Blurred quantum Darwinism across quantum reference frames

Quantum Darwinism describes objectivity of quantum systems via their correlations with their environment—information that hypothetical observers can recover by measuring the environments. However, observations are done with respect to a frame of reference. Here we take the formalism of [Giacomini et al., Nat. Commun. 10, 494 (2019)] and consider the repercussions on objectivity when changing quantum reference frames. We find that objectivity depends on nondegenerative relative separations, conditional state localization, and environment macrofractions. There is different objective information in different reference frames due to the interchangeability of entanglement and coherence, and of statistical mixing and classical correlations. As such, objectivity is subjective across quantum reference frames.

Figure 1

Top: if the peaks for the states are separated much farther than their standard deviations (here $\mathrm{\Delta }\mu =10\sigma$), there is very little overlap and these states are distinguishable. Bottom: Alternatively, if the central peaks are the same or very close, varying greatly standard deviations (here five times or more) allows for good distinguishability, because measurement at further locations will, with high probability, correspond to the wider distributions.

Figure 2

In the frame of $\left(E_1\right)$, the original peaks of environment $E_j$, at ${\mu }_{E_j|i}$ are shifted by $q_C$, which ranges over the entire space, but with a Gaussian envelope centered at $-{\mu }_{E_1|i}$. Every curve corresponds to a different environment state conditioned on a different $q_C$ (for a fixed $i$). Different $q_C$ give curves that overlap significantly and hence are not distinguishable.

Figure 3

Plot of the conditional state fidelity ${‖\sqrt{{\rho }_{F|0}^{\left(E_1\right)}}\sqrt{{\rho }_{F|1}^{\left(E_1\right)}}‖}_1$ versus the amount of localization ($\sigma$ the same for all Gaussian states) and macrofraction size $\left|F\right|$, for the case of the state in Eq. (25). Here the peak positions ${\left\{{\mu }_{E_i|0},{\mu }_{E_i|1}\right\}}_i$ are picked randomly from the interval $\left[-1,1\right]$, and the graph is averaged over 400 collections of random samples. Sharp localization $\sigma \to 0$, and large macrofractions $\left|F\right|$ lead to low conditional state fidelity and hence greater distinguishability.

Figure 4

Top: Curves representing the system conditional states in the original laboratory frame: they are separated and hence distinguishable. Bottom: Both curves are shifted by a different amount as we move to an environment frame, yet the curves overlap and are no longer distinguishable.

Figure 5

Example of an objective SBS state described by Proposition t15. Top: Depiction of the state in frame $C$. The original objective information is $\{p_0,p_1\}$. The environment $E_1$ has mixed, incoherent states conditioned on $i=0$ ($p_0$, top left), and $i=1$ ($p_1$, top right). Bottom: When moving into the quantum reference frame of $E_1$, the system states have been shifted such that they still remain distinguishable. The resulting objective information is $\{p_0{t}_0,p_0{t}_1,p_1{t}_2,p_1{t}_3\}$.

Figure 6

Selected probabilities in the reference frame $E_1$ for Cases 1.1 and 1.2. Note that for Cases 1.3 and 1.4 the probability $p_0^{\left(E_1\right)}=\frac{1}{D}\approx 0.083$ and is constant in time. For Case 1.5 $p_0^{\left(E_1\right)}$ is identical as in Case 1.1. (Cases are given in Table 2.)

Figure 7

Probabilities $p_0^{\left(E_1\right)}$ for cases from the groups 2, 3, and 4 in the reference frame of $E_1$, which have various interactions in addition to central measurement Hamiltonian. Inset: Probabilities during time [0,1] where the measurement-like Hamiltonian dominates. (Cases given are in Table 2.)

Figure 8

Plot of $I_{\text{mean}}^{\left(E_1\right)}$ (the mean mutual information seen from the reference frame of the first environment, $E_1$) for Cases 1.1, 1.2, 1.3, and 1.4. In Case 1.5 the function is constant and equal to 0. This shows the cases when $I_{\text{mean}}^{\left(E_1\right)}(t=1)=0$. The plot for Case 1.1 is identical for Cases 2.1 and 3.1. (Cases are given in Table 2.) When the mutual information is low, the environments have strong independence.

Figure 9

Plot of mean mutual information in the reference frame of $E_1, I_{\text{mean}}^{\left(E_1\right)}$ for Cases 2.2, 3.2, and 4; cf. Table 3. Inset: mean mutual information during times $t=[0,1]$ where the measurement-like Hamiltonian dominates. (Cases given are in Table 2.) As the mutual information is large as time increases, the environments do not have strong independence.

Figure 10

The quantum mutual information between the central system $S$ and subsystem $E_1$ ($C$) in the reference frame of $C$ ($E_1$). A high quantum mutual information often suggests objectivity. (Cases are given in Table 2.)

Figure 11

The Holevo information between central system $S$ and subsystem $E_1$ ($C$) in the reference frame of $C$ ($E_1$). Compared with Fig. 10, the Holevo information is very similar to the quantum mutual information, aside from Case 3.2 in frame $E_1$.

Figure 12

The values of mean mutual information between environments $I_{\text{mean}}^{\left(C\right)}$ as a function of time for cases from groups 3 and 4 (cf. Table 4).

Figure 13

Probabilities ${\left\{p_i^{\left(E_1\right)}\right\}}_{i=0,1,2,10,11}$ for Case 3.1. $p_0^{\left(E_1\right)}$ is dominating at the time $t=1$, afterwards the probabilities neighboring at the 12-dimensional ring, i.e., $p_1^{\left(E_1\right)}$ and $p_{11}^{\left(E_1\right)}$, start to be non-negligible. After some time more distant probabilities, $p_2^{\left(E_1\right)}$ and $p_{10}^{\left(E_1\right)}$, also start increasing.