Generic Emergence of Objectivity of Observables in Infinite Dimensions

Quantum Darwinism posits that information becomes objective whenever multiple observers indirectly probe a quantum system by each measuring a fraction of the environment. It was recently shown that objectivity of observables emerges generically from the mathematical structure of quantum mechanics, whenever the system of interest has finite dimensions and the number of environment fragments is large [F. G. S. L. Brandão, M. Piani, and P. Horodecki, Nat. Commun. 6, 7908 (2015)]. Despite the importance of this result, it necessarily excludes many practical systems of interest that are infinite dimensional, including harmonic oscillators. Extending the study of quantum Darwinism to infinite dimensions is a nontrivial task: we tackle it here by using a modified diamond norm, suitable to quantify the distinguishability of channels in infinite dimensions. We prove two theorems that bound the emergence of objectivity, first for finite mean energy systems, and then for systems that can only be prepared in states with an exponential energy cutoff. We show that the latter class includes any bounded-energy subset of single-mode Gaussian states.

Figure 1

In quantum Darwinism, the objective classical reality (image of a cat, subsystem $A$ in the middle) emerges from an underlying quantum mechanical description (bottom layer, illustrating superposition effects) through the observation of multiple environment fragments (subsystems $B_1\dots B_N$, depicted as painting artists). Here we show that the objectivity of observables is generic even when $A$ is an infinite-dimensional quantum system, subject to suitable energy constraints.

Figure 2

The number of environment fragments, $N$ (horizontal axis, logarithmic scale), is plotted against the upper bound on $\parallel {\mathrm{\Lambda }}_j-{\mathcal{E}}_j{\parallel }_{\diamond \overline{n}}$, denoted $\mathrm{\Upsilon }$ (vertical axis, logarithmic scale), which expresses the ability to discriminate between the channel ${\mathrm{\Lambda }}_j={\mathrm{Tr}}_{\setminus B_j}\circ \mathrm{\Lambda }$ and the measure-and-prepare channel ${\mathcal{E}}_j$, as given in Theorem 1. Here we take $\overline{n}=1$ and $\delta =0.01$. We include the analytical bound in Eq. (2) (blue circles) and a numerically optimized bound (red triangles, see Supplemental Material [26]). A power-law fit of the numerical bound provides $\parallel {\mathrm{\Lambda }}_j-{\mathcal{E}}_j{\parallel }_{\diamond \overline{n}}\le (1/\delta )\beta N^{-(1/\alpha )}$ with $\beta \simeq 6.94$, $\alpha \simeq 15.18$. All plotted quantities are dimensionless.

Figure 3

The number of environment fragments, $N$ (horizontal axis, logarithmic scale), is plotted against the upper bound on $\parallel {\mathrm{\Lambda }}_j-{\mathcal{E}}_j{\parallel }_{\diamond \omega ,\mathrm{\Omega }}$, denoted $\mathrm{\Upsilon }$ (vertical axis, logarithmic scale), given in Theorem 2. We take $\delta =0.01$ as in Fig. 2. The parameters $\omega$, $\mathrm{\Omega }$ are optimized for each $N$ to provide the best possible bound attainable through our methods, assuming a set of Gaussian resource states ${\mathcal{G}}_{\overline{n}}$ with $\overline{n}=1$ (these would be the input states that can be used to discriminate between channels). Blue circles indicate the rhs of Eq. (6), while red triangles refer to a numerically optimized bound (see Supplemental Material [26]). A power-law fit of the latter yields $\parallel {\mathrm{\Lambda }}_j-{\mathcal{E}}_j{\parallel }_{\diamond \omega ,\mathrm{\Omega }}\le (1/\delta )\beta N^{-(1/\alpha )}$ with $\beta \simeq 5839$, $\alpha \simeq 3.20$. All plotted quantities are dimensionless.