A philosophy that underpins many approaches to understanding quantum mechanics (the many worlds interpretation in particular, but collapse models and other related ideas also) is that continuous Schroedinger evolution is somehow `nicer', `preferred', or `more fundamental' than the "damned quantum jumps".(adsbygoogle = window.adsbygoogle || []).push({});

A measurement in QM can be described by a set of Kraus operators [itex]\left\{

K_{i}\right\} [/itex] which satisfy

[tex]

\sum_{j}K_{j}^{\dagger }K_{j}=\mathbb{I}.

[/tex]

For a system initially in some state [itex]\rho,[/itex] the final (collapsed) state after a measurement which yields outcome [itex]j[/itex] with probability [itex]p_{j}=Tr(\rho K_{j}^{\dagger }K_{j})[/itex] is

[tex]

\rho \rightarrow K_{j}\rho K_{j}^{\dagger }/p_j

[/tex]

In standard quantum mechanics the continuous (Schroedinger) evolution takes the form

[tex]

\rho \rightarrow U\rho U^{\dagger }.

[/tex]

where [itex]U[/itex] is a unitary operator.

If the Hamiltonian governing this evolution has spectral decomposition

[tex]

H=\sum_{j=1}^{d}\lambda _{j}|j\rangle \langle j|

[/tex]

then this unitary is given by

[tex]

U=\sum_{j=1}^{d}e^{\lambda _{j}t}|j\rangle \langle j|.

[/tex]

(Note I'm just doing the finite dimensional case here for simplicity).

Alternatively we can imagine that the system is actually undergoing a large number of very frequent measurements as follows. Define the Kraus operators

[tex]

K_{j}=\frac{1}{\sqrt{d}}\left[ \mathbb{I+}\left( e^{id\lambda _{j}\tau

}-1\right) |j\rangle \langle j|\right]

[/tex]

where [itex]\tau [/itex] is a very small time increment, and we presume a measurement occurs approximately every [itex]\tau[/itex] seconds. Since one can readily verify that [itex]K_{j}^{\dagger}K_{j}=\mathbb{I}/d[/itex] we see that regardless of the initial state [itex]\rho [/itex] the outcomes are all equally likely. Thus in a time [itex]t\gg \tau [/itex] roughly [itex]\tfrac{t}{d\tau}[/itex] of each specific outcome will be obtained, and it is easy to see then that the final state will be very close to the one which unitary evolution would have generated. One may think that [itex]\tau[/itex] needs to be very small (say Planck scale), but thinking about it I cannot see that we have experimental evidence of smooth evolution beyond the scale of optical vacumm fluctations ([itex]1/\omega^3[/itex]) with [itex]\omega[/itex] roughly an optical frequency.

Thus we see that the "less fundamental" form of quantum evolution can actually subsume the supposedly more fundamental one. Perhaps our attachment to unitary evolution is simply an historical artifact better dispensed with!

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# Quantum mechanics without unitary evolution

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