Device-Independent Certification of Genuinely Entangled Subspaces

Self-testing is a procedure for characterizing quantum resources with the minimal level of trust. Up to now it has been used as a device-independent certification tool for particular quantum measurements, channels, and pure entangled states. In this work we introduce the concept of self-testing more general entanglement structures. More precisely, we present the first self-tests of an entangled subspace—the five-qubit code and the toric code. We show that all quantum states maximally violating a suitably chosen Bell inequality must belong to the corresponding code subspace, which remarkably includes also mixed states.

Figure 1

A model of self testing. A classical verifier $V$ aims to certify a particular quantum property of the resource shared by noncommunicating provers $P_i$. The verifier sends different classical inputs $x_i$ to the provers and they respond with classical outputs $a_i$. If nonlocal, correlations between the outputs allow the verifier to make nontrivial conclusions about that quantum property.

Figure 2

The toric code. Each edge of the lattice represents a qubit. Stabilizing operators can be divided in two groups: those associated with each lattice vertex $v$ with $X$ acting on every qubit associated with an edge attached to the given vertex, or associated with each plaquette $p$ of the lattice with $Z$ acting on each qubit represented by an edge surrounding the plaquette.

Figure 3

Numerical estimation of the minimal subspace extractability for the target five-qubit Shor’s code subspace as a function of the relative observed violation $(\beta -{\beta }_c)/({\beta }_q-{\beta }_c)$ of the corresponding Bell inequality $I_5$.