Fundamental loss of quantum coherence related to gravity Jorge - - PowerPoint PPT Presentation

Fundamental loss of quantum coherence related to gravity

Jorge Pullin Jorge Pullin Horace Hearne Laboratory for Theoretical Physics Louisiana State University With Rodolfo Gambini (Uruguay) and Rafael Porto (UC Santa Barbara)

It is my guess that everyone who takes a course in quantum mechanics is surprised by the different way in which space and time are treated.

Ψ = ∂ Ψ ∂ − H t i ˆ

• >

< Ψ x x ˆ , ˆ

Clearly, Schrödinger’s equation can only be understood as an idealization. In the real world clocks are quantum mechanical and the variable we choose to call t will be associated with a quantum operator.

So how does one do quantum mechanics with real clocks? You do it “relationally”. First of all you choose some physical variable as your “clock”, let us call it T. Such variable will be represented by a quantum operator. Then you choose the variables that will describe the physical system under study. Generically we call them X. One then computes:

) | ( t T x X P >= < >= <

)) ( ) ( ) ( ( Tr ρ t P t P t P dt

T T x

∫

=

) | ( t T x X P >= < >= <

) ) ( ( Tr ρ t P dt

T T T x

∫

=

That is, the conditional probability that X takes a value x0 when T takes a value t0. Notice that in the right hand side we have the “ideal” t of Schrödinger’s theory. The density matrices (quantum states) evolve with the traditional Schrödinger equation, we only ask different questions about them than usual.

How does quantum evolution look like when one casts it in terms of T rather than t? Here one needs to make some assumptions. We assume that the density matrix can be written as a direct product

• f that of the clock and that of the system under study and that one

has a unitary independent evolution for the clock and the system, We also define the probability density that the clock variable takes the value T when the ideal time takes the value t, the value T when the ideal time takes the value t, And define an evolution in terms of the variable T,

With these identifications we can rewrite the conditional probability as an ordinary probability in quantum mechanics for the density matrix ρ(T) To get something closer to the usual Schroedinger equation, we assume that the probability for the clock is quite peaked, assume that the probability for the clock is quite peaked, Which gives for the evolution,

So the differential equation that gives the time evolution of the density matrix is given by,

... ]] , ˆ [ , ˆ )[ ( ] , ˆ [ + + = ∂ ∂ − ρ σ ρ ρ H H T H T i

Where σ(T) is the rate of spread of the wavefunction of the clock. Where σ(T) is the rate of spread of the wavefunction of the clock. We have assumed one started with a clock in a quantum state such that the variable T has a distribution that is very peaked around t. In fact, the above expression is approximate, as the spreads increase

• ne gets higher order terms with more commutators.

Class.Quant.Grav.21:L51-L57,2004 New J.Phys.6:45,2004

What are the consequences of the extra term? If we assume σ is constant, the equation can be solved exactly and one gets that the density matrix in an energy eigen-basis evolves as, Where the omega’s are the Bohr frequencies associated with the eigenvalues of H. Therefore, the off-diagonal elements of the density matrix decay to zero exponentially, and pure states generically evolve into mixed states. Quantum mechanics with real clocks therefore does not have a unitary evolution.

The effect can be made arbitrarily large simply choosing “lousy clocks” to do physics. This is not usually done, but an interpretation

• f experiments with Rabi oscillations indicate the effect is there,

R . Bonifacio, S . Olivares, P. Tombesi et. al., J. Mod. Optics, 47 2199 ( 2000)

Can the effect be eliminated just by choosing better and better clocks? And if not, how much does reality depart from traditional quantum theory? To estimate this we have to ask ourselves the quantum theory? To estimate this we have to ask ourselves the question “what is the best clock we can build”?

They consider a clock consisting of two mirrors between which a light ray bounces back and This question was considered some time ago by Salecker and Wigner. between which a light ray bounces back and

• forth. Every bounce is a “tick” of the clock.

They note that by the time the light bounced

• ff a mirror and returns, the original mirror’s

wave-function would have spread. The width

• f the spread limits the accuracy of the clock.

M t t ≈ δ ) 1 ( = = c

So if you want a better clock, make it more massive. But there is a catch: If you put too much mass your clock becomes a black hole!

(Ng and Van Dam, Ann. NY Acad. Sci 755, 579 (1995))

A black hole is, in fact, the best clock you can have in this sense. How is a black hole a clock? Black holes have vibrational modes (quasi-normal modes). Although these modes are heavily damped, The frequency of oscillation is inversely proportional to the mass

• f the black hole. Therefore smaller black holes make better clocks.

at least in principle they allow to think of the black hole as an

• scillator. (Think of a bell )

This tension leads to a fundamental limit on how accurate a clock can be if you wish to measure a given time Tmax: Where tP is Planck’s time: 10-44s.

Now that we estimated what the best possible clock can be in nature, we can put a limit on the value of σ(T) in the equation we derived for quantum mechanics with real clocks:

... ]] , ˆ [ , ˆ )[ ( ] , ˆ [ + + = ∂ ∂ − ρ σ ρ ρ H H T H T i

With, So, we have argued that due to the fact that one cannot have a perfectly classical clock in nature, quantum mechanics needs to be modified and we have provided a quantitative estimate of the modification based on the best possible clocks one can construct at least in principle. The above equation conserves energy (good!) but implies that evolution is not unitary (interesting…).

So if we go back to the formula for the decoherence, with our estimate of σ(T), is the decoherence observable experimentally? For a two level system, we get, with the optimal black hole clock, For this effect to be observable, one needs a quantum coherent system with a large separation of energy levels and a long life. system with a large separation of energy levels and a long life. The “Schödinger cat” type experiments are the type of systems we wish to consider. The most promising experiments are Bose-Einstein condensates, (10N atoms, 1027-2N s).

Simon, Jaksch, quant-ph/0406007 Phys. Rev. A70, 052104 (2004).

Interestingly, if one punches in the numbers for LISA, the effect appears observable, but this is fallacious, since classical optical experiments do not measure the off-diagonal elements of the density matrix.

The black hole information paradox: In 1975 Hawking showed that black holes eventually evaporate. The only thing left at the end of the process is an outgoing purely thermal black body radiation. This raises the question of what happened to all the information that went into the creation of the black hole. Information loss is problematic in traditional quantum mechanics because evolution is unitary and this implies information is preserved. In particular a pure state will evolve into a pure state. However, we have argued in this talk that in quantum mechanics with real clocks evolution is not unitary. We also argued that the effect is very small. Could it be large enough to eliminate the black hole information puzzle?

A precise calculation is beyond the current possibilities, since a complete black hole evaporation would require full quantum gravity. To carry out a back-of-the-envelope calculation we make a naïve model of the black hole as a two level system with energy separation given by the temperature. Concretely, One can compute exactly the evolution of the density matrix for One can compute exactly the evolution of the density matrix for such a naïve model. The final result is that the density matrix is given by (approximately, in modulus) as,

3 2 12 max 12

| ) ( | | ) ( |         ≈

BH Planck

M M T ρ ρ

So for an astrophysical black hole the loss of coherence is of the order of 10-28. One could still argue that the puzzle still exists for smaller black holes…

So we are claiming that in real life one could have never observed the black hole information paradox, since quantum states decohere (or in other ways information is lost) due to our lack of perfect clocks at a rate faster than the one an evaporating black hole makes it disappear. The paradox can still be posed at the level of the “idealized” Schrödinger theory in terms of the perfectly classical time t. Schrödinger theory in terms of the perfectly classical time t. But if one adopts the point of view that the true physical theory is the one formulated in terms of T, then the paradox does not arise.

• R. Gambini, R. Porto, JP PRL 93, 240401 (2004)

Limitations to quantum computing: Lloyd (Science 406, 1047 (2000) has argued that there is limit to the power of a quantum computing. He uses the Margolus-Levitin theorem that states that in order to perform a computation in a time ∆t, one needs to expend at least an energy As a consequence, a system with average energy E can perform a maximum of n=

• p/s.

perform a maximum of n=

• p/s.

For an ‘ultimate laptop’ (1Kg, 1 liter) this bound turns out to be ~ 1051 op/s. These results assume evolution is unitary. We have argued it is not. When evolution is not unitary, erroneous computations are carried

• ut. This gets worse the “faster” one wishes to make the computer.

Can’t one error-correct? Sure. But there are limitations to how fast

• ne can error-correct (error correction cannot operate faster than c)

For instance for a NOT gate our effect implies that after a time t~π/(2E) we will have How do the numbers look, say, for the “ultimate laptop”? We need to distinguish between “serial” and “parallel” computers. In a serial computer one achieves a lot of speed by having a lot In a serial computer one achieves a lot of speed by having a lot

• f energy difference in each qubit. In a parallel computer, one can

achieve similar speeds with less energy per qubit. For a machine with L qubits and a number of simultaneous operations dp, we get, Ultimate laptop This is 4 orders of magnitude better than Lloyd’s bound. If one had chosen a serial machine, the bound is even tighter 1042op/s.

• R. Gambini, R. Porto, JP, quant-ph/0507262 Proceedings of GAS@BS (2006)

Other applications: The introduction of a fundamental loss of coherence can have implications for the measurement problem in quantum mechanics. The usual criticism of the solution via interaction with the environment (namely that if one waits long enough one could recover quantum behavior) is eliminated if one assumes that realistic clocks and rods are introduced.

• R. Gambini, JP Found. Phys 37, 1074 (2007)

The framework can be made covariant if one introduces realistic measuring rods in addition to clocks. The effect ends up depending

• n the proper time interval.
• R. Gambini, R. Porto, JP Int. J. Mod. Phys. D14, 2181 (2006)

The same effect leads to loss of entanglement in quantum mechanics

• R. Gambini, R. Porto, JP arxiv:0708.2935 (quant-ph)

Summary

• We have argued that due to the lack of perfect classical clocks,

quantum mechanics should be described by a modified Schrödinger equation.

• Traditional quantum mechanics is still valid, but
• nly as an idealization. The evolution has a fundamental level
• f loss of coherence.
• f loss of coherence.
• The effect is strong enough to eliminate the black hole

information puzzle: information is lost fast enough due to our lack of ideal clocks that the puzzle is unobservable.

• It might one day (ahem…) limit how fast a laptop you can

carry to give a talk!

• It also has implications for entanglement and for the

measurement problem in quantum mechanics.

For a pedagogical review see

• R. Gambini, R. Porto, JP Gen. Rel. Grav. 39, 1143 (2007) gr-qc/060390

Why hasn’t all this been done before? In the context of quantum gravity there appear technical difficulties. When one works out the Hamiltonian theory of general relativity one finds the theory has constraints, that is functions C(q,p)=0. Physical quantities must have vanishing Poisson brackets with the constraints (otherwise there would be further constraints). Unfortunately one of the constraints is the Hamiltonian=0, which implies physical quantities are constants of the motion (cannot be used as clocks). Page and Wootters were mired by this problem when they presented the relational proposal for the first time (Kuchar demolished their proposal in his seminal review on the problem of time). We have recently introduced a discretization procedure that approximates general relativity by a theory without constraints. Therefore the above conceptual problem is not present.

• M. Campiglia, C. Di Bartolo, R. Gambini, JP Phys. Rev. D74 124012 (2006)