Partial orbits of quantum gates and full three-particle entanglement

Abstract

We introduce the notion of partial orbit for a given set of elementary quantum gates, and we study the action of different sets of gates on an initially three-particle disentangled state. We analyze the entanglement of the resulting quantum states by appealing to the violation of Svetlichny inequality. We find that the violation values are concentrated on specific regions. This constitutes evidence for the existence of a structure, which is quite different from that of a randomly generated (using the Haar measure) set of gates. In turn, this fact highlights the relevance of studying the physical properties of the partial orbits associated with different sets of elementary gates.

Appendices

Appendix

In order to study the violation of the Svetlichny inequality, we consider three particles $a, \, b, \, c$ and two possible measurements for each particle. For particle a, we consider the measurements $A = \vec {\alpha }\cdot \vec {\sigma }$ and $A' = \vec {\alpha '}\cdot \vec {\sigma }$; for particle b, the measurements $B = \vec {\beta }\cdot \vec {\sigma }$ and $B'= \vec {\beta '}\cdot \vec {\sigma }$; and for particle c, the measurements $C = \vec {\gamma }\cdot \vec {\sigma }$ and $C' = \vec {\gamma '}\cdot \vec {\sigma }$. The Svetlichny inequality of a state $\rho$ can be expressed in terms of the Svetlichny function ${\mathcal {S}}(\rho ,\vec {\alpha },\vec {\alpha }',\vec {\beta },\vec {\beta }',\vec {\gamma },\vec {\gamma }')$ as follows

$$\begin{aligned} {\mathcal {S}}(\rho ,\vec {\alpha },\vec {\alpha }',\vec {\beta },\vec {\beta }',\vec {\gamma },\vec {\gamma }') = |\text{ tr }(\rho S)| \le 4, \end{aligned}$$

(6)

where S is the Svetlichny operator defined as follows:

$$\begin{aligned} S= & {} A\otimes B\otimes C + A\otimes B'\otimes C + A\otimes B \otimes C' - A\otimes B'\otimes C' + A'\otimes B \otimes C \nonumber \\&-A'\otimes B'\otimes C - A'\otimes B\otimes C' - A'\otimes B'\otimes C'. \end{aligned}$$

(7)

By parameterizing the unit norm vectors $\vec {\alpha }, \, \vec {\alpha '}, \, \vec {\beta }, \,\vec {\beta '}, \, \vec {\gamma }, \, \vec {\gamma '}$ with spherical coordinates, the dependence of the local observables A, B, and C with respect to the angles is given by:

$$\begin{aligned} A(\theta _a,\phi _a)&=\cos \theta _a \sin \phi _a \, \sigma _{x} + \sin \theta _a \sin \phi _a \, \sigma _{y} + \cos \phi _a \, \sigma _{z}, \end{aligned}$$

(8)

$$\begin{aligned} B(\theta _b,\phi _b)&= \cos \theta _b \sin \phi _b \, \sigma _{x} + \sin \theta _b \sin \phi _b \, \sigma _{y} + \cos \phi _b \, \sigma _{z}, \end{aligned}$$

(9)

$$\begin{aligned} C(\theta _c,\phi _c)&= \cos \theta _c \sin \phi _c \, \sigma _{x} + \sin \theta _c \sin \phi _c \, \sigma _{y} + \cos \phi _c \, \sigma _{z}, \end{aligned}$$

(10)

with similar equations for the primed coordinates. For each state $\rho$, obtained by applying the elementary gates to the initial sate $\rho _{0}$, we find the combination of angles that maximizes the Svetlichny function ${\mathcal {S}}(\rho ,\vec {\alpha },\vec {\alpha }',\vec {\beta },\vec {\beta }',\vec {\gamma },\vec {\gamma }')$, i.e., the combination that maximizes the violation of the inequality (6). The maximum value obtained is called Svetlichny maximum value. In this way, we quantify how much entanglement is obtained after the application of a sequence of elementary gates. The Svetlichny maximum values of the states of the partial orbits are used to build the histograms displayed in Section 4.

Appendix

In Fig. 4, we show similar calculations to those of Sect. 4, but starting with a three-qubit GHZ state, i.e., an initially entangled state.

Fig. 4

Histograms in logarithmic scale of the maximum values of the Svetlichny function for partial orbits of different degrees, associated with an initial GHZ state. a Clifford set, degree 7; b Clifford set, degree 6 with noise parameter $=0.2$; c Clifford set, degree 6 with noise parameter $=0.05$; d Cnot + Toff set, degree 7; e Cnot + Toff set, degree 6 with noise parameter $=0.2$; f Cnot + Toff set, degree 6 with noise parameter $=0.05$