Causal Asymmetry in a Quantum World

Causal asymmetry is one of the great surprises in predictive modeling: The memory required to predict the future differs from the memory required to retrodict the past. There is a privileged temporal direction for modeling a stochastic process where memory costs are minimal. Models operating in the other direction incur an unavoidable memory overhead. Here, we show that this overhead can vanish when quantum models are allowed. Quantum models forced to run in the less-natural temporal direction not only surpass their optimal classical counterparts but also any classical model running in reverse time. This holds even when the memory overhead is unbounded, resulting in quantum models with unbounded memory advantage.

Popular Summary

The fundamental laws of physics work in the same way whether time moves forward or backward. Yet, while a glass can fall and scatter shards across the floor, glass shards never gather together and leap back onto the counter to form a complete glass. The source of this temporal asymmetry is one of the deepest mysteries in physics. We tackle this problem by combining two different disciplines, computational and quantum mechanics. Our results illustrate that the asymmetry could emerge from forcing classical causal explanations on observations in a fundamentally quantum world.

Computational mechanics asks the following question: Given a sequence of observations, how many past causes must we postulate to explain future behavior? This quantity is asymmetric when time is reversed. There is an unavoidable memory overhead cost for modeling a process in the “less-natural” temporal direction—one must pay a price to enforce explanations adhering to a less-favored order of events.

We show that quantum models always mitigate this overhead. Not only can we construct quantum models that need less past information than optimal classical counterparts, these models can always be reprogrammed to model the time-reversed process without additional memory cost. This remains true even for observational data where this classical overhead diverges, such that all classical models for the less-natural temporal direction require unbounded memory.

We illustrate scenarios where classical favoritism for particular causal orders vanishes when quantum models are permitted, thus highlighting a new mechanism for the origin of time’s arrow.

Figure 1

A stochastic process can be modeled in either temporal order. (a) A causal model takes information available in the past $\overleftarrow{x}$ and uses it to make statistically accurate predictions about the process’s conditional future behavior $P(\overrightarrow{X}|\overleftarrow{X}=\overleftarrow{x})$. (b) A retrocausal model replicates the system’s behavior, as seen by an observer who scans the outputs from right to left, encountering $X_{t+1}$ before $X_t$. Thus, it stores relevant future information $\overrightarrow{x}$, in order to generate a statistically accurate retrodiction of the past $P(\overleftarrow{X}|\overrightarrow{X}=\overrightarrow{x})$. Causal asymmetry implies a nonzero gap between the minimum memory required by any causal model $C^{+}$ and its retrocausal counterpart $C^{-}$.

Figure 2

(a) The $ϵ$-machine for the process $P_h^{+}(\overleftarrow{X},\overrightarrow{X})$, created by flipping a biased coin and emitting outcome 2 when $H\to T$, 0 when $T\to T$, and 1 when $T/H\to H$. This process has two causal states $s_1^{+}$ and $s_0^{+}$, where the latter includes all pasts ending in either 0 or 2. (b) The time-reversed process $P^{-}(\overleftarrow{Y},\overrightarrow{Y})$. Here, pasts ending in 0, 1, and 2 now all lead to qualitatively different future behavior and must be stored in distinct causal states $s_0^{-}$, $s_1^{-}$, and $s_2^{-}$, respectively, which occur with respective probabilities ${\pi }_0^{-}=(q-pq)/(p+q)$, ${\pi }_1^{-}={\pi }_1^{+}=p/(p+q)$, and ${\pi }_2^{-}=pq/(p+q)$.

Figure 3

Quantum circuits for generating (a) $P_h^{+}(\overleftarrow{X},\overrightarrow{X})$ and (b) $P_h^{-}(\overleftarrow{Y},\overrightarrow{Y})$. Here, $C_U$ (black circle and line) is the standard control gate $C_U:|w⟩|\psi ⟩\to |w⟩U^{(w\text{mod}2)}|\psi ⟩$. Meanwhile, ${\overline{C}}_U$ (white circle, black line) is defined as ${\overline{C}}_U|w⟩|\psi ⟩=|0⟩U^{(w+1\text{mod}2)}|\psi ⟩$. (a) To simulate $P_h^{+}(\overleftarrow{X},\overrightarrow{X})$, we initialize a qubit in state $|s_i^{+}⟩$ and an ancilla in state $|0⟩$. Executing the local unitary $V_p|0⟩\to |s_0^{+}⟩$, followed by the two-qubit gate $C_{V_q}$, where $V_q{V}_p|0⟩=|s_1^{+}⟩$, creates a suitable entangled state—such that a computation basis measurement of the top qubit yields $x_t$ and simultaneously collapses the bottom qubit into the causal state for the next time step. (b) To simulate $P_h^{-}(\overleftarrow{Y},\overrightarrow{Y})$, we prepare state $|s_i^{-}⟩|0⟩|0⟩$ as input. Execution of ${\overline{C}}_{U_p}$, where $U_p|0⟩=\sqrt{1-p}|0⟩+\sqrt{p}|1⟩$, followed by $C_{U_q}$, where $U_q$, satisfies $U_q|0⟩=\sqrt{q}|0⟩+\sqrt{1-q}|1⟩$, and finally, $C_X$, where $X$ is the Pauli $X$ operator, generates a suitable entangled state—such that measuring the first two qubits yields $y_t$ (provided we identify measurement outcome $00\to y_t=0$, $10\to y_t=1$, and $01\to y_t=2$) and collapses the remaining qubit into the quantum causal state for the next time step. In either circuit, retaining only the state of $\mathrm{\Xi }$ (green circle) at each time step is sufficient for generating statistically correct predictions or retrodictions.

Figure 4

Complexity of the heralding coin plotted against $p$ and $q$. The figure illustrates $E\le C_q^{+}=C_q^{-}\le C_{\mu }^{+}\le C_{\mu }^{-}$ across all values of the parameter space ($0\le p$, $q\le 1$). Panel (d) depicts the classical causal asymmetry $\mathrm{\Delta }{C}_{\mu }$, and panel (f) effectively demonstrates $C_q^{+}=C_q^{-}$ and thus $\mathrm{\Delta }{C}_q=0$.

Figure 5

The $n\text{−}m$ flower process, illustrated for the case $m=2$, and $n$ even. Physically, this process can be generated by a set $\{d_1,\dots ,d_n\}$ of $m$-sided dice, where each die $d_i$ is biased so that it lands on side $j\in \{1,\dots ,m\}$ with probability $p_j^i$ (and, in general, the bias on each die is different, such that $p_j^i\ne p_j^k$ for $i\ne k$). We randomly select a die $d_i$, recording the choice $x_t=i$. Afterwards, we role the die, transcribing the outcome $j$ as $x_{t+1}=j+n$.