A Weinberg: Quantum Mechanics Without State Vectors

[A. Neumaier]

Picking such a double cone containing our present planetary system implies that we may assume the algebra of observables currently accessible to mankind to be such a factor of type III_1. Remarkably, such a C^*-algebra has no pure states

More information about this is in the discussion in a related thread, starting here.

 

[greypilgrim]

Sorry if this is a really obvious question. How can the density matrix for each part of two entangled states coordinate in order to produce the bell state measurement results if they are separate?

This question hasn't been answered yet, and it puzzles me as well. In the second paper, Weinberg says
"Whether in open systems in ordinary quantum mechanics or in closed systems in some modified version of quantum mechanics, in order to avoid instantaneous communication at a distance in entangled states, it is important to require that the density matrix at one time depends on the density matrix at any earlier time, but not otherwise on the state vector at the earlier time."

Since the density operators of both subsystems in a Bell experiment are completely mixed, how can local measurements (be it his suggested Lindblad evolution or any other kind) be correlated?

 

[kith]

The reduced density matrix contains all information which is relevant for experiments which are performed on the subsystem alone.

Weinberg stresses that if the outcomes of such a measurement on subsystem A depended on the reduced density matrix of subsystem B, FTL communication would be possible by altering the reduced density matrix of B.

What Weinberg's statement is silent about is whether the reduced density matrices contain all information about the combined system of A and B. And indeed, they don't contain information about how the outcomes of individual measurements on the subsystems are correlated.

The information about the correlations is only contained in the density matrix of the combined system and not in the reduced density matrices of the subsystems.

 

[maline]

So by reformulating quantum theory the weirdness of the double slit experiment and the violation of Bell's inequality (= the negation of local realism) disappear?
That's pretty weird.

Indeed, treating $\rho$ as "real" does nothing to remove the "weirdness". The crucial point is that the statement

With this definition of a physical state, even in entangled states nothing that is done in one isolated system can instantaneously effect the physical state of a distant isolated system

applies only to time-evolution of a given density matrix, but not to the process of the experimenter updating the density matrix in response to measurement results, a.k.a. "collapse", "reduction", etc. - our favorite bugbear.

In this context, the EPR paradox is expressed as follows:
Suppose Alice and Bob, at spacelike separation, share an entangled state described by $\rho$. Alice performs a measurement on her subsystem, views the results, and uses them to describe the post-measurement state with a new density matrix $\rho'$. Bob, on the other hand, although we will assume he knows what measurement Alice planned to perform, cannot know the measurement results. Therefore he must continue to use $\rho$, evolving it with time in accordance with the interactions that take place as part of Alice's experimental protocol. The content of No Signalling Property is that the reduced density matrix $\rho_B$, describing Bob's subsystem, is unaffected by Alice's activities at spacelike separation. But the same cannot be said of $\rho'_B$. In general, since Alice uses new observations to write $\rho'$, $\rho'_B$ will likely contain more information (lower entropy) than $\rho_B$. For the standard EPR case of a singlet pair of spin-1/2 particles, $\rho_B$ is the completely mixed state while $\rho'_B$ is the pure state with spin opposite to Alice's result.

As long as we think of density matrices as describing knowledge- basically the instrumentalist viewpoint- there is nothing strange about this: Alice has more knowledge and so she can write a "better" density matrix. But if density matrices are "real", then which one, $\rho$ or $\rho'$, should we consider to be the "true" state? Alice's new knowledge is certainly correct and "part of reality", so it seems her update must be called a "real change of the state"- good old collapse. And of course, this change is nonlocal. Alice's measurement has instantaneously generated new information constraining Bob's subsystem; information that was not part of the previous "state of reality".

The only way out is MWI, which Weinberg dislikes: $\rho$ indeed remains the true state of the whole system, now a "multiverse", while $\rho'$ is Alice's description of the particular "branch" she now finds herself in. "Probabilites" (whatever that means in MWI) for experiments Bob may perform are described by $\rho_B$, while $\rho'_B$ describes "probabilities" for the branches where those results are eventually compared with Alice's particular result.

All this is well known. I do not understand what Weinberg hopes to gain by assuming the density matrix is real. In particular, in approaches like GRW (objective stochastic collapse) which is the direction he seems to prefer, the Lindblad equation describes the evolution of $\rho$ only from the perspective of one who has not seen the measurement results, so it certainly should not be taken to represent "reality".