Certifying asymmetry in the configuration of three qubits

TL;DR

The work addresses certifying asymmetry in a configuration of three qubit Bloch vectors within a dimension-bounded prepare-and-measure setting by constructing a linear witness I6 from biased I3 blocks. It derives the mirror-symmetric bound Qmirror via QCQP/SDP techniques and defines the witness gap Δ = Qmax − Qmirror as a quantitative measure of asymmetry, with Qmax being the quantum maximum for the target configuration. The authors show how to self-test a triple of pure qubits and demonstrate that any violation of the mirror-symmetric bound certifies asymmetry, including numerical results identifying the most asymmetric configurations and an experimental demonstration on IBM’s public quantum processor. This framework provides a semi-device-independent method to certify geometric properties of quantum state configurations and suggests avenues for extending the approach to larger sets and higher dimensions.

Abstract

Symmetry restrictions limit the types of tasks that can be achieved with a given set of quantum states. Therefore, any breaking of these symmetries could potentially be exploited as a resource for quantum communication. Here we demonstrate this operationally by certifying asymmetry in the configuration of the Bloch vectors of a set of three unknown qubit states within the dimensionally bounded prepare-and-measure scenario. To do this, we construct a linear witness from three simpler witnesses as building blocks, each featuring, along with two binary measurement settings, three preparations; two of them are associated with the certification task, while the third one serves as an auxiliary. The final witness is chosen to self-test some target configuration. We numerically derive a bound $Q_{\text{mirror}}$ for any mirror-symmetric configuration, thereby certifying asymmetry if this bound is exceeded (e.g. experimentally) for the unknown qubit configuration. We also consider the gap $(Q_{\text{max}}-Q_{\text{mirror}})$ between the analytically derived overall quantum maximum $Q_{\text{max}}$ and the mirror-symmetric bound, and use it as a quantifier of asymmetry in the target configuration. Numerical optimization shows that the most asymmetric configuration then forms a right scalene triangle on the unit Bloch sphere. Finally, we implement our protocol on a public quantum processor, where a clear violation of the mirror-symmetric bound certifies asymmetry in the configuration of our experimental triple of qubit states.

Figures (4)

• Figure 1: Prepare-and-measure scenario used in self-testing a pair of qubit states $\rho_1$ and $\rho_2$ along with an auxiliary state $\rho_3$ using two dichotomic measurements. Alice's preparation device randomly generates a state $\rho_x$, $x=1,2,3$, which is sent to Bob. Bob performs a measurement labeled by $y=1,2$ on the received state.
• Figure 2: Maximum quantum value $I_3^Q(\omega)$ and classical bit value $I_3^C(\omega)$ of the witness as a function of the bias parameter $\omega$. The two curves coincide at $\omega=0$ and $\omega=1$, while for intermediate values the quantum maximum exceeds the classical bit value.
• Figure 3: Schematic of the prepare-and-measure scenario illustrating the self-test and certification of the asymmetry of a triple of qubit states. Alice's preparation device randomly generates a qubit state $\rho_x$, chosen from six states labeled by $x=1,\ldots,6$. She then sends the state to Bob, who performs a measurement labeled by $y$ on the received qubit. The inputs $x = 4,\ldots,6$ correspond to auxiliary qubits required for our scenario to self-test the three qubit states labeled by $x = 1,\dots,3$.
• Figure 4: Illustration of the two asymmetric configurations aimed to be implemented on IBMQ with a success in asymmetry certification. For each case, the corresponding optimal symmetric configuration saturating the bound $Q_{\text{mirror}}$ is also depicted. In the plots, the starred Bloch vectors represent the target asymmetric configuration, while the unstarred ones represent the optimal symmetric configuration. Panel (a) corresponds to the starred angles given in Eq. (\ref{['angles_case1']}). Here the optimal symmetric Bloch vectors lie in the same $X$--$Z$ plane. Panel (b) corresponds to the starred angles given in Eq. (\ref{['angles_case2']}). Here, the Bloch vectors of the optimal symmetric configuration span the full three-dimensional space, where $\vec{n}_1$ and $\vec{n}_2$ are mirror images of each other with respect to the $Y$--$Z$ plane, while $\vec{n}_3$ is along the $Z$-axis.

Theorems & Definitions (3)

• Lemma 1
• Lemma 2
• Lemma 3