Measurement and Quantum Dynamics in the Minimal Modal Interpretation of Quantum Theory

Abstract

Any realist interpretation of quantum theory must grapple with the measurement problem and the status of state-vector collapse. In a no-collapse approach, measurement is typically modeled as a dynamical process involving decoherence. We describe how the minimal modal interpretation closes a gap in this dynamical description, leading to a complete and consistent resolution to the measurement problem and an effective form of state collapse. Our interpretation also provides insight into the indivisible nature of measurement—the fact that you can’t stop a measurement part-way through and uncover the underlying ‘ontic’ dynamics of the system in question. Having discussed the hidden dynamics of a system’s ontic state during measurement, we turn to more general forms of open-system dynamics and explore the extent to which the details of the underlying ontic behavior of a system can be described. We construct a space of ontic trajectories and describe obstructions to defining a probability measure on this space.

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Fig. 1

Notes

  1. 1.

    The Ghirardi–Rimini–Weber interpretation is an example of such an approach [3].

  2. 2.

    An in-depth discussion of various types of modal interpretations and their shortcomings is covered in [10].

  3. 3.

    We recognize that there are foundational questions about the precise, rigorous meaning of probability. We wish to disentangle and set aside these deep mysteries from what we take to be an independent set of foundational questions in quantum theory. Thus, we merely require that our probabilities obey Kolmogorov’s axioms, and we remain agnostic about the metaphysical meaning of these probabilities. As we shall see, operationally, the probabilities that define our epistemic states line up with the empirical outcomes one would expect.

  4. 4.

    As we will explain later, objective uncertainty can be characterized as the minimal amount of uncertainty that any observer can attain regarding the state of the system without perturbing the system. This kind of uncertainty arises fundamentally from entanglement.

  5. 5.

    The notion of system-centric ontology can also be thought of as a ‘localization’ of ontology that is reminiscent of the way in which general relativity localizes inertial reference frames. By contrast, an interpretation like many-worlds expands the universal state vector in a preferred basis and defines the ontology of all systems with respect to that seemingly arbitrary choice.

  6. 6.

    See [11] for a more detailed discussion.

  7. 7.

    Our interpretation’s dynamics are more restricted and arise in a different manner from those developed by Bacciagaluppi and Dickson in [18].

  8. 8.

    The minimal modal interpretation can thus be thought of as a hidden-variables interpretation where the actual ontic state of the system plays the role of a hidden variable.

  9. 9.

    This observation was made in earlier work, such as [19].

  10. 10.

    The detailed behavior of these functions depends on how one models the coupling between the systems. Explicit realizations have been studied in [20] and in greater generality in [21]. The latter paper demonstrates that a spin system interacting with an environment will undergo decoherence with approximately Gaussian damping of coherence terms.

  11. 11.

    Deviations from the Born rule arise when measurements are modeled as a decoherence-type quantum process involving a measurement apparatus and environment of finite size and an interaction of finite duration. Such deviations are not a unique feature of the minimal modal interpretation, but also occur in other interpretive frameworks, like the many-worlds interpretation, in which decoherence plays a central role.

  12. 12.

    We will address the more general situation in which correlations always exist in Sect. 4.6.

  13. 13.

    Another area in which such non-probabilistic uncertainty may arise is cosmology. Models exhibiting eternal inflation generically feature causally disconnected regions that conceivably manifest different phases of an underlying physical theory with different empirical properties, such as different masses for elementary particles and different interaction couplings between them. There is no obvious way to define a measure on these empirical attributes. Many attempts have been made and will likely continue, but the possibility of a more fundamental type of uncertainty should not be dismissed.

  14. 14.

    Abrams and Lloyd argue in [34] that the freedom to implement arbitrarily chosen nonlinear dynamics would lead to surprising implications for solving NP-complete problems. We emphasize that the nonlinear dynamics here is not fully under experimental control.

  15. 15.

    From time to time, one reads of proposals that linear open-system dynamics can, in fact, be defined even in the presence of initial subsystem–environment correlations. However, because any such dynamical map has the specific correlations of a particular initial density matrix built into its definition, the dynamical map manifestly cannot be linear in the sense that it can take as inputs general linear combinations of arbitrary initial density matrices.

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Acknowledgements

D. K. thanks Gaurav Khanna, Darya Krym, John Estes, and Paul Cadden-Zimansky for many useful discussions. D. K. has been supported in part by FQXi minigrant Observers in Quantum Theory-#10610. J. A. B. would like to acknowledge helpful conversations with David Albert, Ned Hall, and Jeremy Butterfield. We are both grateful to Brian Greene and Allan Blaer for many discussions and insightful suggestions.

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Quantum Conditional Probabilities

Quantum Conditional Probabilities

In this appendix, we motivate the formula (7) for the quantum conditional probabilities at the heart of the minimal modal interpretation. We start with a parent system \(W=Q_{1}+Q_{2}\) partitioned into subsystems \(Q_{1}\) and \(Q_{2}\) that are mutually disjoint. The reduced density matrix of the subsystem \(Q_{1}\) at time \(t'\) is given by the partial trace

$$\begin{aligned} \hat{\rho }_{Q_{1}}\left( t'\right) =\text {Tr}_{Q_{2}}\left[ \hat{\rho }_{W}\left( t'\right) \right] . \end{aligned}$$
(49)

The reduced density matrix of the subsystem \(Q_{2}\) is similarly defined. At any given time t, the density matrices of W and the subsystems \(Q_{1}\) and \(Q_{2}\) can be expanded in the bases of their respective eigenprojectors \(\left\{ \hat{P}_{W}\left( w;t\right) \right\} _{w}\), \(\left\{ \hat{P}_{Q_{1}}\left( i_{1};t\right) \right\} _{i_{1}},\) and \(\left\{ \hat{P}_{Q_{2}}\left( i_{2};t\right) \right\} _{i_{2}}\):

$$\begin{aligned} \hat{\rho }_{W}\left( t\right)&=\sum _{w}p_{W}\left( w;t\right) \hat{P}_{W}\left( w;t\right) ,\end{aligned}$$
(50)
$$\begin{aligned} \hat{\rho }_{Q_{1}}\left( t\right)&=\sum _{i_{1}}p_{Q_{1}}\left( i_{1};t\right) \hat{P}_{Q_{1}}\left( i_{1};t\right) ,\end{aligned}$$
(51)
$$\begin{aligned} \hat{\rho }_{Q_{2}}\left( t\right)&=\sum _{i_{2}}p_{Q_{2}}\left( i_{2};t\right) \hat{P}_{Q_{2}}\left( i_{2};t\right) . \end{aligned}$$
(52)

According to the minimal modal interpretation, the probability of the subsystem \(Q_{1}\) being in the ontic state \(\Psi _{i_{1}}\left( t'\right)\) at time \(t'\) is

$$\begin{aligned} p_{Q_{1}}\left( i_{1};t'\right) =\text {Tr}_{Q_{1}}\left[ \hat{P}_{Q_{1}}\left( i_{1};t'\right) \hat{\rho }_{Q_{1}}\left( t'\right) \right] . \end{aligned}$$
(53)

This expression can be rewritten as a formula that explicitly involves the disjoint subsystem \(Q_{2}\) and the parent system W by expanding the trace to encompass the entire parent system’s Hilbert space and inserting an identity operator for the Hilbert space of the subsystem \(Q_{2}\):

$$\begin{aligned} p_{Q_{1}}\left( i_{1};t'\right) =\text {Tr}_{W}\left[ \left( \hat{P}_{Q_{1}}\left( i_{1};t'\right) \otimes \hat{1}_{Q_{2}}\right) \hat{\rho }_{W}\left( t'\right) \right] . \end{aligned}$$
(54)

The identity operator \(\hat{1}_{Q_{2}}\) can be expanded in terms of the eigenprojectors \(\hat{P}_{Q_{2}}\left( i_{2};t'\right)\),

$$\begin{aligned} \hat{1}_{Q_{2}}=\sum _{i_{2}}\hat{P}_{Q_{2}}\left( i_{2};t'\right) , \end{aligned}$$
(55)

and this summation can be pulled out of the trace to yield

$$\begin{aligned} p_{Q_{1}}\left( i_{1};t'\right) =\sum _{i_{2}}\text {Tr}_{W}\left[ \left( \hat{P}_{Q_{1}}\left( i_{1};t'\right) \otimes \hat{P}_{Q_{2}}\left( i_{2};t'\right) \right) \hat{\rho }_{W}\left( t'\right) \right] . \end{aligned}$$
(56)

We now suppose that the parent system’s evolution is well-approximated by a linear CPTP map \(\mathcal {E}_{W}^{t'\leftarrow t}\). Linearity implies that

$$\begin{aligned} \hat{\rho }_{W}\left( t'\right) =\mathcal {E}_{W}^{t'\leftarrow t}\left\{ \hat{\rho }_{W}\left( t\right) \right\} =\sum _{w}p\left( w;t\right) \mathcal {E}_{W}^{t'\leftarrow t}\left\{ \hat{P}_{W}\left( w;t\right) \right\} , \end{aligned}$$
(57)

thereby allowing us to rewrite (56) as

$$\begin{aligned} p\left( w';t'\right) =\mathop {\sum }\limits _{{i_{2},w}}\text {Tr}_{W}\left[ \left( \hat{P}_{Q_{1}}\left( i_{1};t'\right) \otimes \hat{P}_{Q_{2}}\left( i_{2};t'\right) \right) \mathcal {E}_{W}^{t'\leftarrow t}\left\{ \hat{P}_{W}\left( w;t\right) \right\} \right] p\left( w;t\right) . \end{aligned}$$
(58)

This last expression can be interpreted as a Bayesian propagation formula in its familiar sense,

$$\begin{aligned} p\left( w';t'\right) =\mathop {\sum }\limits _{{i_{2},w}}p_{Q_{1},Q_{2}|W}\left( i_{1},i_{2};t'|w;t\right) p\left( w;t\right) , \end{aligned}$$
(59)

provided that we adopt Axiom 4 and make the identification

$$\begin{aligned} p_{Q_{1},Q_{2}|W}\left( i_{1},i_{2};t'|w;t\right) =\text {Tr}_{W}\left[ \left( \hat{P}_{Q_{1}}\left( i_{1};t'\right) \otimes \hat{P}_{Q_{2}}\left( i_{2};t'\right) \right) \mathcal {E}_{W}^{t'\leftarrow t}\left\{ \hat{P}_{W}\left( w;t\right) \right\} \right] . \end{aligned}$$
(60)

Our last step is not strictly necessary—we choose to interpret the trace formula in (60) as a conditional probability. In keeping with the minimalist spirit of our interpretation of quantum theory, note that we have constructed this new set of conditional probabilities out of standard ingredients without introducing any exotic elements or assumptions.

The conditional probabilities defined by (60) can be generalized to the case of a parent system \(W=Q_{1}+\cdots +Q_{n}\) consisting of n disjoint subsystems \(Q_{1},\ldots ,Q_{n}\) by replacing \(Q_{2}\rightarrow Q_{2}+\cdots +Q_{n}\) and \(\hat{1}_{Q_{2}}\rightarrow \hat{1}_{Q_{2}}\otimes \cdots \otimes \hat{1}_{Q_{n}}\). Following steps analogous to those detailed above for the bipartite case, one derives the n-subsystem joint conditional probabilities

$$\begin{aligned} p_{Q_{1},\ldots ,Q_{n}|W}\left( i_{1},{\ldots ,}i_{n};t'|w;t\right)= & {} \text {Tr}_{W}\left[ \left( \hat{P}_{Q_{1}}\left( i_{1};t'\right) \otimes \cdots \otimes \hat{P}_{Q_{n}}\left( i_{n};t'\right) \right) \right. \nonumber \\&\quad \left. \mathcal {E}_{W}^{t'\leftarrow t}\left\{ \hat{P}_{W}\left( w;t\right) \right\} \right] . \end{aligned}$$
(61)

Of course, in order for these quantities to qualify as proper conditional probabilities, they should be non-negative and sum to unity. What follows is a proof that our quantum conditional probabilities indeed have these properties.

  1. 1.

    Non-negativity: The tensor-product operator

    $$\begin{aligned} \hat{P}_{Q_{1}}\left( i_{1};t'\right) \otimes \cdots \otimes \hat{P}_{Q_{n}}\left( i_{n};t'\right) \end{aligned}$$
    (62)

    and the time-evolved projection operator \(\mathcal {E}_{W}^{t'\leftarrow t}\left\{ \hat{P}_{W}\left( w;t\right) \right\}\) are both manifestly positive semidefinite. If we call the first positive semidefinite operator \(\hat{A}\) and the second \(\hat{B}\), then \(\sqrt{\hat{A}}\) and \(\sqrt{\hat{B}}\) are also positive semidefinite and we have

    $$\begin{aligned} \text {Tr}\left[ \hat{A}\hat{B}\right]= & {} \text {Tr}\left[ \sqrt{\hat{A}}\sqrt{\hat{A}}\sqrt{\hat{B}}\sqrt{\hat{B}}\right] =\text {Tr}\left[ \sqrt{\hat{B}}\sqrt{\hat{A}}\sqrt{\hat{A}}\sqrt{\hat{B}}\right] \\= & {} \text {Tr}\left[ \left( \sqrt{\hat{A}}\sqrt{\hat{B}}\right) ^{\dagger }\left( \sqrt{\hat{A}}\sqrt{\hat{B}}\right) \right] \ge 0. \end{aligned}$$

    Therefore, our conditional probabilities are non-negative, as claimed:

    $$\begin{aligned} p_{Q_{1},\ldots ,Q_{n}|W}\left( i_{1},\ldots ,i_{n};t'|w;t\right) \ge 0. \end{aligned}$$
    (63)
  2. 2.

    Unit measure: Taking a fixed parent-system ontic state w and summing over all the final subsystem states \(i_{1},\ldots ,i_{n}\), we find

    $$\begin{aligned} \mathop {\sum }\limits _{{i_{1},\ldots ,i_{n}}}p_{Q_{1},\ldots ,Q_{n}|W}\left( i_{1},\ldots ,i_{n};t'|w;t\right)&=\mathop {\sum }\limits _{{i_{1},\ldots ,i_{n}}}\text {Tr}_{W}\left[ \left( \hat{P}_{Q_{1}}\left( i_{1};t'\right) \otimes \cdots \otimes \hat{P}_{Q_{n}}\left( i_{n};t'\right) \right) \right. \\&\quad \left. \mathcal {E}_{W}^{t'\leftarrow t}\left\{ \hat{P}_{W}\left( w;t\right) \right\} \right] \\&=\text {Tr}_{W}\left[ \left( \hat{1}_{Q_{1}}\otimes \cdots \otimes \hat{1}_{Q_{n}}\right) \mathcal {E}_{W}^{t'\leftarrow t}\left\{ \hat{P}_{W}\left( w;t\right) \right\} \right] \\&=\text {Tr}_{W}\left[ \mathcal {E}_{W}^{t'\leftarrow t}\left\{ \hat{P}_{W}\left( w;t\right) \right\} \right] \\&=1. \end{aligned}$$

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Barandes, J.A., Kagan, D. Measurement and Quantum Dynamics in the Minimal Modal Interpretation of Quantum Theory. Found Phys 50, 1189–1218 (2020). https://doi.org/10.1007/s10701-020-00374-0

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Keywords

  • Quantum theory
  • Quantum mechanics
  • Many-body quantum systems
  • Foundations of physics
  • Quantum foundations
  • Decoherence
  • Interpretations of quantum theory
  • Quantum dynamics
  • Quantum measurement problem