Yakir Aharonov's time-symmetric formulation of quantum mechanics

[gentzen]

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:

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 like
Weak measurement: The Aharonov-Albert-Vaidman effect:

Phys. Rev. Lett. 60, 1351 (1988) (https://doi.org/10.1103/PhysRevLett.60.1351)​

or

Y. 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":

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.

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.

 

[Morbert]

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,

Are you referring to this?
https://arxiv.org/pdf/gr-qc/9304023.pdf
(See eqn 3.1a)

 

[gentzen]

Are you referring to this?
https://arxiv.org/pdf/gr-qc/9304023.pdf
(See eqn 3.1a)

No, Aharonov said something like "including an article by myself and Benny Reznik".

But in a certain sense, your suggestion is spot-on regarding what I want to know: Did Aharonov essentially used some time-symmetric version of CH for the generalization to density matrices, or did he come up with a completely different approach.

 

[gentzen]

No, Aharonov said something like "including an article by myself and Benny Reznik".

There is a wikipedia article for the two-state vector formalism. It says:

Building on the notion of two-state, Reznik and Aharonov constructed a time-symmetric formulation of quantum mechanics that encompasses probabilistic observables as well as nonprobabilistic weak observables.[6]

6. Reznik, B.; Aharonov, Y. (1995-10-01). "Time-symmetric formulation of quantum mechanics". Physical Review A. American Physical Society (APS). 52 (4): 2538–2550. arXiv:quant-ph/9501011. Bibcode:1995PhRvA..52.2538R. doi:10.1103/physreva.52.2538. ISSN 1050-2947. PMID 9912531. S2CID 11845457.

But in a certain sense, your suggestion is spot-on regarding what I want to know: Did Aharonov essentially used some time-symmetric version of CH for the generalization to density matrices, or did he come up with a completely different approach.

I got the impression that Reznik and Aharonov came up with a completely different approach, namely instead of two independent hermitian positive semidefinite matrices $\rho_i$ and $\rho_f$, the uncertainties in the initial and final state can be correlated:

When the conditions are determined only “partially” the system is initially and finally in a mixed state. In the context of our formalism this can be interpreted as a situation with correlations between the initial and final conditions.

But they were well aware of time-symmetric versions of CH:

More recently the formalism was re-discovered independently by Griffiths [2], Unruh [3], and Gell-Mann and Hartle [4].‡
‡The relation between the approach developed in this article, and the decoherent histories approach is studied elsewhere

 

[DrChinese]

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:

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)​

I have a few earlier references on this line of thinking (TSVF) that may be useful:

Properties of a quantum system during the time interval between two measurements (behind a paywall!)
Yakir Aharonov, Lev Vaidman (1990)

On a Time Symmetric Formulation of Quantum Mechanics
B. Reznik, Y. Aharonov (1995)

The Two-State Vector Formalism of Quantum Mechanics: an Updated Review
Yakir Aharonov, Lev Vaidman (2001-2007)

New Insights on Time-Symmetry in Quantum Mechanics
Yakir Aharonov, Jeff Tollaksen (2007)

---

Interestingly: The 1990 article was one of the earliest to discuss weak measurements. This was part of a series of articles by Aharonov, Vaidman and others. I believe both Vaidman and Tollaksen did PhD work with Aharonov - who himself did some pioneering work on Time Symmetry in physics going all the way back to 1964! That's pre-Bell...

Both Vaidman and Aharonov are also extremely versed in Bohmian theory. And Vaidman, of all things, is also a leading proponent of MWI as well as the TSVF. If you want to learn a bit about that, he wrote the Stanford/Plato article on that. This paper is also an amazing work, even if you don't agree with all of it. His knowledge of both sides of many issues is considerable:

Quantum Theory and Determinism
Lev Vaidman (2013)

 

[gentzen]

But let me add that I just mentioned this trouble with inexact zeros in CH to avoid that my comment gets misinterpreted as an attack on MWI or some of its proponents. I didn't want to attack CH or open an unrelated discussion about CH in this thread.

It's hard to see why small numbers should entail robustness issues. Unless you refer it to some other state, an amplitude in QM is meaningless.

The robustness issues of CH can for example be observed in the context of the above observation that the time-symmetric formulation of CH allows to analyse the situation where the initial preparation and the final (postselection) measurement are described by density matrices. (The time-symmetric formulation of CH with two hermitian positive semidefinite matrices $\rho_i$ and $\rho_f$ satisfying $\operatorname{Tr}(\rho_i\rho_f)=1$ such that the decoherence functional reads $D(\alpha,\beta)=\operatorname{Tr}(C_\alpha\rho_iC_\beta^\dagger\rho_f)$.) Even so the bound $\operatorname{rank}(\rho_i)\operatorname{rank}(\rho_f) \geq m$ for the number $m$ of histories $\alpha$ with non-zero probability is interesting, it suffers from the fact that even negligibly small eigenvalues of $\rho_i$ and $\rho_f$ contribute to the left-hand-side, and histories $\alpha$ with negligibly small probability contribute to the right-hand-side. More robust and useful would be a bound on the entropy of the probabilities of possible sets of histories. In fact, what the bound seems to suggest is that this entropy would be bounded by the sum of the entropies of $\rho_i$ and $\rho_f$. But even if that bound should be true, I am not aware of a proof or a CH paper that would prove or investigate anything related.

Another "robustness" related observation is that the decoherence functional $D(\alpha,\beta)$ is positive semidefinite with trace 1, even if non-zero non-diagonal elements are present. So it actually makes sense to compute its entropy even in that case. The attempt to interpret that entropy by first diagonalizing the decoherence functional, and the construct modified $\tilde{C}_\alpha$ corresponding to the diagonalized functional (by linear combination of the $C_\alpha$) fails, because those $\tilde{C}_\alpha$ in general fail to arise from a "projectors at specific time instances" structure. (This is one reason why the prospects look dubious, for Griffiths' hope that slightly tweaking the input data could turn all those annoying near-zeros into exact zeros.)

 

[gentzen]

More robust and useful would be a bound on the entropy of the probabilities of possible sets of histories. In fact, what the bound seems to suggest is that this entropy would be bounded by the sum of the entropies of $\rho_i$ and $\rho_f$. But even if that bound should be true, I am not aware of a proof or a CH paper that would prove or investigate anything related.

The bound is not true. At least not if one defines "the sum of the entropies of $\rho_i$ and $\rho_f$" as $S(\rho_i)+S(\rho_f)$ with $S(\rho):=-\operatorname{Tr}(\frac{\rho}{\operatorname{Tr}(\rho)}\log(\frac{\rho}{\operatorname{Tr}(\rho)}))$ (for a positive semidefinite Hermitian matrix $\rho$). There is simply not enough interaction of the condition $\operatorname{Tr}(\rho_i\rho_f)=1$ with $S(\rho_i)$ and $S(\rho_f)$. An expression with appropriate interaction is $S(\rho_i,\rho_f):=-\operatorname{Tr}(\rho_i(\log(\frac{\rho_i}{\operatorname{Tr}(\rho_i)})+\log(\frac{\rho_f}{\operatorname{Tr}(\rho_f)}))\rho_f)$. But again, I am not aware of a proof (that this is a bound for the entropy of the probabilities of possible sets of histories) or any related CH paper.

To see that $S(\rho_i)+S(\rho_f)$ cannot bound the entropy, one can take any set of consistent histories, and construct another set of consistent histories with identical probabilities on a slightly enlarged Hilbert space, with arbitrarily small $S(\tilde{\rho}_i)$ and $S(\tilde{\rho}_f)$. Simply take the direct sum of the given Hilbert space with a two dimensional Hilbert space, and extend $\rho_i$ by $D_i:=\begin{bmatrix}d_i & 0 \\ 0 & 0\end{bmatrix}$ and $\rho_f$ by $D_f:=\begin{bmatrix}0 & 0 \\ 0 & d_f\end{bmatrix}$ to this Hilbert space, i.e. $\tilde{\rho}_i:=\begin{bmatrix}\rho_i & 0 \\ 0 & D_i\end{bmatrix}$ and $\tilde{\rho}_f:=\begin{bmatrix}\rho_f & 0 \\ 0 & D_f\end{bmatrix}$. Extend the projectors $P_{\alpha_k}(t_k)$ to $\tilde{P}_{\alpha_k}(t_k):=\begin{bmatrix}P_{\alpha_k}(t_k) & 0 \\ 0 & Q_{\alpha_k}\end{bmatrix}$ with $Q_1:=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ and $Q_{\alpha_k}:=0$ for $\alpha_k \neq 1$. The $\tilde{P}_{\alpha_k}(t_k)$ are a valid family of projectors iff the $P_{\alpha_k}(t_k)$ are. Because $D_iD_f=0$ we have $\operatorname{Tr}(\tilde{\rho}_i\tilde{\rho}_f)=\operatorname{Tr}(\rho_i\rho_f)=1$ and $\tilde{D}(\alpha,\beta)=D(\alpha,\beta)$. Now observe that $S(\tilde{\rho}_i)\to 0$ for $d_i\to\infty$ and $S(\tilde{\rho}_f)\to 0$ for $d_f\to\infty$.

 

[gentzen]

Because my intuition regarding $S(\rho_i)+S(\rho_f)$ as a bound had been so wrong, I started to wonder whether I actually have a proof for the bound $\operatorname{rank}(\rho_i)\operatorname{rank}(\rho_f) \geq m$ for the number $m$ of histories $\alpha$ with non-zero probability. (Maybe that proof contains hints of how to prove that $S(\rho_i,\rho_f)$ is an upper bound for the entropy of the probabilities of possible sets of histories.) Proofs were mentioned in

... force the Hilbert space to be big enough, at least if the initial state is pure ($\rho = \ket{\phi}\bra{\phi}$). The consistency condition $\operatorname{Tr}(C_\alpha\rho C_\beta^\dagger)=0$ then reduce to $(C_\beta\ket{\phi},C_\alpha\ket{\phi})=0$, i.e. the vectors $C_\alpha\ket{\phi}$ are orthogonal. So if there are $m$ histories $\alpha$ with non-zero probability, then the dimension $N$ of the Hilbert space satisfies $N \geq m$.

If the initial state $\rho$ has rank $r$ instead of being pure, then we only get $r N \geq m$. (One proves this by replacing $\rho$ with a pure space in a suitably enlarged Hibert space, see Diósi for details.) This bound can really be achieved, ...

Here is this proof for the case $\rho_f=I$ from "On maximum number of decoherent histories" by Lajos Diósi
https://arxiv.org/abs/gr-qc/9409028

Let us start initially with pure state $\rho = \ket{\psi}\bra{\psi}$ and introduce the unnormalized vectors $$\ket{\varphi_\alpha} = C_\alpha\ket{\psi}. \qquad (7)$$ From the decoherence condition (2) it follows that they are orthogonal to each other: $$ \langle \varphi_\beta | \varphi_\alpha \rangle = 0 \quad \text{ for all } \alpha \neq \beta. \qquad (8)$$ The maximum number of (nonzero) orthogonal vectors in $\mathcal{H}$ is $N$.
Let us allow mixed states, too. Consider the orthogonal expansion $\rho =\sum^N_{r=1} w_r \ket{\psi_r}\bra{\psi_r}$ which is always possible with nonnegative normalized weights $w_r$. Consider the trivial embedding of our system into a larger one whose state space is $\mathcal{H}\otimes\mathcal{H'}$ where $\mathcal{H'}$ is also $N$-dimensional Hilbert space. Construct the following state vector in $\mathcal{H}\otimes\mathcal{H'}$: $$\ket{\Psi}=\sum_{r=1}^N \sqrt{w_r}\ket{\psi_r}\otimes\ket{\psi'_r} \qquad (9)$$ where $\{\ket{\psi'_r}, r = 1, . . . , N \}$ form a complete orthonormal system in $\mathcal{H'}$. Introduce the following unnormalized vectors in $\mathcal{H}\otimes\mathcal{H'}$: $$\ket{\Phi_\alpha} = (C_\alpha \otimes 1) \ket{\Psi} \qquad (10)$$ From Eqs. (7-10) one can prove that these vectors are also orthogonal to each other: $$\langle \Phi_\beta | \Phi_\alpha \rangle = \langle \varphi_\beta | \varphi_\alpha \rangle = 0 \quad \text{ for all } \alpha \neq \beta. \qquad (11)$$ In Hilbert space $\mathcal{H}\otimes\mathcal{H'}$, the maximum number of orthogonal vectors is equal to the number $N \times N$ of dimensions. In such a way we have proven that the maximum number of histories in a given decohering family is $N^2$. (The general limit is $\operatorname{rank}(\rho)N$, as one could easily prove.)

So how can this proof be adapted to the general time symmetric case? A first step is to change (8) and (11) to
$$ \langle \varphi_\beta | \varphi_\alpha \rangle = D(\alpha,\beta) = 0 \quad \text{ for all } \alpha \neq \beta. \qquad (8)$$ $$\langle \Phi_\beta | \Phi_\alpha \rangle = D(\alpha,\beta) = 0 \quad \text{ for all } \alpha \neq \beta. \qquad (11)$$ because this clarifies things (especially (11) was strange before). Next step is to replace $\langle \varphi_\beta | \varphi_\alpha \rangle$ by $\bra{\varphi_\beta} \rho_f \ket{\varphi_\alpha}$ and $\langle \Phi_\beta | \Phi_\alpha \rangle$ by $\bra{\Phi_\beta} \tilde{\rho}_f \ket{\Phi_\alpha}$ with $\tilde{\rho}_f:=\rho_f\otimes I$, and get rid of the $\alpha \neq \beta$ restriction:
$$\bra{\varphi_\beta} \rho_f \ket{\varphi_\alpha} = D(\alpha,\beta) \quad \text{ for all } \alpha, \beta. \qquad (8')$$ $$\bra{\Phi_\beta} \tilde{\rho}_f \ket{\Phi_\alpha} = D(\alpha,\beta) \quad \text{ for all } \alpha, \beta. \qquad (11')$$
The orthogonal expansion $\rho =\sum^N_{r=1} \ldots$ must be replaced by $\rho_i =\sum^{\operatorname{rank}(\rho_i)}_{r=1} w_r \ket{\psi_r}\bra{\psi_r}$ and the statement "$\mathcal{H'}$ is also $N$-dimensional Hilbert space" by "$\mathcal{H'}$ is a $\operatorname{rank}(\rho_i)$-dimensional Hilbert space". This leads to $\operatorname{rank}(\tilde{\rho}_f)=\operatorname{rank}(\rho_i)\operatorname{rank}(\rho_f)$. Because it is "relatively easy" to conclude $\operatorname{rank}(\tilde{\rho}_f)\geq m$ from (11'), all that remains is to check that (11') is actually true.
In the spirit of Diósi's proof, one first checks that (8') holds: $$D(\alpha,\beta)=\operatorname{Tr}(C_\alpha\ket{\psi}\bra{\psi}C_\beta^\dagger\rho_f)=\operatorname{Tr}(\bra{\psi}C_\beta^\dagger\rho_fC_\alpha\ket{\psi})=\operatorname{Tr}(\bra{\varphi_\beta} \rho_f \ket{\varphi_\alpha})$$
The $\operatorname{Tr}(...)$ on the RHS can be omitted, because it is applied to a scalar (i.e. a 1x1 matrix).
This calculation is then repeated for (11'):
$$D(\alpha,\beta)=\operatorname{Tr}(C_\alpha \left( \sum_r w_r \ket{\psi_r}\bra{\psi_r} \right) C_\beta^\dagger\rho_f)=\sum_r w_r\operatorname{Tr}(\bra{\psi_r}C_\beta^\dagger\rho_fC_\alpha\ket{\psi_r})$$
where again the $\operatorname{Tr}(...)$ on the RHS can be omitted.
$$\bra{\Phi_\beta} \tilde{\rho}_f \ket{\Phi_\alpha} =\left( \sum_s\sqrt{w_s}(\bra{\psi_s}C_\beta^\dagger)\otimes\bra{\psi'_s} \right) \tilde{\rho}_f \left( \sum_r\sqrt{w_r}(C_\alpha\ket{\psi_r})\otimes\ket{\psi'_r} \right)$$ $$= \sum_{r,s}\sqrt{w_s}\sqrt{w_r}(\bra{\psi_s}C_\beta^\dagger \rho_f C_\alpha\ket{\psi_r}) \cdot (\bra{\psi'_s} I \ket{\psi'_r}) =\sum_r w_r \bra{\psi_r}C_\beta^\dagger\rho_fC_\alpha\ket{\psi_r}$$ where $\bra{\psi'_s} I \ket{\psi'_r} =\delta_{s,r}$ turned $\sum_{r,s}\sqrt{w_s}\sqrt{w_r}$ into $\sum_r w_r$. This concludes the proof.

---

This adapted proof is nearly identical to Diósi's proof. The explicit checking of (11') above is not really more complicated than the omitted explicit checking of (11). A real difference is the observation that it is "relatively easy" to conclude $\operatorname{rank}(\tilde{\rho}_f)\geq m$ from (11').
What about the "hints of how to prove that $S(\rho_i,\rho_f)$ is an upper bound for the entropy"? The proof suggests how one can reduce it to the case $\operatorname{rank}(\rho_i)=1$ (or also to the case $\operatorname{rank}(\rho_f)=1$ if that feels more intuitive). The realization that this reduction didn't work for $S(\rho_i)+S(\rho_f)$ as bound actually helped me to "see" why that bound had to fail.