Envariance and equivalence

I was having doubts about selecting the topic of this week. We had the exciting detection of gravitational waves and I was thinking to write about it, but then I decided not to break the logical sequence of the current series. I will do my best to make this post exciting...

So back to the measurement problem. When the wavefunction collapses, it projects to a subspace of the Hilbert space, and this means we need to use a basis. But the choice of a basis is arbitrary and we need a preferred basis. Do we have such a thing at our disposal? If we consider only the Hilbert space of the quantum system the eigenvectors come to mind, but the eigenvectors of which Hermitean operator? Since the measurement result does not exist before measurement, we need to consider the Hilbert space of the measurement device as well. And when we have two Hilbert spaces we have at our disposal the Schmidt decomposition which is unique when the coefficients are distinct. Now this decomposition depends on the original wavefunction, but if we remember last time, the original Cartesian pair contains a distinguished wavefunction as well: the wavefunction of the measurement device in the "ready" state, and so the Schmidt decomposition basis is indeed a preferred basis.

In the Grotendieck approach, we have pairs of wavefunctions which are physically indistinguishable: there is no physical device which can be constructed to distinguish one pair from another. (Why? Because quantum mechanics is probabilistic and not deterministic). What if I act with a unitary transformation on the physical system? Can I undo this operation by acting with a unitary transformation on the measurement device? This is not always possible but in the Schmidt basis there are such unitaries (which I will call Schmidt unitaries). If instead of the measurement device we consider the environment, this is Zurek's idea of envariance (environment assisted invariance):

“When a state \( |\psi_{SE} \rangle \) of a pair system S, E can be transformed by \(U_S = u_S \otimes 1_E \) acting soley on S, \(U_S |\psi_{SE}\rangle = (u_S ⊗ 1_E )|\psi_{SE} \rangle = |\eta_{SE}\rangle\) but the effect of \(U_S\) can be undone by acting solely on E with an appropriately chosen \( U_E = 1_S \otimes u_E : U_E |\eta_{SE} \rangle = (1_S \otimes u_E )|\eta_{SE} \rangle = |\psi_{SE}\rangle\)

I will not use directly Zurek's envariance, but this was a great inspiration for me on constructing the equivalence I need to solve the measurement problem.

Without ado here is my definition of the equivalence:

Suppose we have two wavefunctions \( |\psi_p \rangle, |\phi_p \rangle \in H_p \) corresponding to a physical system, and another two wavefunctions \( |\psi_n \rangle , |\phi_n \rangle \in H_n \) corresponding to another physical system, where \(H_p, H_n\) are Hilbert spaces. Suppose also that \( |\psi_p \rangle \otimes |\psi_n \rangle \) has the same biorthogonal decomposition base as \( |\phi_p \rangle \otimes |\phi_n \rangle \)and in this base m, n, p, q are the subspace dimensionality for \(|\psi_p \rangle, |\psi_n \rangle, |\phi_p \rangle, |\psi_n \rangle \), respectively.

We call two pairs of a Cartesian product of wavefunctions equivalent:

\( (|\psi_p \rangle, |\psi_n \rangle ) \sim (|\phi_p \rangle, |\phi_n \rangle )\)

if the energy level of the composite system \( |\psi_p \rangle \otimes |\psi_n \rangle \) is equal with the energy level of \( |\phi_p \rangle \otimes |\phi_n \rangle \) and if there are unitary transformations \(U_p\) acting on the left element \((|\psi \rangle, \cdot )\) and unitary transformations \(U_n \) acting on the right element \( (\cdot, |\psi \rangle)\) such that: 

\((U_p|\psi_p \rangle ) \otimes |\phi_n \rangle = |\phi_p \rangle \otimes (U_n |\psi_n \rangle ) \)

subject to the constraint that: m + q + k = p + n + k 

Under those conditions one can easily prove the reflexivity, symmetry, and transitivity properties needed to construct an equivalence relationship.

Before measurement, the Cartesian pairs (measurement outcomes, device states) corresponding to potential measurement outcomes respect this equivalence relationship. When an outcome is registered, the distinguished Cartesian pair corresponding to the outcome breaks the equivalence. For example, in a bubble chamber, an alpha particle interacts with an atom ionizing it. The wavefunction of the free electron prevents the existence of the unitaries which would transform the Cartesian pair corresponding to the outcome into another Cartesian pair. There is nothing non-unitary in the process. This is like we do integer arithmetic and the final answer is say 10. But which 10? Is it: (10,0) or (14,4), or perhaps (273, 263)? "Measurement" in this case would correspond to picking a particular representation say: (23, 13).  Similarly, in the quantum case, after the interaction of the quantum system with the measurement device we are forced to pick a single representation, because the others are no longer valid. 

Now we can consider very interesting questions: 

• Can we erase the measurement outcome by restoring the equivalence relationship?
• Can we change the outcome by changing the equivalence relationship?
• Is the measurement process irreversible?

I will try to work out those issue in detail in the next post when I will analyze the quantum eraser experiment in the Cartesian pair formalism. There is a lot to cover and perhaps I will need more than one post for this.