Theory-independent monitoring of the decoherence of a superconducting qubit with generalized contextuality

Abstract

Characterizing the nonclassicality of quantum systems under minimal assumptions is an important challenge for quantum foundations and technology. Here we introduce a theory-independent method of process tomography and perform it on a superconducting qubit. We demonstrate its decoherence without assuming quantum theory or trusting the devices by modelling the system as a general probabilistic theory. We show that the superconducting system is initially well-described as a quantum bit, but that its realized state space contracts over time, which in quantum terminology indicates its loss of coherence. The system is initially nonclassical in the sense of generalized contextuality: it does not admit of a hidden-variable model where statistically indistinguishable preparations are represented by identical hidden-variable distributions. In finite time, the system becomes noncontextual and hence loses its nonclassicality. Moreover, we demonstrate in a theory-independent way that the system undergoes non-Markovian evolution at late times. Our results extend theory-independent tomography to time-evolving systems, and show how important dynamical physical phenomena can be experimentally monitored without assuming quantum theory.

Figures (10)

• Figure 1: Diagram of the prepare-transform-measure setup of the experiment. We regard the actual state preparation and the subsequent time evolution as a single preparation procedure, which is simply a convenient convention (analogous to the choice of Schrödinger versus Heisenberg picture). As a result, the prepared states will depend on the waiting time $\tau$, but the measurements will not.
• Figure 2: Frequency data collected in the experiment. Experimentally measured frequency of occurrence of the outcome $a=0$ for the $m\times n$ table of preparations and measurements, for $\tau=0$.
• Figure 3: Relations between the state spaces of the qubit, stabilizer qubit lilystone_contextuality_2019 and classical four level system. (a) the state space of the stabilizer qubit (in green) is embedded in the state space of the qubit, given by the grey ball; (b) the state space of the stabilizer qubit (in green) is embeddable within a tetrahedron (in grey), the state space of a classical four level system; (c) the embedding of the stabilizer qubit into the tetrahedron pictured here is such that any valid effect on the stabilizer qubit is also a valid effect on the tetrahedron. This requirement prevents, for example, the embeddability of the qubit into the tetrahedron (we know that the qubit is nonembeddable because it is contextual). Here the orange planes represent the effect corresponding to the $+1$ outcome of the Pauli $Z$ measurement: the upper plane intersects all states giving probability $1$ for the outcome $+1$, the central plane those giving probability $\frac{1}{2}$ and the lower plane those giving probability 0.
• Figure 4: Errors of the optimal GPT fits for different ranks. (a) Test and train errors for the optimal GPT fits for ranks $k \in \{2,...,9\}$. Inset: zoomed in for ranks $k\geq4$. The training error is evaluated on 10 frequency tables $\{F^\alpha\}_{\alpha =1}^{10}$, whilst the test error is evaluated on the 90 pairs of frequency tables $(F^\alpha,F^\beta)$ with $\alpha \neq \beta$. The error bars for the training error and test error are given by the statistical uncertainty over the 10 tables and 90 pairs respectively. (b) For each pair $(F^\alpha, F^\beta)$ with $\alpha,\beta \in \{1,...,10\}$, $\alpha \neq \beta$ with $F^\alpha$ serving as the training data and $F^\beta$ as the testing data we plot $\chi^2_k(F^\beta, D_k^\alpha) - \chi^2_{k-1}(F^\beta, D_{k-1}^\alpha)$ which is the change in the test error between the best fit rank $k$ and best fit rank $k-1$ models. For $k \leq 4$ it is strictly negative, showing that the test error decreases, while for $k >4$ it is strictly positive, showing that the test error increases.
• Figure 5: Reconstructed state and effect spaces for zero time delay. Plots showing the realized (blue) and consistent (green) spaces for the rank $k=4$ GPT for $\tau=0$. (a) The normalized state spaces are three-dimensional and similar to Bloch balls; (b) effect space projection onto dimensions 1,2,3; (c) effect space projection onto dimensions 0,1,2 (or similarly for 0,2,3 and 0,1,3) containing the zero and unit effects $0$ and $\mathbf{1}$.

Theorems & Definitions (2)

• Lemma 1
• proof