Measuring the size of a quantum superposition of many-body states

We propose a measure for the “size” of a quantum superposition of two many-body states with (supposedly) macroscopically distinct properties by counting how many single-particle operations are needed to map one state onto the other. This definition gives sensible results for simple, analytically tractable cases and is consistent with a previous definition restricted to Greenberger-Horne-Zeilinger-like states. We apply our measure to the experimentally relevant, nontrivial example of a superconducting three-junction flux qubit put into a superposition of left- and right-circulating supercurrent states, and we find the size of this superposition to be surprisingly small.

Figure 1

(Color online) (a) Example of normal-state persistent currents mentioned in the text, where $D=3$ single-particle operations are needed to turn state $\mid A⟩$ into $\mid B⟩$. (b) Hilbert spaces ${\mathcal{H}}_d$ generated by repeated application of single-particle operators. (c) Probability distribution $P_{\theta }(D=d)$ for the distance $D$ between two BEC states or between the two components of a generalized GHZ state, as a function of the angle between the corresponding single-particle states, for $N=10$ particles; see Eq. (4).

Figure 2

(Color online) (a) Circuit diagram of the three-junction flux qubit. (b) Equivalent representation in the charge basis. (c) Energy-level diagram for $E_J∕E_C=20$ and $\alpha =1$, as a function of magnetic frustration. At $f=0.5$, the ground and first excited state are superpositions of left- and right-going current states, $\mid \pm I⟩$, the states between which we calculate the “distance” $D$. The current distribution in the ground state is displayed in the inset.

Figure 3

(Color online) (a) Average many-body distance $\overline{D}$ between the left- and right-going current states forming the ground state of a three-junction flux qubit at $f=0.5$, plotted as a function of $E_J∕E_C$, for various asymmetry parameters $\alpha$. (b) Corresponding probability distribution for $\alpha =0.8$. (c) Magnitude $I$ of the average current in the two current states, and average charge fluctuation $\delta N$ in the ground state [symbols as in (a)].