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An Evolutionary Formalism for Weak Quantum Measurements

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An Evolutionary Formalism for Weak Quantum Measurements An Evolutionary Formalism for Weak Quantum Measurements Apoorva Patel∗ and Parveen Kumar† Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012 mechanics. Nonetheless, the formulation misses some- Unitary evolution and projective measurement thing: While the set of projection operators Pi is fixed { } are fundamental axioms of quantum mechanics. by the measured observable, only one “i” occurs in a par- Even though projective measurement yields one ticular experimental run, and there is no prediction for of the eigenstates of the measured operator as the which “i” that would be. outcome, there is no theory that predicts which What appears instead in the formulation is the prob- eigenstate will be observed in which experimental abilistic Born rule, requiring an ensemble interpretation run. There exists only an ensemble description, for verification. Measurement of an observable on a col- which predicts probabilities of various outcomes lection of identically prepared quantum states gives: over many experimental runs. We propose a dy- namical evolution equation for the projective col- prob(i)= ψ Pi ψ = T r(Piρ) , ρ PiρPi . (3) h | | i −→ lapse of the quantum state in individual exper- Xi imental runs, which is consistent with the well- This rule evolves pure states to mixed states. All pre- established framework of quantum mechanics. In dicted quantities are expectation values obtained as av- case of gradual weak measurements, its predic- erages over many experimental runs. The appearance of tions for ensemble evolution are different from a mixed state also necessitates a density matrix descrip- those of the Born rule. It is an open question tion, instead of a ray in the Hilbert space description for whether or not suitably designed experiments can a pure state. observe this alternate evolution. Over the years, many attempts have been made to Keywords: Born rule, Decoherence, Density matrix, combine these two distinct quantum evolution rules in Fixed point, Quantum trajectory, State collapse. a single framework. Although the problem of which “i” will occur in which experimental run has remained un- solved, progress has been achieved in understanding the I. THE PROBLEM “ensemble evolution” of a quantum system. This talk is about filling a gap in the existing frame- A. Environmental Decoherence work of quantum mechanics. At its heart, quantum me- chanics contains two distinct dynamical rules for evolv- ing a state. One is unitary evolution, specified by the The system, the measuring apparatus as well as the Schr¨odinger equation: environment—all are ultimately made from the same set of fundamental building blocks. With quantum theory d d successfully describing the dynamical evolution of all the i ψ = H ψ , i ρ = [H,ρ] . (1) dt| i | i dt fundamental blocks, it is logical to consider the proposi- tion that the whole universe is governed by the same set It is continuous, reversible and deterministic. The other of basic quantum rules. The essential difference between is the von Neumann projective measurement, which gives the system and the environment is that the degrees of arXiv:1412.1312v2 [quant-ph] 23 Jun 2015 one of the eigenvalues of the measured observable as the freedom of the system are observed while those of the measurement outcome and collapses the state to the cor- environment are not. (In a coarse-grained view, unob- responding eigenvector. With Pi denoting the projection served degrees of freedom of the system can be treated operator for the eigenvalue λi, in the same manner as those of the environment.) All the unobserved degrees of freedom then need to be “summed ψ P ψ / P ψ , P 2 = P = P †, P = I. (2) over” to determine how the remaining observed degrees | i −→ i| i | i| i| i i i i Xi of freedom evolve. No physical system is perfectly isolated from its sur- This change is discontinuous, irreversible and probabilis- roundings. Interactions between the two, with a unitary tic in the choice of “i”. It is consistent on repetition, evolution for the whole universe, entangles the observed i.e. a second measurement of the same observable on the system degrees of freedom with the unobserved environ- same system gives the same result as the first one. mental degrees of freedom. When the unobserved degrees Both these evolutions, not withstanding their dissim- of freedom are summed over, a pure but entangled state ilar properties, take pure states to pure states. They for the universe reduces to a mixed state for the system: have been experimentally verified so well that they are ac- cepted as axioms in the standard formulation of quantum ψ U ψ , ρ = T r (ρ ) , ρ2 = ρ . (4) | iSE −→ SE| iSE S E SE S 6 S 2 In general, the evolution of a reduced density matrix is Some of them are physical, e.g. introduction of hidden linear, Hermiticity preserving, trace preserving and pos- variables with novel dynamics, and breakdown of quan- itive, but not unitary. Using a complete basis for the tum rules due to gravitational interactions. Some others environment µ E , such a superoperator evolution can are philosophical, e.g. questioning what is real and what be expressed{| in thei } Kraus decomposition form: is observable, in principle as well as by human beings with limited capacity. Given the tremendous success of ρ M ρ M † , (5) S −→ µ S µ quantum theory, realised with a “shut up and calculate” Xµ attitude, and the stringent constraints that follow, none M = µ U 0 , M †M = I. of the theoretical approaches have progressed to the level µ Eh | SE| iE µ µ Xµ where they can be connected to readily verifiable exper- imental consequences. This description explains the probabilistic ensemble evo- Perhaps the least intrusive of these approaches is the lution of quantum mechanics in a language similar to “many worlds interpretation”5. It amounts to assign- that of classical statistical mechanics. But it still has no ing a distinct world (i.e. an evolutionary branch) to each mechanism to explain the projective collapse of a quan- probabilistic outcome, while we only observe the outcome tum state. (Ensemble averaging is often exchanged for corresponding to the world we live in. (It is amusing to ergodic time averaging in equilibrium statistical mechan- note that this discussion meeting is being held in a place ics, but that option is not available in unitary quantum where the slogan of the department of tourism is “One mechanics.) state, Many worlds”6.) Such an entanglement between Generically the environment has a much larger num- the measurement outcomes and the observers does not ber of degrees of freedom than the system. Then, in the violate any quantum principle, although the uncount- Markovian approximation which assumes that informa- able proliferation of evolutionary branches it supposes tion leaked from the system does not return, the evolu- is highly ungainly. Truly speaking, the many worlds in- tion of the reduced density matrix can be converted to a terpretation bypasses the measurement problem instead differential equation. With the expansion of solving it. M0 = I+dt( iH+K)+..., M 6=0 = √dtL +..., (6) With the technological progress in making quantum − µ µ 1,2 devices, we need a solution to the measurement problem, Eq.(5) leads to the Lindblad master equation : not only for formal theoretical reasons, but also for in- d creasing accuracy of quantum control and feedback4. A ρ = i[ρ,H]+ [L ]ρ , (7) 7 dt L µ practical situation is that of the weak measurement , typ- Xµ ically realised using a weak system-apparatus coupling, 1 1 [L ]ρ = L ρL† ρL† L L† L ρ . where information about the measured observable is ex- L µ µ µ − 2 µ µ − 2 µ µ tracted from the system at a slow rate. Such a stretch- The terms on the r.h.s. involving sum over µ modify the ing out of the time scale allows one to monitor how the unitary Schr¨odinger evolution, while T r(dρ/dt) = 0 pre- system state collapses to an eigenstate of the measured serves the total probability. When H = 0, the fixed point observable, and to track properties of the intermediate of the evolution is a diagonal ρ, in the basis that diago- states created along the way by an incomplete measure- nalises Lµ . This preferred basis is determined by the ment. Knowledge of what really happens in a particular system-environment{ } interaction. (When there is no diag- experimental run (and not the ensemble average) would onal basis for Lµ , the evolution leads to equipartition, be invaluable in making quantum devices more efficient i.e. ρ I.) Furthermore,{ } the off-diagonal components and stable. of ρ decay,∝ due to destructive interference among envi- ronmental contributions with varying phases, which is known as decoherence. II. A WAY OUT This modification of a quantum system’s evolution, due to its coupling to unobserved environmental degrees of freedom, provides the correct ensemble interpretation, Let us assume that the projective measurement results and a quantitative understanding of how the off-diagonal from a continuous geodesic evolution of the initial quan- 3,4 tum state to an eigenstate i of the measured observable: components of ρ decay . Still the quantum theory is in- | i complete, and we need to look further to solve the “mea- ψ Q (s) ψ / Q (s) ψ , Q (s)=(1 s)I +sP , (8) surement problem” till it can predict the outcome of a | i −→ i | i | i | i| i − i particular experimental run. where the dimensionless parameter s [0, 1] repre- sents the “measurement time”. The density∈ matrix then B. Going Beyond evolves as, maintaining T r(ρ) = 1, 2 2 A wide variety of theoretical approaches have been pro- (1 s) ρ + s(1 s)(ρPi + Piρ)+ s PiρPi ρ − − .
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