arXiv:1412.1312v2 [quant-ph] 23 Jun 2015 lrpoete,tk uesae opr tts They quantum of formulation standard states. the ac- in pure are axioms they as to that cepted well so states verified experimentally pure been have take one. properties, first the ilar as result the same on observable the same gives the system of same measurement second a i.e. i ntecoc f“ of choice the probabilis- in and tic irreversible discontinuous, is change This eigenvalue the for cor- operator the With to the eigenvector. state as responding the observable collapses measured and the outcome of measurement eigenvalues the gives which of other measurement, one projective The Neumann von deterministic. the and is reversible continuous, is It collapse. State trajectory, Quantum point, Fixed question open Keywords: an is evolution. It alternate can this experiments observe designed from rule. suitably not different Born or whether are the predic- evolution of its those ensemble measurements, for weak tions well- gradual In the of mechanics. with exper- case quantum consistent individual of framework is in established which state runs, quantum imental the col- dy- of projective the a lapse for propose equation We evolution namical runs. outcomes experimental description, various many ensemble of over an probabilities only predicts exists which which There experimental predicts which in run. that observed theory be will no the eigenstate as is operator there measured the outcome, one of yields eigenstates the measurement mechanics. of projective quantum though of measurement Even axioms projective fundamental and are evolution Unitary n tt.Oei ntr vlto,seie ythe by specified evolution, unitary is One evolv- Schr¨odinger for equation: state. rules me- a dynamical quantum ing distinct heart, two its contains At chanics mechanics. quantum of work | ohteeeouin,ntwtsadn hi dissim- their withstanding not evolutions, these Both hstl saotfiln a nteeitn frame- existing the in gap a filling about is talk This ψ −→ i P i i | ψ dt d onrl,Dchrne est matrix, Density Decoherence, rule, Born i / | nEouinr omls o ekQatmMeasurements Quantum Weak for Formalism Evolutionary An ψ | .TEPROBLEM THE I. P i i = | ψ etefrHg nryPyis ninIsiueo Science of Institute Indian Physics, Energy High for Centre i| H i .I scnitn nrepetition, on consistent is It ”. P , | ψ i i 2 i , λ = i , P P dt d i i eoigteprojection the denoting ρ = [ = P i † ,ρ H, , pov Patel Apoorva X i ] . P i = I. (2) (1) ∗ n ave Kumar Parveen and iua xeietlrn n hr sn rdcinfor prediction no is there “ and which run, experimental ticular eto fietclypeae unu ttsgives: col- states a quantum on prepared observable identically an of of lection interpretation Measurement ensemble verification. an for requiring rule, Born abilistic ytemaue bevbe nyoe“ one only observable, measured the by ehnc.Nntees h omlto isssome- misses formulation the thing: Nonetheless, mechanics. igefaeok lhuhtepolmo hc “ which of problem in the rules Although evolution framework. quantum single distinct a two these combine for description space Hilbert the state. in descrip- pure ray matrix a a density of a instead necessitates of tion, also appearance state pre- The mixed av- All runs. a as experimental obtained states. many values over mixed erages expectation to are states quantities pure dicted evolves rule This esml vlto”o unu system. quantum the un- a understanding of remained in evolution” achieved has “ensemble been run has experimental progress which solved, in occur will o h nvrerdcst ie tt o h system: the for state mixed state a entangled to but reduces pure universe a the over, for summed are degrees unobserved freedom the environ- of When unobserved freedom. the of with degrees observed mental freedom the of entangles degrees universe, unitary system whole a the with for two, evolution the between Interactions roundings. degrees observed evolve. remaining freedom the of “summed how be determine to the to need All treated over” then environment.) be freedom the of can of degrees those system unobserved as unob- the manner the view, same of the of coarse-grained freedom in those a of of while (In degrees degrees served observed not. the are are that system environment is the environment of the betweenfreedom and difference set essential same system The the the by rules. governed quantum is proposi- basic universe the of whole consider the to that logical tion is the it all theory of blocks, quantum evolution fundamental dynamical With the describing blocks. successfully set building same fundamental the from of made ultimately are environment—all | ψ prob htapasisedi h omlto steprob- the is formulation the in instead appears What vrteyas ayatmt aebe aeto made been have attempts many years, the Over opyia ytmi efcl sltdfo t sur- its from isolated perfectly is system physical No the as well as apparatus measuring the system, The i SE ( hl h e fpoeto operators projection of set the While i i −→ = ) htwudbe. would that ” .EvrnetlDecoherence Environmental A. h U ψ SE | P | i † ψ | aglr 560012 Bangalore , ψ i SE i = ρ , r T S ( P = i ρ r T ) ρ , E ( ρ −→ SE i cusi par- a in occurs ” ) X ρ , i { S 2 P P i 6= ρP i } ρ i sfixed is S . . (4) (3) i ” 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 − − . (9) posed to get around the quantum measurement problem. −→ (1 s)2 + (2s s2)T r(P ρ) − − i 3

Expansion around s = 0 yields the differential equation: preserves Hermiticity, trace and positivity, but is nonlin- ear. Because of its non-stochastic nature, it can model d the single quantum trajectory specific to a particular ex- ρ = ρPi + Piρ 2ρ T r(Piρ)= ρ, Pi ρ T r( ρ, Pi ) . ds − { }− { } perimental run. (10) Although Eq.(10) does fill a gap in solving the mea- This simple equation describing an individual quantum surement problem, with the preferred basis Pi fixed by trajectory has several remarkable properties. We can ex- the system-apparatus interaction, we still need{ a} separate plore them, putting aside the argument that led to the criterion to determine which Pi will occur in a particular equation. Explicitly, experimental run. This is a situation reminiscent of spon- taneous symmetry breaking8, where a tiny external field In addition to maintaining T r(ρ) = 1, the nonlin- • 2 (with a smooth limit to zero) picks the direction, and the ear evolution preserves pure states. ρ = ρ implies evolution is unique given that direction (stability of the ρPiρ = ρ T r(Piρ), and then direction depends on the thermodynamic size of the sys- d d d d tem). We do not have a prescription for such a choice, ρ2 ρ = ρ ρ + ρ ρ ρ =0 . (11) also referred to as a “quantum jump”. The stochastic ds − ds ds − ds

ensemble interpretation of quantum measurements is re- For pure states, with ψ d ψ = 0, we can also produced, as per the Born rule, when a particular Pi is h | ds | i picked with probability T r(P ρ(s = 0)). write d ψ = (P ψ P ψ ) ψ . So the component i ds | i i−h | i| i | i of the state along Pi grows at the expense of the other orthogonal components. A. Combining Trajectories Each projective measurement outcome is the fixed • point of the deterministic evolution: The probabilistic Born rule for measurement outcomes, Eq.(3), is rather peculiar despite being tremendously suc- d ρ =0 at ρ∗ = P ρP /T r(P ρ) . (12) cessful. The reason is that the probabilities are deter- ds i i i i mined by the initial state ρ(s = 0), and do not depend on the subsequent evolution of the state ρ(s = 0). Any The fixed point nature of the evolution makes attempt to describe projective measurement as6 continu- the measurement consistent on repetition. Note ous evolution would run into the problem that the sys- that one-dimensional projections satisfy PiρPi = tem would have to remember its state at the instant the Pi T r(Piρ). measurement started until the measurement is complete. In a bipartite setting, the complete set of projection This is a severe constraint for any theory of weak mea- • surement, where the measurement time scale is stretched operators can be labeled as Pi = Pi1 Pi2 , { } { ⊗ } out, and we can rightfully question whether the Born rule with Pi = I. Since the evolution is linear in i would hold in such a case. the projectionP operators, a partial trace over the unobserved degrees of freedom produces the same It is possible to reconcile the Born rule with continuous equation (and hence the same fixed point) for the projective measurements, by invoking retardation effects reduced density matrix for the system. Purification arising from special relativity for the speed of informa- is thus a consequence of the evolution; for example, tion travel between the system and the apparatus. Then a qubit state in the interior of the Bloch sphere the Born rule will be followed by sudden impulsive mea- evolves to the fixed point on its surface. surements with a duration shorter than the retardation time, but it may be violated by gradual weak measure- At the start of measurement, we expect the param- ments with a duration longer than the retardation time. • eter s to be proportional to the system-apparatus We look beyond the Born rule satisfying possibility that interaction, s H t. To understand the ap- the evolution trajectory corresponding to Pi is chosen ∼ || SA|| proach towards the fixed point, letρ ˜ ρ Pi for a at the start of the measurement and remains unaltered one-dimensional projection, which satisfies≡ − thereafter, in order to look for more general evolutionary choices that may be suitable for weak measurements. d ρ˜ =ρP ˜ + P ρ˜ 2˜ρ 2ρ T r P ρ˜ . (13) Let wi be the probability weight of the evolution tra- ds i i − − i jectory for P , with w 0 and w = 1. We have
i i ≥ i i wi(s =0)= ρii(s = 0) in accordanceP with the Born rule, It follows that towards the end of measurement while w (s = 0) are some functions of ρ(s). Then the tra- s , and convergence to the fixed point is ex- i 6 → ∞ −2s jectory averaged evolution of the density matrix during ponential, with ρ˜ e , similar to the charging measurement is given by: of a capacitor. || || ∼ d ρ = w [ρP + P ρ 2ρ T r(P ρ)] . (14) These properties make Eq.(10) a legitimate candidate for ds i i i − i Xi describing the collapse of a quantum state during pro- jective measurement. It represents a superoperator that It still preserves pure states, as per Eq.(11). In terms of a 4 complete set of projection operators, we can decompose terms in the evolution Hamiltonian, which have been ig- ρ = jk Pj ρPk. The projected components evolve as nored throughout in our measurement description and P whose contribution would have to be added to Eq.(14) d in describing complete evolution of the system, can sta- P ρP = P ρP w +w 2 w T r(P ρ) . (15) ds j k j k j k − i i bilise them, and make the evolution converge towards an
Xi
n-dim degenerate subspace. This evolution obeys the identity, independent of the An appealing choice for the trajectory weights is the “instantaneous Born rule”, i.e. w = wB T r(P ρ(s)) choice of wi , j j j { } throughout the measurement process. That≡ avoids logi- 2 d 1 d cal inconsistency in weak measurement scenarios, where Pj ρPk = Pj ρPj Pj ρPk ds Pj ρPj ds one starts the measurement, pauses somewhere along the

way, and then restarts the measurement. In this situa- 1 d + PkρPk , (16) tion, the trajectory averaged evolution is: PkρPk ds
d P ρP = P ρP (wB + wB 2 (wB )2) . (19) with the consequence that the diagonal projections of ds j k j k j k − i ρ completely determine the evolution of all the off-
Xi diagonal projections. The diagonal projections are all This evolution converges towards the n dimensional − non-negative, Pj ρ(s)Pj = dj (s)Pj with dj 0, and we subspace specified by the dominant diagonal projections ≥ obtain: of the initial ρ(s = 0). It is deterministic and does not

1/2 follow Eq.(3). The measurement result remains consis- dj (s) dk(s) tent under repetition though. Pj ρ(s)Pk = Pj ρ(0)Pk . (17) dj (0) dk(0) The evolution can be made stochastic, in a manner similar to the Langevin equation, by adding noise to the In particular, phases of the off-diagonal projections weights wi while still retaining i wi = 1. The weak Pj ρPk do not evolve, in sharp contrast to what happens measurement process is expectedP to contribute such a during decoherence. Also, their asymptotic values, i.e. noise9,12,13. The resultant evolution, and its dependence Pj ρ(s )Pk, may not vanish, whenever more than one on the magnitude of the noise, needs to be investigated. → ∞ diagonal Pj ρ(s )Pj remain nonzero. In a sense, de- cohering measurements→ ∞ select the Cartesian components of the quantum state in the eigenbasis provided by the B. Relation to the Master Equation system-apparatus interaction and lose information about the angular coordinates, while the collapse equation se- The master equation is obtained assuming that the lects the radial components of the quantum state around environmental degrees of freedom are not observed, and the measurement fixed points leaving the angular compo- hence are summed over. On the other hand, the de- nents unchanged. Mathematically speaking, both mea- grees of freedom corresponding to the measured observ- surement schemes are consistent. able are observed in any measurement process with a def- It is easily seen that when all the wi are equal, no infor- inite outcome, and cannot be summed over. In analysing mation is extracted from the system by the measurement the measurement process, we need to keep track of only and ρ does not evolve. More generally, the diagonal pro- those degrees of freedom of the apparatus that have a jections evolve according to: one to one correspondence with the system’s eigenstates, and the rest can be kept aside. The crucial difference be- d d =2d w w d =2d (w w )d . (18) tween the states of the system and the apparatus is that ds j j j − i i j j − i i Xi
Xi6=j
the system can be in a superposition of the eigenstates but the apparatus has to end up in one of the pointer Here, with d = 1, w d w is the weighted states only (and not their superposition). i i i i i ≡ average of Pwi . Clearly,P the diagonal projections with In the traditional description, at the start of the mea- { } wj > w grow and the ones with wj < w decay. Any dj surement, the joint state of the system and the apparatus that is zero initially does not change, and the evolution can be chosen to be c i 0 , with c 2 = 1. The i i| iS| iA i | i| is therefore restricted to the subspace spanned by all the system-apparatus interactionP then unitarilyP evolves it to Pj ρ(s = 0)Pj = 0. Also, all the measured observable the entangled state c i ˜i . This evolution is a 6 i i| iS| iA eigenstates, i.e. ρ = Pj with dj = 1, are fixed points controlled unitary transformationP (and not a copy oper- of the evolution. These features are stable under small ation). The preferred measurement basis is the Schmidt perturbations of the density matrix. decomposition basis, ensuring a perfect correlation be- Other fixed points of Eq.(18) correspond to “degener- tween the system eigenstate i and the measurement | iS ate” situations where some of the wj (say n> 1 in num- pointer state ˜i A. In particular, the reduced density ma- 1 | i ber) are equal and all the others vanish, i.e. wj 0, n . trices of the system and the apparatus are identical at These fixed points are unstable under asymmetric∈{ per-} this stage. Thereafter, the state collapse picks one of the turbations that lift the degeneracy. It may be that other components i ˜i , without losing the perfect correlation. | i| i 5

The algebraic structure of the collapse equation, Our analysis is more general and is applicable to any Eq.(10), is closely related to that of the master equa- quantum system. tion, Eq.(7). Expansion of Eq.(10) around the fixed point For a single transmon qubit undergoing Rabi oscil- ρ = Pi gives, with [Pi]Pi = 0, lations, perturbations caused by its weak measurement L have been observed, and then successfully cancelled by a d ρ˜ =2 [Pi]˜ρ 2PiρP˜ i 2(1 Pi)˜ρ(1 Pi) 2˜ρ T r(˜ρPi) . measurement result dependent feedback shift of the Rabi ds L − − − − − frequency12, all consistent with the description provided (20) by Eqs.(23-25). We have verified the same behaviour for The term [P ]˜ρ = [P ]ρ on the r.h.s. decouples P ρP i i i i two qubit systems using numerical simulations based on from the restL of theL density matrix by making the off- Eqs.(18) and (17), and trajectory weights chosen anal- diagonal components (P ρP and P ρP ) decay, but does i j j i ogous to Eq.(25), e.g. perturbations due to weak mea- not alter the diagonal components. The next two terms surements of J J on a Bell state undergoing joint on the r.h.s. make the diagonal components ofρ ˜ decay, z z Rabi oscillations⊗ can be cancelled by a measurement re- leading the evolution to the fixed point. The last term sult dependent Rabi frequency shift. We could not make on the r.h.s. is of higher order inρ ˜. this cancellation strategy work, however, with simultane- The Lindblad operators also satisfy the relation, ous measurement of more than one commuting operators [ρ]P + [P ]ρ = [˜ρ]P = O(˜ρ2) . (21) and independent shifts of individual Rabi frequencies. L i L i L i [P ]ρ is the influence of the apparatus pointer state on L i the system density matrix, while [ρ]Pi is the influence III. OPEN QUESTIONS of the system density matrix onL the apparatus pointer state. For pure states, the collapse equation is just Our proposed collapse equation, Eq.(10), is quadrat- d ically nonlinear. Nonlinear Schr¨odinger evolution is in- ρ = 2 [ρ]P . (22) ds − L i appropriate in quantum mechanics, because it violates the well-established superposition principle. Nonlinear These expressions suggest an inverse relationship be- superoperator evolution for the density matrix is also tween the processes of decoherence and collapse. Such avoided in quantum mechanics, because it conflicts with an action-reaction relationship can follow from a con- the probability interpretation for mixtures of density ma- servation law. Initially, the combined system-apparatus trices. Nevertheless, nonlinear quantum evolutions need state evolves unitarily, establishing perfect correlation not be unphysical, and have been invoked in attempts to and without any decoherence. The subsequent new de- solve the measurement problem10,11. Eq.(10) can be a scription would be that during the measurement process, valuable intermediate step in such attempts to interpret when [ρ]P decoheres the apparatus pointer state P (it L i i collapse as a consequence of some unknown underlying cannot remain in superposition), there is an equal and dynamics. It is definitely worth keeping in mind that opposite effect [ρ]P on the system density matrix ρ, −L i non-abelian gauge theories and general relativity are ex- resulting in the state collapse. amples of well-established theories with quadratic non- linearities in their dynamical equations. C. The Qubit Case and Some Tests Irrespective of the underlying dynamics that may lead to the collapse equation, Eq.(10), it is worthwhile to test it at its face value. It readily produces an eigenstate of All the previous results can be expressed in a consider- the measured observable as the measurement outcome, ably simpler form in case of the smallest quantum system, but predictions of its ensemble version, Eq.(14), are not i.e. the 2-dimensional qubit with 0 and 1 as the mea- | i | i stochastic and do not reproduce the Born rule. So to surement basis vectors. Evolution of the density matrix judge its validity, it is imperative to ask the question: during the measurement, Eqs.(18) and (17), is given by: Do quantum systems exhibit this alternate evolution, and d if so under what conditions? Experimental tests would ρ00(s)=2(w0 w1) ρ00(s) ρ11(s) , (23) require determination of the density matrix evolution, ds − in presence of weak measurements and with highly sup- pressed decoherence effects. Such tests are now techno- 1/2 ρ00(s) ρ11(s) logically feasible! It is indeed possible to generalise and ρ01(s)= ρ01(0) . (24) ρ00(0) ρ11(0) extend the Bayesian measurement formalism tests for a single qubit, by R. Vijay et al.12 and by K.W. Murch Selecting the trajectory weights as addition of Gaussian et al.13, to larger quantum systems. They would clarify white noise to the instantaneous Born rule results in: what trajectory weights wi (including stochastic noise) are appropriate for describing ensemble dynamics of weak w0 w1 = ρ00 ρ11 + ξ . (25) − − measurements. The same evolution has been obtained by Korotkov9, us- Finally, it is useful to note that, if the fixed point col- ing the Bayesian measurement formalism for a qubit. lapse dynamics of Eq.(10) can be realised in practice, 6 it would provide an unusual strategy for quantum error orthogonal projections. The state would then return to- correction. After encoding the logical Hilbert space as wards the logical subspace as long as its projection on the a suitable subspace of the physical Hilbert space, one logical subspace is larger than that on the error subspace, only has to perform measurements in an eigenbasis that even if no feedback operations based on the measurement separates the logical subspace and the error subspace as results are carried out!

∗ Electronic address: [email protected] arXiv:1008.2510[quant-ph]. † Electronic address: [email protected] 9 Korotkov, A.N., Continuous quantum measurement of a 1 Lindblad, G., On the generators of quantum dynamical double dot. Phys. Rev. B, 1999, 60(8), 5737-5742; Selec- subgroups. Comm. Math. Phys., 1976, 48(2), 119-130. tive quantum evolution of a qubit state due to continuous 2 Gorini, V., Kossakowski, A. and Sudarshan, E.C.G., Com- measurement. Phys. Rev. B, 2001, 63, 115403-1 to 15. pletely positive semigroups of N-level systems. J. Math. 10 Gell-Mann, M. and Hartle, J.B., Classical equations for Phys., 1976, 17(5), 821-825. quantum systems. Phys. Rev. D, 1993, 47(8), 3345-3382. 3 Giulini, D., Joos, E., Kiefer, C., Kuptsch, J., Stamatescu, 11 Penrose, R., On gravity’s role in quantum state reduction. I.-O. and Zeh, H.D., Decoherence and the Appearance of a General Relativity and Gravitation, 1996, 28(5), 581-600. Classical World in Quantum Theory, Springer, 1996. 12 Vijay, R., Macklin, C., Slichter, D.H., Weber, S.J., Murch, 4 Wiseman, H.M. and Milburn, G.J., Quantum Measure- K.W., Naik, R., Korotkov, A.N. and Siddiqi, I., Stabilizing ment and Control, Cambridge University Press, 2010. Rabi oscillations in a superconducting qubit using quan- 5 The Many-Worlds Interpretation of Quantum Mechanics, tum feedback. Nature, 2012, 490, 77-80. (eds. DeWitt, B.S. and Graham, N.), Princeton University 13 Murch, K.W., Weber, S.J., Macklin, C. and Siddiqi, I., Press, 1973. Observing single quantum trajectories of a superconduct- 6 Bangalore is the capital of the state of Karnataka in India. ing quantum bit. Nature, 2013, 502, 211-214. See http://www.karnatakatourism.org/ 7 Aharonov, Y., Albert, D.Z. and Vaidman, L., How the result of a measurement of a component of the spin of a ACKNOWLEDGEMENTS. We thank Michel Devoret spin-1/2 particle can turn out to be 100. Phys. Rev. Lett., and Rajamani Vijayaraghavan for helpful conversations. 1988, 60(14), 1351-1354. The slogan of the Karnataka Tourism Department was 8 Ghose, P., Measurement as spontaneous symme- pointed out to us by Raymond Laflamme. PK is sup- try breaking, non-locality and non-Boolean holism. ported by a CSIR research fellowship.