7.15, the Fourier transform π· 2 πΉ ^ π‘ = 2π 2 π‘ 2 πΉ ^ , by example 7.16.3, example 7.16.5 and theorem 7.2a) πΏπ^ = 2π β1 π βπππ‘ and π·πΏπ ^ = π‘π βπππ‘ , thus (2)
πΉ^ =
ππ ^ β π½0 +2ππΎ0 π‘ π βπππ‘ . 2π 2 π‘ 2
As πΉ is continuous on a compact set (the closure of πΎ), the Fourier series (3)
πΉ=
ππ π βπππ
exists, and its Fourier transform (4)
πΉ^ π‘ =
ππ
sin π π‘βπ π π‘βπ
.
Combined with (6) this yields (5)
ππ = πΉ ^ π =
π½ π β β1 π π½ 0 +2ππ πΎ0 2π 2 π 2
.
The singularity at the origin is removable; ππ ^ is analytic, as ππ has support in πΎ and π‘ β1 ππ ^ π‘ = π ^ π‘ = πΎ0 π‘ = 0 . Therefore π0 is well defined. We insert (5) into (3) and sum the terms on the r.h.s. of (5) up separately into the absolutely convergent series (6)
π΅ π
=
π½π πβ 0 π 2
π βπππ
π
and the two sums considered next. As π ππ
πβ 0
β1 π π2
π βπππ
=
πβ 0
β1 π π
π βπππ
= ln
ln π βππ
= βππ
π
< π , the absolutely convergent series on the l.h.s. equals π β constant π. Redefining π = π + π0 we obtain (7.1) (7.2)
2π 2 πΉ = π΅ π
+ π β 2πππΎ0 π
β π½0 2ππ β π = π
π½π πβ 0 π
π
2 2
π βπππ
β π + 2πππΎ0 + π½0 π
β π
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1+π βππ
1+π ππ
π
2 2
=
for some
Quantum Evolution and Anticipation
ππ β π =
(7.3)
π ππ
πβ 0 π½π
π βπππ
β π + π½0 β π
(7.4)
2πππΎ0 =
π΅ βπ + π½0 π
(7.5)
π = βπ΅ βπ β 2π 2 ππΎ0 + π½0
π2 2
where (7.1) is defined for π
β€ π due to the continuity of πΉ, (7.2) and (7.3) for test functions π with support in πΎ. (7.1) follows from the norm convergence of π΅, (7.2) and (7.3) from 6.17, as π΅ β π = π½π βπππ
β π exists for every π due to the norm convergence of π΅ and boundedness of πβ 0 2 π π
π βπππ
.
This proves part b-c). (7.4) and (7.5) are determined from the boundary condition π βπ = π βπ = 0. πΉ is absolutely continuous if π is continuous, but may not even be one-sided differentiable in the singular-continuous spectrum and at points of accumulation of the discontinuous spectrum. In that case the r.h.s. of (7.4) cannot be evaluated. πΎ0 may be approximated from (7.2) using a sequence of test functions converging to π»βπ β π»π . ππ is always well defined by (7.3). β The following examples demonstrate the power of lemma 1. Examples 1) Infinite orthogonal evolution π½π = πΏπ . 1
2π
π
2
1
π
π΅ = 0, πΉ = β1 β β 2 , π = 1 + , ππ = 1. 8 π π 2 π 2) Periodic evolution π½π = πΏπ mod π . For 0 β€ π = π
+ π β 2πππβ1 < 2ππβ1 , 0 β€ π < π: π π΅ ππ
β1
=
βπ πβ 0 ππ
1βπ βπππ
π βππππ
= ππβ1 ln 1βπ +πππ
= ππβ1 ln π βπ
π+ππ
= ππβ1 ln π βπ
π+ππ
=π+
π π. Inserting this into (11.2) yields π = ππβ1 , as the π term is cancelled out and the boundary condition at β π is met. (11.3) yields π΅ = πβ 0 π βππππ
which for ππ
β π βπππ
π πππ
0 mod 2π equals 1βπ βπππ
+ 1βπ πππ
= β1, i.e. ππ = π΅ + 1 = 0, but at the singularities equals (by lemma A.2.c below) the πΏ functional, so ππ = 0β€π