On the Dynamics of Local Hidden-Variable Models

TL;DR

The paper investigates whether the time evolution of locally correlated quantum states can be captured by dynamical hidden variables in a local hidden-variable framework. It formalizes LHV models for sets of states and defines dynamical LHV models via a state-independent velocity field on hidden variables, distinguishing between noninteracting and interacting dynamics. The authors show that nice dynamical LHV models exist for noninteracting unitary evolution, but provide a concrete two-qubit Heisenberg counterexample where no state-independent velocity field can reproduce the full dynamics, and they prove a general no-go theorem: for sufficiently many particles, smooth, deterministic, microscopic LHV dynamics under the full unitary group cannot exist. This reveals a new form of dynamical nonlocality tied to interactions and has implications for classical simulation of quantum dynamics and foundational aspects of quantum mechanics.

Abstract

Bell nonlocality is an intriguing property of quantum mechanics with far reaching consequences for information processing, philosophy and our fundamental understanding of nature. However, nonlocality is a statement about static correlations only. It does not take into account dynamics, i.e. time evolution of those correlations. Consider a dynamic situation where the correlations remain local for all times. Then at each moment in time there exists a local hidden-variable (LHV) model reproducing the momentary correlations. Can the time evolution of the correlations then be captured by evolving the hidden variables? In this light, we define dynamical LHV models and motivate and discuss potential additional physical and mathematical assumptions. Based on a simple counter example we conjecture that such LHV dynamics does not always exist. This is further substantiated by a rigorous no-go theorem. Our results suggest a new type of nonlocality that can be deduced from the observed time evolution of measurement statistics and which generically occurs in interacting quantum systems.

Figures (4)

• Figure 1: LHV dynamics. The grey disc represents all quantum states and the orange subset represents the local states. Suppose we have some local quantum states which remain local during time evolution. Then, the quantum measurement statistics at each moment in time, that is, the probabilities $\mathbf{P}(a|x, \rho(t))$ to observe outcome $a$ upon measuring $x$, can be reproduced by LHV models. Can the time-evolution of the measurement statistics be captured by evolving the hidden variables?
• Figure 2: (a) Assuming a local hidden-variable model as the fundamental microscopic description for local states that remain local under time evolution, the physics of the hidden variables cannot depend on the quantum state. The possible single-particle hidden variables $\lambda_j\in\Lambda_1$ and the way they determine the measurement outcome $a_j$ under a local measurement $x_j$ are independent of the quantum state $\rho$ and the number of particles $N$. (b) A given time-independent Hamiltonian $H$ should correspond to a time- and state-independent velocity field $V$ on the hidden-variable level. A particular instance of the hidden variables $\lambda$ evolves according to this velocity field. For every quantum state $\rho$ this leads to time evolution of the hidden-variable distribution $p_\rho$ according to the continuity equation.
• Figure 3: The space of quantum states. In this sketch, unitary evolution is represented by angular motion (indicated by the example Hamiltonians $H_1,H_2,H_3$). Hence, the set of all states corresponds to a disc (grey), with the white dot at the center representing the maximally mixed state. All pure states lie on the boundary of this disc. The separable states (blue) are convex combinations of the pure product states. The local states (bright orange) are a convex superset of the separable states. The largest disc contained in the separable states contains all the states that remain separable under arbitrary unitary time evolution (dark blue). Likewise, the local states contain a maximal disc of states that remain local under arbitrary unitary time evolution (orange). These are the states that produce local correlations which remain local under arbitrary unitary evolution.
• Figure 4: (a) The full unitary group $\text{U}(\mathcal{H})$ is large. For a faithful action on the hidden-variable space $\Lambda$, sufficiently many dimensions of that space are required. In this pictorial example we imagine a four-dimensional group acting on a two-dimensional space. Locally there are only two shifts (represented by $T_{U_1}$ and $T_{U_2}$) and one rotation (represented by $T_{U_3}$), that is, there are three independent available transformations (we only consider isometries, i.e., no ' stretching' or ' shearing', see the proof of the dimensionality constraint). Hence, the four-dimensional group cannot act faithfully, ' there is no fourth independent transformation which $U_4$ could map to'. (b) Hidden-variable dynamics in the sense of an action of the unitary group on the hidden-variable space requires $B_{LHV}\ge B_{QM}$, where $B_{QM} = D^{2N}-1$ for the unitary group of $D$-dimensional qudits (blue) and $B_{LHV} = Nd(Nd+1)/2$ for a $d$-dimensional single-particle hidden-variable space $\Lambda_1$ (red). We observe that the Bell-LHV base model can only accommodate dynamics for a single qubit. A much more expressive LHV with $d=20$ could allow for dynamics for up to $6$ qubits or $3$ qutrits, although there is no guarantee that such dynamics actually exists even in these cases.

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Theorem 1: Nice dynamical LHV models for noninteracting unitaries
• proof
• Conjecture 1
• Theorem 2: Dimensionality constraint
• proof