Wigner Function Negativity and Contextuality in Quantum Computation on Rebits

We describe a universal scheme of quantum computation by state injection on rebits (states with real density matrices). For this scheme, we establish contextuality and Wigner function negativity as computational resources, extending results of M. Howard et al. [Nature (London) 510, 351 (2014)] to two-level systems. For this purpose, we define a Wigner function suited to systems of $n$ rebits and prove a corresponding discrete Hudson’s theorem. We introduce contextuality witnesses for rebit states and discuss the compatibility of our result with state-independent contextuality.

Popular Summary

Contextuality is a fundamental property of quantum mechanics that distinguishes quantum theories from classical theories. Specifically, it demonstrates that a quantum measurement of an observable cannot be viewed as merely revealing a preexisting value. Contextuality is also a resource for quantum computation; universal quantum computation could not proceed without it. This fact has recently been established for the model of quantum computation with magic states (i.e., states that cannot be created by the measurements and unitary gates of the scheme), for essentially everything except two-level systems. We address the open case of two-level systems, which is the most relevant from an algorithmic perspective. We show that contextuality remains a necessary resource for quantum states with real density matrices.

The two-level scenario differs from its higher-dimensional counterparts in one essential way: It involves the phenomenon of state-independent contextuality, which poses an obstacle to viewing—a close quantum analog to probability distributions over phase space. Negative values of this function are an indicator of quantumness.

Contextuality has also been established for the scheme of quantum computation by measurement but, in this case, only for qubits. Our work opens up the possibility of comparing two computational schemes to determine how they make use of contextuality.

Figure 1

Wigner function of the three-rebit graph state $\mid K_3⟩$ corresponding to the complete graph $K_3$. The Wigner function takes negative values, and furthermore this negativity is strong enough to witness contextuality of $\mid K_3⟩$ with respect to CSS-ness preserving Pauli measurements.

Figure 2

Phase diagram for the families of states $\tilde{\rho }(x,y)$ of Eq. (38), with $x,y\in \mathbb{R}$. The medium-shaded area shows the physical states; the dark-shaded area shows the states classified as noncontextual by Theorem 3. The states classified as contextual by Theorem 4 lie outside the square of $|x|,|z|\le 1$ and are thus not physical.

Figure 3

Phase diagram for the states $\rho (a,b)$ of Eq. (39), with $a,b\in \mathbb{R}$. The medium-shaded area shows the physical states; the dark-shaded are shows the states classified as noncontextual by Theorem 3.