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I recently watched "A New Approach to Quantum Mechanics" by Dr. Yakir Aharonov Part 1 (12.12.2023 | Institute for Quantum Studies) on YouTube (and also Part 2 and 3). I was naive enough to believe it was really new. Only when I searched for a valid reference in order to be allowed to ask about it here on PF, I learned that most of the content was already present in:
Weak measurement: The Aharonov-Albert-Vaidman effect:
One of my first thoughts triggered by this two-state vector perspective was that it offers a nice way to look at the “n qubits in (quantum) context” can contain up to 2n classical bits riddle. A completely different thought was that I found the prefect initial preparation and perfect final (postselection) measurement used for the explanation of that perspective unrealistic regarding what you can actually do in a laboratory. So I wondered what would change when you described the initial preparation and the final (postselection) measurement by density matrices instead. Turns out somebody asked exactly this question in another video at 58:30, but I was unable to find the reference where it was done, which Aharonov mentioned in his answer. Then I remembered that I once used the time-symmetric formulation of CH to get "at least some intuition why there is that unexpected product in the bound":
Aharonov, Yakir, Sandu Popescu, and Jeff Tollaksen. "A Time-symmetric Formulation of Quantum Mechanics." Physics Today 63.11 (2010). doi: 10.1063/1.3518209 (https://typeset.io/pdf/time-symmetric-formulation-of-quantum-mechanics-2gvmqarts5.pdf)
and all the content already got mentioned in another YouTube video J. Tollaksen: The Time-Symmetric Formulation of Quantum Mechanics, Weak Values and the Classical Limit of Quantum Mechanics (EmQM13) which also mentions much earlier publications likeWeak measurement: The Aharonov-Albert-Vaidman effect:
Phys. Rev. Lett. 60, 1351 (1988) (https://doi.org/10.1103/PhysRevLett.60.1351)
orY. Aharonov, P. G. Bergmann, J. L. Lebowitz, Phys. Rev. 134, 1410 (1964)
One of my first thoughts triggered by this two-state vector perspective was that it offers a nice way to look at the “n qubits in (quantum) context” can contain up to 2n classical bits riddle. A completely different thought was that I found the prefect initial preparation and perfect final (postselection) measurement used for the explanation of that perspective unrealistic regarding what you can actually do in a laboratory. So I wondered what would change when you described the initial preparation and the final (postselection) measurement by density matrices instead. Turns out somebody asked exactly this question in another video at 58:30, but I was unable to find the reference where it was done, which Aharonov mentioned in his answer. Then I remembered that I once used the time-symmetric formulation of CH to get "at least some intuition why there is that unexpected product in the bound":
I didn't notice before that this bound is actually a nicely rigorous occurence of that unexpected “n qubits in context <= 2n classical bits" phenomenon. And I also didn't notice before that using ##\rho_i## and ##\rho_f## with ##\operatorname{rank}(\rho_i)=\operatorname{rank}(\rho_f)=1## leaves me with ##m \leq 1##, i.e. at most one possible history. In a certain sense, this is even a confirmation of Aharonov's claim that he got completely rid of randomness by his two-state vector formalism.gentzen said:For thinking about the sharp bound itself, the time-symmetric formulation of CH with two hermitian positive semidefinite matrices ##\rho_i## and ##\rho_f## satisfying ##\operatorname{Tr}(\rho_i \rho_f)=1## seems well suited to me. The decoherence functional then reads ##D(\alpha,\beta)=\operatorname{Tr}(C_\alpha\rho_i C_\beta^\dagger\rho_f)## and the bound on the number ##m## of histories ##\alpha## with non-zero probability becomes ##\operatorname{rank}(\rho_i)\operatorname{rank}(\rho_f)\geq m##. Interpreting ##\rho_i## as corresponding to pre-selection ("preparation") and ##\rho_f## as post-selection ("measurement") gives at least some intuition why there is that unexpected product in the bound.