Is ‘Quantum Darwinism’ Really a Darwinism? Florian J. Boge IZWT, BU Wuppertal The Generalized Theory of Evolution D¨ usseldorf University February 3rd, 2018
QT in a Nutshell Decoherence & Q. Darwinism Whence the Darwinism? Conclusions References Rough Guide QT in a nutshell decoherence and ‘Quantum Darwinism’ three steps to dispute Darwinian character: if interpretation neutral, no Darwinism because selection and reproduction apply at di ff erent levels if this is fixed, tied to Everett interpretation if Everett interpretation accepted, no Darwinism after all, due to lack of a resource Quantum Darwinism?, F. J. Boge 2 / 29
QT in a Nutshell Decoherence & Q. Darwinism Whence the Darwinism? Conclusions References Quantum Theory in a Nutshell state | ψ S � of system S is element of a linear vector space H over complex numbers C ‘observables’ O represented by self-adjoint (linear) operators ˆ O acting on H some non-commuting ; e.g. ˆ x ˆ p − ˆ p ˆ x = ı � h � 0 unitary (linear, bijective, norm preserving) operators ˆ U represent state transformations U ( t ; t f ) = e − ı h ˆ e.g. dynamics: ˆ U ( t ; t f ) | ψ S ( t ) � = | ψ S ( t f ) � with ˆ H ( t f − t ) � (simplest case) Quantum Darwinism?, F. J. Boge 3 / 29
QT in a Nutshell Decoherence & Q. Darwinism Whence the Darwinism? Conclusions References Quantum Theory in a Nutshell ‘ambiguity’ 1: there can be physical reasons to write | ψ S � = α 1 | o 1 � + α 2 | o 2 � + α 3 | o 3 � + . . . = � j α j | o j � , where ˆ O | o j � = o j | o j � , meaning that S has definite value o j for O ‘ambiguity’ 2: a state from H can always be written as a superposition in some arbitrary basis of H : | A 2 � | B 1 � | A 1 � | B 2 � Quantum Darwinism?, F. J. Boge 4 / 29
QT in a Nutshell Decoherence & Q. Darwinism Whence the Darwinism? Conclusions References Quantum Theory in a Nutshell for | ψ S � = � O ( o j ) = | α j | 2 ( Born’s rule ) j α j | o j � we have Pr ψ S α j = � o j | ψ S � ( inner product ) � 1 if i = j � o i | o j � = δ ij = ( orthonormal basis ; ONB) 0 else | o j � vs. | o j � � o j | , | ψ S � vs. | ψ S � � ψ S | (‘projectors’, repres. pure states) ρ S = � k p k | ψ ( k ) � ψ ( k ) density operator ˆ S � S | ( mixed state, so long as p k � 1 for some k ) O = � ˆ j o j | o j � � o j | Quantum Darwinism?, F. J. Boge 5 / 29
QT in a Nutshell Decoherence & Q. Darwinism Whence the Darwinism? Conclusions References Interpretive Problems ˆ U h ˆ ı → | o k � |M o j � , where ˆ H int ∆ t | o j � |M 0 � �− U = e � ˆ � → � j , k α jk | o k � |M o j � = � U j α j | o j � |M 0 � o j � |M o j � =: | ψ SM � �− j ˜ α j | ˜ o j � | ˜ entangled state, i.e., cannot be written as | ˜ ˜ M o j � in any basis of H S ⊗ H M ambiguity 2 (again): 2 ( | ↑ z � + | ↓ z � ) |M 0 � �→ | ψ f 1 | ψ SM � = SM � = √ 1 2 ( | ↑ z � |M ↑ z � + | ↓ z � |M ↓ z � ) √ | ψ f 1 SM � = . . . = 2 ( | ↑ x � |M ↑ x � + | ↓ x � |M ↓ x � ) √ projection postulate (Dirac, 1958; von Neumann, 1932): | ψ SM � �→ | ˜ o ℓ � |M o ℓ � ad hoc: how / when / where / why does the change occur? What causes it? (“Heisenberg cut”; “Wigner’s friend”) Quantum Darwinism?, F. J. Boge 6 / 29
QT in a Nutshell Decoherence & Q. Darwinism Whence the Darwinism? Conclusions References Decoherence Theory � U SM , E | ψ SM � |E 0 � = ˆ ˆ U SM , E j α j |S j � |M j � |E 0 � = � j α j | ˜ S j � | ˜ M j � |E j � partial tracing : � ψ SME | = � i , j α j α ∗ ρ SME = | ψ SME � i |S j � � S i | ⊗ |M j � � M i | ⊗ |E j � � E i | ˆ ρ SM = � i , j α j α ∗ Tr E ( ˆ ρ SME ) =: ˆ i |S j � � S i | ⊗ |M j � � M i | � E j |E i � t ρ SM ≈ � j | α j | 2 |S j � if � E i |E j � ≈ 0 for i � j , we obtain ˆ � S j | ⊗ |M j � � M j | often perfectly valid for t → ∞ and good approximation after a short decoherence time (e.g. Joos et al. 2003, p. 67; Schlosshauer 2007, pp. 70 ff .) Quantum Darwinism?, F. J. Boge 7 / 29
QT in a Nutshell Decoherence & Q. Darwinism Whence the Darwinism? Conclusions References Decoherence Theory ρ SM ≈ � j | α j | 2 |S j � ˆ � S j | ⊗ |M j � � M j | approximate & improper mixture of eigenstates ( |S j � � S j | ⊗ |M j � � M j | ) M = � S = � of preferred observables ˆ � M j | , ˆ j m j |M j � j s j |S j � � S j | stable under influence of environment (‘preferred basis’) typically approximately localized states with approximately well-defined velocity / momentum ( quasi-classical ) but : no ‘or’ from an ‘and’ (Bell, 1990) Quantum Darwinism?, F. J. Boge 8 / 29
QT in a Nutshell Decoherence & Q. Darwinism Whence the Darwinism? Conclusions References ‘Quantum Darwinism’ Zurek (2009, p. 182): “Monitoring by the environment means that information about S is deposited in E . [...] Decoherence theory ignores it [the information – FJB]. The environment is ‘traced out’. [...] Quantum Darwinism recognizes that [...] observers eavesdrop on the environment. Most of our data come from fragments of E .” Quantum Darwinism?, F. J. Boge 9 / 29
QT in a Nutshell Decoherence & Q. Darwinism Whence the Darwinism? Conclusions References ‘Quantum Darwinism’ How much information? Shannon entropy H ( X ) = − � x p x log 2 ( p x ) of a variable X with distribution p x “as a measure of how much information we have gained after we learn the value of X ” (Nielsen and Chuang, 2010, p. 500) � � von Neumann entropy S ( S ) = − Tr ρ S log 2 ˆ ˆ ρ S ρ S = � if ˆ x p x | x � � x | (perfect decoherence), the two will coincide how to exploit environment? use mutual information I ( S : F ) = H ( S ) + H ( F ) − H ( S , F ) , where F is a fraction of E (a set of subsystems of E ), and H ( S , F ) is evaluated ρ SE ) ≈ � x p x | x � � x | ⊗ |F x � � F x | after short w.r.t. ˆ ρ SF = Tr E\F ( ˆ decoherence time Quantum Darwinism?, F. J. Boge 10 / 29
QT in a Nutshell Decoherence & Q. Darwinism Whence the Darwinism? Conclusions References ‘Quantum Darwinism’ Diagram cf. Zurek (2009, p. 183) Quantum Darwinism?, F. J. Boge 11 / 29
QT in a Nutshell Decoherence & Q. Darwinism Whence the Darwinism? Conclusions References Whence the Darwinism? three modules of generalized evolution (cf. Schurz 2011, p. 131; Lewontin 1970, p. 1): reproduction : some entities will reproduce (in generations, w.r.t. certain traits) variation : reproduced traits will vary; variation will get reproduced selection : some entities / traits will reproduce faster than others, hence spreading and pushing other entities / traits aside in the long run Quantum Darwinism?, F. J. Boge 12 / 29
QT in a Nutshell Decoherence & Q. Darwinism Whence the Darwinism? Conclusions References Whence the Darwinism? Zurek (2009, p. 182): “only states that produce multiple informational o ff spring—multiple imprints in the environment—can be found out from small fragments of E [variation – FJB]. The origin of the emergent classicality is then not just survival of the fittest states (the idea already captured by [decoherence]) [selection – FJB], but their ability to ‘procreate’, to deposit multiple [...]copies of themselves[...] throughout E [reproduction – FJB].” Quantum Darwinism?, F. J. Boge 13 / 29
QT in a Nutshell Decoherence & Q. Darwinism Whence the Darwinism? Conclusions References Whence the Darwinism, Really? (Step 1) decoherence: � t → ∞ ,Tr E � ⊗ |M i � i , j α i α ∗ � � � � � S i S j � M j | ⊗ |E i � � E j | − − − − −→ j � i | α i | 2 |S i � � S i | ⊗ |M i � � M i | ( selection of the { |S i � � S i |} and { |M i � � M i |} ) before decoherence, there is a range of bases on equal footing ( variation ) ‘multiple copies’ in E of states stable under decoherence ( reproduction ): � → � ˆ U j α j |S j � | ε ( 1 ) j α j |S j � | ε ( 1 ) 0 � �− � j ˆ � → � j α j |S j � | ε ( 1 ) � | ε ( 2 ) U j α j |S j � | ε ( 1 ) � | ε ( 2 ) 0 � �− � j j j . . . observation: selection of observable (and associated basis ) due to decoherence, but reproduction of values (and associated states ) Quantum Darwinism?, F. J. Boge 14 / 29
QT in a Nutshell Decoherence & Q. Darwinism Whence the Darwinism? Conclusions References Whence the Darwinism, Really? (Step 2) fix: let all the eigenstates / projectors of � j s j |S j � � S j | be selected! variations refer to variations of states of S due to prior interaction with E corresponds to selection of the values s k on a set of ‘branches’ |S k � | ε ( 1 ) k � | ε ( 2 ) k � . . . | ε ( N ) � resolving k | ψ SE � = � j α j |S j � | ε ( 1 ) � | ε ( 2 ) � . . . | ε ( N ) � j j j Quantum Darwinism?, F. J. Boge 15 / 29
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