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.

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. 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. 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

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