Recovering full coherence in a qubit by measuring half of its environment

When a quantum system interacts with its environment it may incur in decoherence. Quantum erasure makes it possible to restore coherence in a system by gaining information about its environment, but measuring the whole of it may be prohibitive: Realistically, one might be forced to address only an accessible subspace and neglect the rest. In such a case, under what conditions will quantum erasure still be effective? In this work we compute analytically the largest recoverable coherence of a random qubit plus environment state and we show that it approaches 100% with overwhelmingly high probability as long as the dimension of the accessible subspace of the environment is larger than $\sqrt{D}$, where $D$ is the dimension of the whole environment. Additionally, we find a sharp transition between a linear behavior and a power-law behavior as soon as the dimension of the inaccessible environment exceeds the dimension of the accessible one. Our results imply that the typical states of a qubit plus environment system admit a measurement spanning only about $\sqrt{D}$ degrees of freedom, any outcome of which projects the qubit on a maximally coherent state. This suggests, for instance, that in the dynamics of open quantum systems, if the interactions are known, it would in principle be possible to gain sufficient information and restore coherence in a qubit by dealing with a fraction of the physical resources.

Figure 1

We imagine a qubit $\mathcal{Q}$ within an ensemble of $n$ environment qubits, where $a$ of them are accessible. The rest $k=n-a$ are inaccessible. We find that if $a\ge k$ (i.e., if we can access at least half of them) there exists an optimal measurement on the accessible qubits whose outcomes project $\mathcal{Q}$ onto states with coherence $\langle \mathcal{C}\rangle \sim 1-\frac{1}{4}{2}^{k-a}$, which approaches 1 exponentially fast as $a\to n$.

Figure 2

The $A\times A$ cross term $X$ in the random matrix $\rho$ of Eqs. (5) and (7) is proportional to the product between two independent random matrices ${\mu }_1$ and ${\mu }_2$ that make up $\mu$.

Figure 3

Average coherence $\langle \mathcal{C}\rangle$ for $A=100$ (solid orange line), together with the high-$K$ and low-$K$ approximations (dashed lines). Up to $K=A$, the behavior of $\langle \mathcal{C}\rangle$ is purely linear $1-\frac{K}{4A}$ (see the inset). For $K>A$ the behavior changes dramatically and is asymptotic to $O(1/\sqrt{K})$. The shaded indicates the linear region.

Figure 4

Average recoverable coherence $\langle \mathcal{C}\rangle$ of a qubit as a function of the number $a$ of qubits of the environment that we can access, where the total $n$ is displayed on top. In blue we show the 50th, 90th, and 99th percentiles around the mean (red line) and all plots are from 0 to 1. We see that $\langle \mathcal{C}\rangle$ transitions from a value close to 0 to a value close to 1 as we gain access to more than half of the environment qubits. Notice that as the total number of environment qubits grows, the mean becomes a better representative of the whole ensemble.

Figure 5

The recoverable coherence $\langle \mathcal{C}\rangle$ of a qubit $\mathcal{Q}$ for an environment of $n=200$ qubits displays a very sharp increase from 0 to 1 as soon as one can control more than 100 of them, meaning that there exists an environment observable with elements $\{{\pi }_j^{\mathcal{A}}\otimes {\mathbb{1}}^{\mathcal{K}}\}$, each of which projects $\mathcal{Q}$ onto maximally coherent states.