Entropic Dynamics and the Quantum Measurement Problem
aa r X i v : . [ qu a n t - ph ] A ug Entropic Dynamics and the QuantumMeasurement Problem ∗ David T. Johnson and Ariel Caticha
Department of Physics, University at Albany-SUNY,Albany, NY 12222, USA.
Abstract
We explore the measurement problem in the entropic dynamics ap-proach to quantum theory. The dual modes of quantum evolution—either continuous unitary evolution or abrupt wave function collapse dur-ing measurement—are unified by virtue of both being special instancesof entropic updating of probabilities. In entropic dynamics particles havedefinite but unknown positions; their values are not created by the act ofmeasurement. Other types of observables are introduced as a convenientway to describe more complex position measurements; they are not at-tributes of the particles but of the probability distributions; their valuesare effectively created by the act of measurement. We discuss the Bornstatistical rule for position, which is trivially built into the formalism, andalso for generic observables.
Quantum mechanics introduced several new elements into physical theory. Oneis indeterminism, another is the superposition principle embodied in both thelinearity of the Hilbert space and the linearity of the Schr¨odinger equation. Be-tween them they dealt a very severe blow to the classical conception of reality.The founders faced the double challenge of locating the source of indeterminismand of explaining why straightforward consequences of the superposition prin-ciple are not observed in the macroscopic world. Despite enormous progress thechallenge does not appear to have been met yet—at least as evidenced by thenumber of questions that stubbornly refuse to go away.The quantum measurement problem embodies most of these questions. Oneis the problem of macroscopic entanglement; another is the problem of definite ∗ Presented at MaxEnt 2011, The 31st International Workshop on Bayesian Inferenceand Maximum Entropy Methods in Science and Engineering, (July 10–15, 2011, Waterloo,Canada). A clear formulation of the problem is [1]; see also [2]. Modern reviews with references tothe literature appear in [3] and [4]. This opens theopportunity of explaining all other “observables” in purely informational terms.After a brief review of background material on ED (section 2) we discuss themeasurement of observables other than position and derive the correspondingBorn rule (section 3). The issue of amplification is addressed in section 4 andwe summarize our conclusions in section 5. A more detailed treatment of thequantum measurement problem is given in [12].
To set the context for the rest of the paper we briefly review the three mainideas that form the foundation of entropic dynamics. Several important topicsand most technical details are not discussed here. For a detailed account of, forexample, how time is introduced into an essentially atemporal inference scheme,or the entropic nature of the phase of the wave function, or the introduction ofconstants such as ~ or m , see [8]. For simplicity here we discuss a single particle. The first idea is about the subject matter: the goal is to predict the position x of a particle on the basis of some limited information. We assume that inaddition to the particle the world contains other variables—we call them y .Not much needs to be known about the y except that they are described bya probability distribution p ( y | x ) that depends on the particle position. Theentropy of the y variables is given by S [ p, q ] = − Z dy p ( y | x ) log p ( y | x ) q ( y ) = S ( x ) . (1)Neither the underlying measure q ( y ) nor the distribution p ( y | x ) need to bespecified further. Note that x enters as a parameter in p ( y | x ) and therefore itsentropy is a function of x : S [ p, q ] = S ( x ). The second idea concerns the method of inference: we use the method ofmaximum entropy subject to appropriate constraints to calculate the probabil-ity P ( x ′ | x ) that the particle takes a short step from x to a nearby point x ′ .The constraints reflect the relation between x and y given by p ( y | x ), and thefact that motion happens gradually—a large step is the result of many infinites-imally short steps. Thus entropic dynamics does not assume any underlyingsub-quantum mechanics whether it be classical or not . The successive accumu-lation of many such short steps results in a probability distribution ρ ( x, t ) thatsatisfies the Fokker-Planck equation ∂ρ∂t = − ~ ∇ · ( ρ~v ) (2) The case of momentum is discussed in [13]. And this is why the y variables are not hidden variables. The technical term ‘hidden vari-ables’ refers to variables introduced to explain the emergent quantum behavior as a reflectionof an essentially classical dynamics – whether stochastic or not – operating at a deeper level.The y variables do not play this role because in ED there is no underlying classical dynamics. ~v is ~v = ~ m ~ ∇ φ with φ ( x, t ) = S ( x, t ) − log ρ / ( x, t ) . (3)These equations show how the entropy S ( x, t ) guides the evolution of ρ ( x, t ). The third idea is an energy constraint: the time evolution of S ( x, t ) is deter-mined by imposing that a certain “energy” be conserved. Thus, we require thediffusion to be non-dissipative. To this end introduce an energy functional, E [ ρ, S ] = Z d xρ ( x, t ) [ ~ m ( ~ ∇ φ ) + ~ m ( ~ ∇ log ρ ) + V ] . (4)Note that this energy is a statistical concept; it is not assigned to the particlebut to ρ and S . Imposing that the energy be conserved for arbitrary initialchoices of ρ and S leads to the quantum Hamilton-Jacobi equation, ~ ˙ φ + ~ m ( ~ ∇ φ ) + V − ~ m ∇ ρ / ρ / = 0 . (5)This equation shows how the distribution ρ ( x, t ) affects the evolution of theentropy S ( x, t ).Finally, by combining the quantities ρ and S into a single complex function,Ψ = ρ / e iφ , the equations, (2) and (5), can be rewritten into the Schr¨odingerequation, i ~ ∂ Ψ ∂t = − ~ m ∇ Ψ + V Ψ . (6)The fact that the Schr¨odinger equation turned out to be linear and unitarymakes the language of Hilbert spaces and Dirac’s bra-ket notation particularlyconvenient—so from now we write Ψ( x ) = h x | Ψ i .To conclude this brief review we emphasize that the Fokker-Planck equation(2), the expression (3) for the current velocity as a gradient, and the relationbetween phase φ and entropy S are derived and not postulated. In practice the measurement of position can be technically challenging because itrequires the amplification of microscopic details to a macroscopically observablescale. However, no intrinsically quantum effects need be involved: the position ofa particle has a definite, albeit unknown, value x and its probability distributionis, by construction, given by the Born rule, ρ ( x ) = | Ψ( x ) | . We can thereforeassume that suitable position detectors are available; in ED the measurementof position can be considered as a primitive notion. This is not in any waydifferent from the way information in the form of data is handled in any other There is a close parallel to statistical mechanics which also requires a clear specificationof the subject matter (the microstates), the inference method (MaxEnt), and the constraints. ρ ( x ) dx = |h x | Ψ i| dx → p i = |h x i | Ψ i| . (7)Since position is the only objectively real quantity there is no reason to defineother observables except that they may turn out to be convenient when consid-ering more complex experiments in which before the particles reach the positiondetectors they are subjected to additional appropriately chosen interactions, saymagnetic fields or diffraction gratings. Suppose the interactions within the com-plex measurement device A are described by the Schr¨odinger eq.(6), that is, bya particular unitary evolution ˆ U A . The particle will be detected at position | x i i with certainty provided it was initially in state | a i i such thatˆ U A | a i i = | x i i . (8)Since the set {| x i i} is orthonormal and complete, the corresponding set {| a i i} is also orthonormal and complete, h a i | a j i = δ ij and P i | a i ih a i | = ˆ I . (9)Now consider the effect of this complex detector A on some arbitrary initialstate vector | Ψ i which can always be expanded as | Ψ i = P i c i | a i i , (10)where c i = h a i | Ψ i are complex coefficients. The state | Ψ i will evolve accordingto ˆ U A so that as it approaches the position detectors the new state isˆ U A | Ψ i = P i c i ˆ U A | a i i = P i c i | x i i . (11)which, invoking the Born rule for position measurements, implies that the prob-ability of finding the particle at the position x i is p i = | c i | . (12)Thus, the probability that the particle in state ˆ U A | Ψ i is found at position x i is | c i | . But we can describe the same outcome from the point of viewof the more complex detector. The particle is detected in state | x i i as if ithad earlier been in the state | a i i . We adopt a new language and say, perhaps5nappropriately, that the particle has effectively been “detected” in the state | a i i , and therefore, the probability that the particle in state | Ψ i is “detected”in state | a i i is | c i | = |h a i | Ψ i| —which reproduces Born’s rule for a genericmeasurement device. The shift in language is not particularly fundamental—itis a merely a matter of convenience but we can pursue it further and assert thatthis complex detector “measures” all operators of the form ˆ A = P i λ i | a i ih a i | where the eigenvalues λ i are arbitrary scalars. Born’s rule is a postulate inthe standard interpretation of quantum mechanics; within ED we see that it isderived as the natural consequence of unitary time evolution.Note that it is not necessary that the operator ˆ A have real eigenvalues, butit is necessary that its eigenvectors | a i i be orthogonal. This means that theHermitian and anti-Hermitian parts of ˆ A will be simultaneously diagonalizable.Thus, while ˆ A does not have to be Hermitian ( ˆ A = ˆ A † ) it must certainly be normal , that is ˆ A ˆ A † = ˆ A † ˆ A .Note also that if a sentence such as “a particle has momentum ~p ” is usedonly as a linguistic shortcut that conveys information about the wave functionbefore the particle enters the complex detector then, strictly speaking, thereis no such thing as the momentum of the particle: the momentum is not anattribute of the particle but rather it is a statistical attribute of the probabilitydistribution ρ ( x ) and entropy S ( x ), a point that is more fully explored in [13].The generalization to a continuous spectrum is straightforward. Let ˆ A | a i = a | a i . For simplicity we consider a discrete one-dimensional lattice a i and x i and take the limit as the lattice spacing ∆ a = a i +1 − a i →
0. The discretecompleteness relation, eq. (9), P i ∆ a | a i i (∆ a ) / h a i | (∆ a ) / = ˆ I becomes Z da | a ih a | = ˆ I , (13)where we defined | a i i (∆ a ) / → | a i . (14)We again consider a measurement device that evolves eigenstates | a i of ˆ A into unique position eigenstates | x i , ˆ U A | a i = | x i . The mapping from x to a can be represented by an appropriately smooth function a = g ( x ). In the limit∆ x →
0, the orthogonality of position states is expressed by a Dirac deltadistribution, h x i | ∆ x / | x j i ∆ x / = δ ij ∆ x → h x | x ′ i = δ ( x − x ′ ) . (15)An arbitrary wave function can be expanded as | Ψ i = P i ∆ a | a i i ∆ a / h a i | Ψ i ∆ a / or | Ψ i = Z da | a i h a | Ψ i . (16)6he unitary evolution ˆ U A of the wave function leads toˆ U A | Ψ i = P i ∆ a | x i i ∆ a / h a i | Ψ i ∆ a / = P i ∆ x | x i i ∆ x / h a i | Ψ i ∆ a / (cid:18) ∆ a ∆ x (cid:19) / → Z dx | x i h a | Ψ i| dadx | / , (17)so that p i = |h x i | ˆ U A | Ψ i| = |h a i | Ψ i| → ρ ( x ) dx = |h a | Ψ i| | dadx | dx = ρ A ( a ) da . (18)Thus, “the probability that the particle in state ˆ U A | Ψ i is found within the range dx is ρ ( x ) dx ” can be rephrased as “the probability that the particle in state | Ψ i is found within the range da is ρ A ( a ) da ” where ρ A ( a ) da = |h a | Ψ i| da , (19)which is the continuum version of the Born rule for an arbitrary observable ˆ A . The technical problem of amplifying microscopic details so they can becomemacroscopically observable is usually handled with a detection device set up inan initial unstable equilibrium. The particle of interest activates the amplifyingsystem by inducing a cascade reaction that leaves the amplifier in a definitemacroscopic final state described by some pointer variable α .An eigenstate | a i i evolves to a position x i and the goal of the amplifica-tion process is to infer the value x i from the observed value α r of the pointervariable. The design of the device is deemed successful when x i and α r are suit-ably correlated and this information is conveyed through a likelihood function P ( α r | x i )—an ideal amplification device would be described by P ( α r | x i ) = δ ri .Inferences about x i follow from a standard application of Bayes rule, P ( x i | α r ) = P ( x i ) P ( α r | x i ) P ( α r ) . (20)The point of these considerations is to emphasize that there is nothing in-trinsically quantum mechanical about the amplification process. The issue isone of appropriate selection of the information (in this case α r ) that happens tobe relevant to a certain inference (in this case x i ). This is, of course, a matterof design: a skilled experimentalist will design the device so that no spuriouscorrelations—whether quantum or otherwise—nor any other kind of interferingnoise will stand in the way of inferring x i .It may seem that we are simply redrawing von Neumann’s line between theclassical and the quantum with our treatment of the amplifying system. In somesense, we are doing just that. However, the line here is not between a classical7reality” and a quantum “reality”—it is between the microscopic particle witha definite but unknown position and an amplifying system skillfully designed soits own microscopic degrees of freedom turn out to be of no interest. In fact, in[12] we showed that such an amplifier can be treated as a fully quantum systembut it makes no difference to the inference. The solution of the problem of measurement within the entropic dynamicsframework hinges on two points: first, entropic quantum dynamics is a the-ory of inference not a law of nature. This erases the dichotomy of dual modes ofevolution—continuous unitary evolution versus discrete wave function collapse.The two modes of evolution turn out to correspond to two modes of updating—continuous entropic and discrete Bayesian—which, within the entropic inferenceframework, are unified into a single updating rule.The second point is the privileged role of position—particles have definitepositions and therefore their values are not created but merely ascertained dur-ing the act of measurement. All other “observables” are introduced as a matterof linguistic convenience to describe more complex experiments. These observ-ables turn out to be attributes of the probability distributions and not of theparticles; their values are indeed “created” during the act of measurement.
References [1] E. P. Wigner, Am J. Phys. , 6 (1963).[2] L. E. Ballentine, Quantum Mechanics: a Modern Development (World Sci-entific, 1998).[3] M. Schlosshauer, Rev. Mod. Phys. , 1267 (2004).[4] G. Jaeger, Entanglement, Information, and the Interpretation of QuantumMechanics (Springer-Verlag, Berlin Heidelberg 2009).[5] A. Komar,
Phys. Rev. , 365 (1962).[6] L. E. Ballentine,
Found. Phys. , 1329 (1990).[7] A. Caticha, “From Entropic Dynamics to Quantum Theory”, AIP Conf.Proc. , 48 (2009) (arXiv.org:0907.4335v3).[8] A. Caticha,
J. Phys. A : Math. Theor. (2011) 225303(arXiv.org:1005.2357).[9] A. Caticha, Lectures on Probability, Entropy, and Statistical Physics (Max-Ent 2008, S˜ao Paulo, Brazil) (arXiv.org:0808.0012).810] An excellent argument for the pragmatic elements underlying the Copen-hagen interpretation is given in H. P. Stapp,
Amer. J. Phys. , 1098(1972).[11] A. Caticha and A. Giffin, “Updating Probabilities,” in Bayesian In-ference and Maximum Entropy Methods in Science and Engineer-ing , ed. by A. Mohammad-Djafari, AIP Conf. Proc. , 31 (2006)(arXiv.org:physics/0608185).[12] D. T. Johnson, “Generalized Galilean Transformations and the Measure-ment Problem in the Entropic Dynamics Approach to Quantum Theory”,Ph.D. thesis, University at Albany (2011) (arXiv:1105.1384).[13] S. Nawaz and A. Caticha, “Momentum and Uncertainty Relations in theEntropic Approach to Quantum Theory” , in these proceedings (MaxEnt,2011).[14] A. Caticha, Found. Phys.30