www.mdpi.org/entropy/ Full paper Time in Quantum Measurement

Based on a model of quantum measurement we derive an estimate for the external measurement-time. Some interesting consequences will be analyzed.


Introduction
Given an apparatus or memory A, a system S and the environment E, the measurement process M can be represented by the following sequence: The …rst step is the entanglement between the system S and the apparatus A. The second step, d; is the decoherence of the entangled AS system by the perturbation through the environment E and the creation of classical alternatives.The third step, p, is the projection to one of the eigenstates and the record of a de…nite outcome.At this stage other states of the memory are being erased.In the course of this process the entropy of the AS system is going to ‡uctuate which re ‡ects the expansion and compression of the abstract phase space spanned by the eigenvectors of a corresponding observable: There is no algorithm (e.g di¤erential equation) known to describe M exactly in time.The ‡uctuation of entropy alone makes it clear that the process M cannot be described in …nite time by a Hamiltonian ‡ow and the projection postulate does not refer to time at all.If we describe the decohered AS system by its density matrix AS , there holds for the entropy S AS = tr AS log AS 0: If the environment is supposed to be in thermal equilibrium of temperature T; the generalized Landauer principle [10] ; [13] tells us that during the measurement process an amount of energy whose average over di¤erent measurement outcomes E satis…es is being dissipated.

Dissipation
In the last paragraph we stated that the measurement process is accompanied by the dissipation of energy to the environment: So far no time parameter has entered the discussion.In the projection postulate quantum mechanics supposes that the realization of an eigenstate happenes instantaneously.The natural question arises whether the process M does not need some minimal amount of time to happen.Note that we are considering here the notion of external time [12] ; i.e. the time valid outside the system AS.
Let us rewrite sequence (1) for a representative model case where the apparatus A, the system S and the environment E are one bit systems each and let = 1 j1i + 2 j2i be the measured state, A i ; i = 1; 2 the states of the apparatus and " i ; i = 1; 2 the states of the environment.We may assume that h1j 2i = hA 0 j A 1 i = h" 0 j " 1 i = 0: Let E be in thermal equilibrium of temperature T: For the simplicity of notation we will just write down the coe¢ cients i 2 C and their conjugates i 2 C; i = 1; 2 of the density matrix and omit the basis vectors in each step of sequence (1): This leaves us with In step d and p a total entropy of S d + S p = P i=1;2 By the second law of thermodynamics there must hold for the corresponding energies Note that to each outcome there corresponds an energy E i ; i = 1;

Scope
In the sequel we would like to look at the question of minimal measurement time from an intrinsic angle.By intrinsic we mean that we abstract from a conscious observer.We solely base on elements given by our model, namely on the system AS, the environment E , the sequence (1) and the corresponding physical consequences (2).For the sequence (1) the word "realization" would probably be more accurate than "measurement" in this context.The projection and registering of the de…nite outcome with the necessary erasure of other/former states of the memory is an integral part of this measurement process.The information gained by decoherence has to be o¤set.
The connection between the environement E and the system AS which, from the intrinsic perspective of E, gives proof of the de…nite outcome of M, is only the dissipated energy.In …rst order approximation the dissipation can be described by some Hamiltonian H ASE which is in our intrinsic sense unknown to the detector.In this situation we can apply the time-energy inequality [1].Let us assume that the true energy is E 0 : Following [1] ; the intrinsic energy uncertainty on the basis of quantities given in our model is where c AS > 0 depends upon the system AS.
By the time-energy inequality [1] and (3) there holds for the lower bound on t t ~ E 0 cAS E : The lower bound holding for all possible measurements is therefore by (2) This result can be interpreted that the process M takes, from the perspective of a generic detector in the environment E, minimally a time interval t = cAS kT S AS to happen.

Application
Let us assume that in empty Euclidean space there is a free particle, represented by a wave packet (x) 2 L 2 (R 3 ) with fairly well de…ned momentum.Since its momentum is well known, by the Heisenberg uncertainty relation, its position is highly unprecise in R 3 .Classical theory, however, would say that this particle moves with constant velocity along a straight line in the direction of the momentum vector.There is no way, given the uncertainty principle, to reconstruct directly the concept of classical trajectory and hence velocity de…ned by = x t in the quantum realm.By the help of our model, however, we will de…ne a quantitiy which can be interpreted as the time t the particle needs to "reach" a point at distance R: This way we will come close to a reconstruction of classical velocity = R t : We will use some technical facts: 1) c x ln x x 2 for some …xed c 2 R and all x 0; 2) L 4 (R 3 ) L 2 (R 3 ); 3) lim x!0 x s ln x !0; s; x > 0: Let there be a free particle, (x) (r); in an environment which is in thermal equilibrium of temperature T: According to our model, the minimal time t R needed to measure our particle at a distance of at least R would by (4) be where S R is de…ned as and c > 0 denotes the constant depending upon the set up (3).The intuitive interpretation of t R is the minimal time it takes for the particle to "reach" a point at distance R or further.Note that, since S R ! 0 as R ! 1; the intuitive fact that it takes longer to reach more distant regions, is assured.
Next we want to better understand the dependency of S R from R: Since 2 L 2 (R 3 ) we have for some R 0 > 0; " > 0 By the technical facts we have for any s 2 R; s > 0 R j j 1+s j j 1 s ln j j dr: If we put the last result into the expression for time we get for R > 0 big enough t R ~R c 2kT : We now can de…ne the analouge of classical velocity by (6) gives us an upper bound which is independent of the distance R; for R big enough.
Interpretation We try to reconstruct as closely as possible an analogue of classical velocity from quantum mechanics.Thereby we focus on the reconstruction of the most simple situation where there is a free particle in space of temperature T .Due to the probabilistic nature of quantum mechanics it is clear that the reconstruction can only be done by using quantities which are at most analogous to the classical notions of "distance passed on a straight line" and "time to pass through that distance" which form the de…nition of classical velocity.
For the analogue of a classical particle moving on a straight line we use the radial symmetry of a free wave packet with fairly well de…ned momentum.Our model gives an estimate for the minimal time it takes to measure the position of this particle at a distance bigger or equal to R: It turns out that, if R is big enough, we …nd an estimate for the minimal time needed which is proportionate to the distance R; exactly what we would expect in the classical case.Note also that if R is big the value of the integral ( 5) is concentrated in an annulus around R. So the probability that the particle is detected outside an annulus [R; R + ] ; > 0 gets smaller as R gets bigger.We then interpret this time as the minimal time needed for our particle to "reach" a distance R and can reconstruct classical velocity.The fact that classically the particle moves with constant velocity is mirrored by an upper bound for the reconstructed velocity.Note that estimate (6) can certainly be re…ned and is probably not the best estimate possible.
If we could …nd an upper bound for c ' ; (6) would give a general bound on the velocity of a free particle.

Comments
Our ideas open some interesting perspectives on the philosophy of time.If conscious observers develop mathematical models of reality they carve out the systems under consideration and base their models on known energy functions or operators.Hence they do necessarily not account for the continuous interaction with the environement and the corresponding ‡uctuations of entropy.At the same time they are able to measure energies within arbitrarily short periods of time [1].
Consequently they conclude that time is a) reversible and b) continuous.Our intrinsic model tries to abstract as much as possible from the meta standpoint of a conscious observer and under these circumstances measurement, as de…ned in our model, needs some interval of time.Furthermore the measurement process is irreversible and could be a true source for the arrow of time.