RRepeatable measurements and the collapsepostulate
Michael Zirpel
Abstract
J. v. Neumann justified the collapse postulate by the empirical fact of the re-peatability of a measurement at a single quantum system. However, in his quantummechanical treatment of the measurement process repeatability emerges withoutcollapse. The entangled state of the measurement device and the measured systemafter their interaction ensures it already. Furthermore, this state gives the samepredictions for the measured system alone as the description demanded by the col-lapse postulate.Keywords: measurement process, repeatable measurement, v. Neumann mea-surement, collapse, reduction, projection
None of the fundamental postulates of quantum mechanics is as controversial as thecollapse (reduction, projection) postulate. It demands for a measurement an instanta-neous, non-deterministic state transition, which is seemingly in conflict with the con-tinuous, deterministic state evolution governed by the Schrödinger equation. In pilotwave (Bohm, 1952) or ensemble (Ballentine, 1998) interpretations this postulate isomitted, but it is part of the orthodox Copenhagen interpretation (Heisenberg, 1958)and of the modern quantum computing (Nielsen and Chuang, 2000).v. Neumann (1932) founded his reasoning for this postulate on the Compton-Simonexperiment, where the measurement results for the momentum of a scattered photonand the associated recoil electron determine each other (Compton, 1927). He explainedthis “sharp (causal) correlation” by the state reduction caused by the first measurement.However, in v. Neumann’s quantum mechanical treatment of the measurement pro-cess the repeatability of the measurement emerges without collapse, just by applicationof Born’s probabilistic interpretation of the wave function. The entangled state of themeasurement device and the measured system after their interaction ensures already,that an immediate repetition of the process with a second measurement device of thesame type will give with probability 1 the same result as the first. Furthermore, thisentangled state of the compound system gives the same probabilities as the descriptiondemanded by the collapse postulate for all following measurements at the measuredsystem. 1 a r X i v : . [ qu a n t - ph ] N ov Basic notions and notations
Let S be a quantum mechanical system, described with Hilbert space H , and A = ∑ k a k (cid:12)(cid:12) α k (cid:11)(cid:10) α k (cid:12)(cid:12) ∈ L ( H ) a discrete, non-degenerate observable. For a measurement ofthe observable A at the system S in the state (cid:12)(cid:12) ψ (cid:11) ∈ H • the probability postulate (Born’s rule) demands, that the probability, to get themeasurement result a j , is (cid:12)(cid:12)(cid:10) α j | ψ (cid:11)(cid:12)(cid:12) , • the collapse postulate demands, that, if the measurement result is a j , the stateimmediately after the measurement will be (cid:12)(cid:12) α j (cid:11) . As a consequence of both postulates the system S has to be described after the mea-surement, if the result is unknown, by a mixture of states represented by the statisticaloperator W = ∑ j (cid:12)(cid:12)(cid:10) α j | ψ (cid:11)(cid:12)(cid:12) (cid:12)(cid:12) α j (cid:11)(cid:10) α j (cid:12)(cid:12) (2.1)v. Neumann’s quantum mechanical treatment of the measurement process describesa measurement as an interaction between the system S and a measurement device M ,which is itself a quantum system with Hilbert space H M . Pairwise orthogonal pointerstates (cid:12)(cid:12) ϕ ( M ) k (cid:11) ∈ H M indicate the measurement results. The interaction of the measuredsystem and the measurement device is described by an unitary transformation U ( SM ) inthe tensor product Hilbert space H ⊗ H M of the compound system SM . The assump-tion, that an ideal measurement of an eigenstate (cid:12)(cid:12) α k (cid:11) of the measured observable A should give exactly the corresponding pointer state (cid:12)(cid:12) ϕ ( M ) k (cid:11) as result, without disturbingthe system S , can be expressed by U ( SM ) (cid:12)(cid:12) α k (cid:11)(cid:12)(cid:12) ϕ ( M ) (cid:11) = (cid:12)(cid:12) α k (cid:11)(cid:12)(cid:12) ϕ ( M ) k (cid:11) (2.2)Therefore, with the initial state (cid:12)(cid:12) ψ (cid:11) of the system S the unitary transformation U ( SM ) will give the entangled final state (cid:12)(cid:12) Φ (cid:11) = U ( SM ) (cid:12)(cid:12) ψ (cid:11)(cid:12)(cid:12) ϕ ( M ) (cid:11) = ∑ k (cid:10) α k | ψ (cid:11)(cid:12)(cid:12) α k (cid:11)(cid:12)(cid:12) ϕ ( M ) k (cid:11) The reading of the pointer can be considered as a secondary measurement. The proba-bility of the result A ( M ) j = ⊗ (cid:12)(cid:12) ϕ ( M ) j (cid:11)(cid:10) ϕ ( M ) j (cid:12)(cid:12) , that the pointer state (cid:12)(cid:12) ϕ ( M ) j (cid:11) is observed,is according the probability postulate for the final state (cid:12)(cid:12) Φ (cid:11) given by p ( A ( M ) j ) = (cid:10) Φ (cid:12)(cid:12) ϕ ( M ) j (cid:11)(cid:10) ϕ ( M ) j (cid:12)(cid:12) Φ (cid:11) = (cid:12)(cid:12)(cid:10) α j | ψ (cid:11)(cid:12)(cid:12) (2.3)As long as the compound system SM is in the state (cid:12)(cid:12) Φ (cid:11) , the system S alone has tobe described by the statistical operator given by the partial tracetr H M ( (cid:12)(cid:12) Φ (cid:11)(cid:10) Φ (cid:12)(cid:12) ) = ∑ k (cid:10) α k | ψ (cid:11)(cid:10) ψ | α k (cid:11)(cid:12)(cid:12) α k (cid:11)(cid:10) α k (cid:12)(cid:12) = W (2.4)which is identical with (2.1). But this description gives no statement about the correla-tion with the measurement result. 2 Repeatability of the measurement
The wave function of the compound system can be used to compute the conditionalprobabilities of the results of further measurements. When a second measurement de-vice M (cid:48) of the same type interacts with the measured system S as part of the compoundsystem SM in state (cid:12)(cid:12) Φ (cid:11) in the enlarged Hilbert space H ⊗ H M ⊗ H M (cid:48) , this gives thestate (cid:12)(cid:12) Φ (cid:48) (cid:11) = U ( SM (cid:48) ) (cid:12)(cid:12) Φ (cid:11)(cid:12)(cid:12) ϕ ( M (cid:48) ) (cid:11) = ∑ k (cid:10) α k | ψ (cid:11)(cid:12)(cid:12) α k (cid:11)(cid:12)(cid:12) ϕ ( M ) k (cid:11)(cid:12)(cid:12) ϕ ( M (cid:48) ) k (cid:11) The pointer results of both measurement devices define a Boolean event algebra anda common probability space, because all projections onto the pointer states A ( M ) j = ⊗ (cid:12)(cid:12) ϕ ( M ) j (cid:11)(cid:10) ϕ ( M ) j (cid:12)(cid:12) ⊗ ( M (cid:48) ) and A ( M (cid:48) ) k = ⊗ ( M ) ⊗ (cid:12)(cid:12) ϕ ( M (cid:48) ) k (cid:11)(cid:10) ϕ ( M (cid:48) ) k (cid:12)(cid:12) commute pairwise.Therefore, the conditional probability to get the pointer result A ( M (cid:48) ) k in the second mea-surement, given the pointer result A ( M ) j in the first, is well-defined p ( A ( M (cid:48) ) k | A ( M ) j ) = p ( A ( M (cid:48) ) k ∧ A ( M ) j ) p ( A ( M ) j ) = (cid:10) Φ (cid:48) (cid:12)(cid:12) ϕ ( M ) j (cid:11)(cid:10) ϕ ( M ) j (cid:12)(cid:12) ⊗ (cid:12)(cid:12) ϕ ( M (cid:48) ) k (cid:11)(cid:10) ϕ ( M (cid:48) ) k (cid:12)(cid:12) Φ (cid:48) (cid:11)(cid:10) Φ (cid:48) (cid:12)(cid:12) ϕ ( M ) j (cid:11)(cid:10) ϕ ( M ) j (cid:12)(cid:12) Φ (cid:48) (cid:11) = δ j , k (3.1)This means: The probability is , that an immediate repetition of the measurementgives the same result as the first. The same statement is valid for further repetitions and, when the initial state of the system is a mixture or when the measured observableis degenerate .This repeatability of the measurement is a consequence of condition (2.2). Theweaker condition U ( SM ) (cid:12)(cid:12) α k (cid:11)(cid:12)(cid:12) ϕ ( M ) (cid:11) = (cid:12)(cid:12) ψ k (cid:11)(cid:12)(cid:12) ϕ ( M ) k (cid:11) with some not further specified (cid:12)(cid:12) ψ k (cid:11) ∈ H , gives the same probability distribution forthe pointer results (2.3), but in general without repeatability. With v. Neumann’s description of a repeatable measurement all further measure-ments at the measured system have without collapse the same conditional probabil-ity distributions as demanded by the collapse postulate. To see that, let instead of M (cid:48) another device M (cid:48)(cid:48) for the measurement of an arbitrary non-degenerate observable B = ∑ k b k (cid:12)(cid:12) β k (cid:11)(cid:10) β k (cid:12)(cid:12) ∈ L ( H ) interact with the measured system S as part of the com-pound system SM in state (cid:12)(cid:12) Φ (cid:11) . With the resulting state (cid:12)(cid:12) Φ (cid:48)(cid:48) (cid:11) = ∑ k , j (cid:10) α k | ψ (cid:11)(cid:10) β j | α k (cid:11)(cid:12)(cid:12) β j (cid:11)(cid:12)(cid:12) ϕ ( M ) k (cid:11)(cid:12)(cid:12) ϕ ( M (cid:48)(cid:48) ) j (cid:11) This gives a simple explanation of avalanche effects, which are part of some measurement devices,where many molecules interact as measurement devices with a measured particle and the resulting state ismacroscopic visible, because all molecules are observed in the same pointer state. But the spectrum of the observable has to be discrete (Busch et al., 1991). B ( M (cid:48)(cid:48) ) k = ⊗ ( M ) ⊗ (cid:12)(cid:12) ϕ ( M (cid:48)(cid:48) ) j (cid:11)(cid:10) ϕ ( M (cid:48)(cid:48) ) j (cid:12)(cid:12) , if the first measurement result is A ( M ) j , is also well-defined p ( B ( M (cid:48)(cid:48) ) k | A ( M ) j ) = (cid:10) Φ (cid:48)(cid:48) (cid:12)(cid:12) ϕ ( M ) j (cid:11)(cid:10) ϕ ( M ) j (cid:12)(cid:12) ⊗ (cid:12)(cid:12) ϕ ( M (cid:48)(cid:48) ) k (cid:11)(cid:10) ϕ ( M (cid:48)(cid:48) ) k (cid:12)(cid:12) Φ (cid:48)(cid:48) (cid:11)(cid:10) Φ (cid:48)(cid:48) (cid:12)(cid:12) ϕ ( M ) j (cid:11)(cid:10) ϕ ( M ) j (cid:12)(cid:12) Φ (cid:48)(cid:48) (cid:11) = (cid:12)(cid:12)(cid:10) β k | α j (cid:11)(cid:12)(cid:12) This exactly the same value as with the collapse postulate; (3.1) is just a special casewith B = A .The total probability of the result B ( M (cid:48)(cid:48) ) k p ( B ( M (cid:48)(cid:48) ) k ) = ∑ j p ( B ( M (cid:48)(cid:48) ) k | A ( M ) j ) p ( A ( M ) j ) = ∑ j (cid:12)(cid:12)(cid:10) α j | ψ (cid:11)(cid:12)(cid:12) (cid:10) β k (cid:12)(cid:12) α j (cid:11)(cid:10) α j (cid:12)(cid:12) β k (cid:11) = tr ( W (cid:12)(cid:12) β k (cid:11)(cid:10) β k (cid:12)(cid:12) ) is the same as the probability of the result b k for the mixture W in (2.1) and (2.4).So far we have considered only the situation immediately after the measurement M .With the collapse postulate it is possible to describe the system S by a pure state (cid:12)(cid:12) α j (cid:11) ∈ H , whose evolution is governed by a group of unitary transformations U t ∈ L ( H ) ,as long as S is isolated. If the measurement result was a j , the state at time t after themeasurement will be (cid:12)(cid:12) α j ( t ) (cid:11) = U t (cid:12)(cid:12) α j (cid:11) If S is isolated and the pointer state (cid:12)(cid:12) ϕ ( M ) k (cid:11) is an eigenstate of the evolution of M ,the state of the compound system SM at the time t after the measurement will be (cid:12)(cid:12) Φ t (cid:11) = ∑ k (cid:10) α k | ψ (cid:11)(cid:12)(cid:12) α k ( t ) (cid:11)(cid:12)(cid:12) ϕ ( M ) k (cid:11) With this state the conditional probability, to get in a second measurement M (cid:48)(cid:48) t at time t the result B ( M (cid:48)(cid:48) t ) k , if the first measurement result at time t = A ( M ) j , is p ( B ( M (cid:48)(cid:48) t ) k | A ( M ) j ) == (cid:10) Φ t (cid:12)(cid:12) ϕ ( M ) j (cid:11)(cid:10) ϕ ( M ) j (cid:12)(cid:12) ⊗ (cid:12)(cid:12) ϕ ( M (cid:48)(cid:48) t ) k (cid:11)(cid:10) ϕ ( M (cid:48)(cid:48) t ) k (cid:12)(cid:12) Φ t (cid:11)(cid:10) Φ t (cid:12)(cid:12) ϕ ( M ) j (cid:11)(cid:10) ϕ ( M ) j (cid:12)(cid:12) Φ t (cid:11) = (cid:12)(cid:12)(cid:10) β k | α j ( t ) (cid:11)(cid:12)(cid:12) (4.1)This is the same value as with the collapse postulate.Interactions of other systems with the measurement device M can destroy the mea-surement result and change the state of the system S remotely. It is an open questionhow a measurement result becomes irreversible. However, pointer readings by a re-peatable measurement at M with another device (cid:101) M will give a state (cid:12)(cid:12) (cid:101) Φ t (cid:11) = ∑ k (cid:10) α k | ψ (cid:11)(cid:12)(cid:12) α k ( t ) (cid:11)(cid:12)(cid:12) ϕ ( M ) k (cid:11)(cid:12)(cid:12) (cid:101) ϕ ( (cid:101) M ) k (cid:11) which does not change the conditional probabilities for the results of a further measure-ment M (cid:48)(cid:48) t (4.1). A similar mechanism is assumed by decoherence theory (Schlosshauer,2004). 4 Conclusions
Applying the probability postulate we have shown that v. Neumann’s treatment of themeasurement process describes repeatable measurements without collapse of the wavefunction. However, this description has to take into account not only the measured sys-tem but also the measurement devices. A simpler description of the measured systemalone is possible, which gives the same conditional and total probabilities for all furthermeasurement results: It is the description demanded by the collapse postulate.One can regard this as a derivation of the collapse postulate, without contradictingimpossibility proofs like (Bassi and Ghirardi, 2000). From this point of view the wavefunction is merely a computational tool. And the collapse is no physical process; itis only the change to an equivalent but simpler probabilistic description with a cut between the observed system and the rest of the world. The dynamical conditionsduring and after the measurement interaction determine, if this cut is possible.Of course, our analysis does not explain how definite measurement results arise.That is already presupposed by the probability postulate. Nevertheless, it explainswhy the collapse of the wave function gives a correct probabilistic description of themeasured system, whenever a definite measurement result was obtained by a repeatablemeasurement. And it explains how it is possible to omit the collapse postulate at all,without losing the capability to describe sequential measurements.
References
D. Bohm, Phys. Rev. (1952).L. E. Ballentine, Quantum Mechanics: A modern Development (World Scientific, Sin-gapore, 1998).W. Heisenberg,
Physics and Philosophy (Penguin Classics, Reprint, 1958).M. Nielsen and I. L. Chuang,
Quantum Computation and Quantum Information (Cam-bridge University Press, 2000).J. v. Neumann,
Mathematische Grundlagen der Quantenmechanik (Springer-Verlag,Berlin, 1932).A. H. Compton, Nobel lecture December 12, 1927 (1927).P. Busch, P. J. Lahti, and P. Mittelstaedt,
The Quantum Theory of Measurement (Springer-Verlag, Berlin, 1991).M. Schlosshauer, Rev. Mod. Phys. 76 (2004), arXiv:quant-ph/0312059 .A. Bassi and G. Ghirardi, Phys. Lett. A (2000), arXiv:quant-ph/0009020arXiv:quant-ph/0009020