If I go to the church of the greater Hilbert space, can I have Unitary Collapse?

Question

Actually, unitary pseudo-collapse?
Von Neuman said quantum mechanics proceeds by two processes: unitary evolution and nonunitary reduction, also now called projection, collapse and splitting.

Collapse is non-unitary by definition, and the impossibility of unitary collapse is also a basic mathematical theorem. But we no longer believe, as Bohr did, in two Kingdoms, one classical and one quantum. Instead, we now talk about the emergence of the pseudo-classical. Similarly, I suggest, we can now talk about a pseudo-collapse, which is pseudo-unitary.

In a collapse two things happen: First, the matrix diagonalizes, and then, it becomes one dimensional, ie one element of the main diagonal becomes 1 and all the others become 0. This second part is the nonunitary projection part. (In a slight variation, we have two corresponding matrices, labelled system and apparatus, and the coresponding "pointer state" in the apparatus goes to 1, and all the other pointer states go to 0.) Now along comes decoherence, and we have three matrices, S,A, and E: System, Apparatus and Environment. Actually, these three matices are submatrices, ie diagonal blocks in one bi gger matrix. You can still diagonalize everything by unitary operations. But you still can't project it to a one dimensional definite outcome by unitary means. However, I think you can make the S block one dimensional by unitary operations if you push the off diagonal terms into the S-E quadrant. Is this correct?

This seems almost obvious to me, but I can't find confirmation. In fact, as I searched, I found lots of discussion on the diagonalization process, but almost nothing about the selection step, which is the real crux of the process, as far as `I am concerned. Why is this? Can someone point me to a good discussion of the selection process? And confirm that my pseudo-unitary pseudo-collapse is mathematically possible?

Answer 1

Answer

Rigorous adherence to the liturgical rituals of the "Church of the Larger Hilbert Space" is feasible in principle yet exponentially inefficient in practice.

Exercise

One way to answer this question is by reference to a feasible numerical computation.

So fire-up MatLab; specify the dynamical system as (say) $n\sim 10$ interacting qubits; specify any desired Hamiltonian; choose some starting energy $E$; then integrate a dynamical trajectory $\psi(\,t\,|\,E\,)$.

Now perform the following exercise:

Exercise I  Solely from operator expectation values associated to an $n$-qubit unitary trajectory $\psi(\,t\,|\,E\,)$, estimate the single-qubit Bloch relaxation parameters $T_1(E)$ and $T_2(E)$.

If you are more ambitious:

Exercise II (extra credit)  Follow in the footsteps of Urey, Onsager, Dirac, Feynman (etc.) and estimate the thermodynamic transport coefficients of larger systems of interacting qubits, again wholly by reference to the unitary dynamical trajectory $\psi(\,t\,|\,E\,)$.

Then we have the following assertion

Assertion  Without loss of accuracy in estimating $T_1(E)$ and $T_2(E)$, all but a fraction $\mathcal{O}(e^{-n})$ of the unitary dynamical trajectory $\psi(\,t\,|\,E\,)$ can be discarded.

The portion of the unitary trajectory that can be discarded — for the practical purpose of estimating thermodynamic parameters — is of course associated to qubits in "cat" states.

That is the practical reason why the "Church of the Larger Hilbert Space" is not popular among systems engineers … its liturgical rituals are exponentially inefficient!

Alternate doctrines

Efficient computational recipes for estimating thermodynamic parameters are extant in the literature; these — necessarily non-unitary — quantum dynamical recipes are reviewed in an answer to the question "Reversing gravitational decoherence."

Open questions

We have seen that rigorous adherence to the liturgical rituals of the "Church of the Larger Hilbert Space" is feasible in principle yet exponentially inefficient in practice. It is natural to wonder:  "Does Nature herself embrace the strictly unitary yet exponentially inefficient liturgical rituals of the Larger Hilbert Space? Or does she resort to computationally efficient yet non-unitary dynamical trajectories similar to those of mortal engineers?"

These questions are open.

Answer 2

I believe "a good discussion of the selection process" can be found in http://arxiv.org/abs/1107.2138 . However, I prefer just to reject collapse, based, e.g., on Schlosshauer's analysis of experimental data: "no positive experimental evidence exists for physical state-vector collapse;" (M. Schlosshauer, Annals of Physics, 321 (2006) 112-149). Collapse is an approximation in the best case. A more general reason to reject collapse - it contradicts unitary evolution.

Answer 3

Yes, this is correct. Let $\rho$ be the initial state of the system S and let $|0\rangle$ be the initial state of the apparatus. Take $P_m$ to be a set of rank-1 projectors you measure, with $m$ denoting the outcome. Then $\rho \otimes |0\rangle \mapsto \sum_m P_m \rho \otimes |m\rangle$, which we can denote as $U$, can be extended to a unitary operation for any initial state $\rho$. This is because for any two pure initial states $|\psi\rangle, |\phi\rangle$ we have that $(\langle\phi| \otimes \langle 0|) U^\dagger U (|\psi\rangle \otimes |0\rangle) = \langle\phi|\psi\rangle$. In words, $U$ acts as a unitary operation on this particular subspace. For a proof that there exists a unitary extension to the entire Hilbert space, see for example Nielsen and Chuang, Quantum Computation and Quantum Information.

At the conclusion of this process, the system state is of the form $\sum_m p_m |m\rangle\langle m|$, which is what I think you meant by "block one-dimensional", unless I misunderstood your question.

Also, a good review of decoherence is this article by Zurek. If this is what interests you, I recommend checking this out http://arxiv.org/abs/quant-ph/0105127