Entanglement-Assisted Weak Value Amplification

Large weak values have been used to amplify the sensitivity of a linear response signal for detecting changes in a small parameter, which has also enabled a simple method for precise parameter estimation. However, producing a large weak value requires a low postselection probability for an ancilla degree of freedom, which limits the utility of the technique. We propose an improvement to this method that uses entanglement to increase the efficiency. We show that by entangling and postselecting $n$ ancillas, the postselection probability can be increased by a factor of $n$ while keeping the weak value fixed (compared to $n$ uncorrelated attempts with one ancilla), which is the optimal scaling with $n$ that is expected from quantum metrology. Furthermore, we show the surprising result that the quantum Fisher information about the detected parameter can be almost entirely preserved in the postselected state, which allows the sensitive estimation to approximately saturate the relevant quantum Cramér-Rao bound. To illustrate this protocol we provide simple quantum circuits that can be implemented using current experimental realizations of three entangled qubits.

Figure 1

Quantum circuit that simulates the WVA of a small parameter $\phi$. A meter qubit is prepared in the state $|+⟩=R_y(\pi /2)|0⟩=(|0⟩+|1⟩)/\sqrt{2}$. An ancilla qubit is prepared in the same state $|{\mathrm{\Psi }}_i⟩=|+⟩$. The ancilla is used as a control for a $Z$ rotation $R_z(2\phi )$ of the meter, which simulates the unitary $\widehat{U}=\exp (-i\phi \widehat{A}\otimes {\widehat{\sigma }}_z/2)$ with $\widehat{A}={\widehat{\sigma }}_z$. The ancilla is then postselected in the nearly orthogonal state $⟨{\mathrm{\Psi }}_f|=⟨-|R_z^{†}(2\epsilon )=⟨0|R_y^{†}(-\pi /2)R_z^{†}(2\epsilon )=(⟨0|e^{\mathrm{i}\epsilon }-⟨1|e^{-i\epsilon })/\sqrt{2}$ with probability $P_s\approx {\epsilon }^2$ by performing two rotations, measuring in the $Z$ basis, and keeping only the $⟨0|$ events. Finally, the meter qubit is measured in the $Z$ basis, which yields the linear response $⟨{\widehat{\sigma }}_z{⟩}_{{+}^{\prime }}\approx \phi \mathrm{Im}{A}_w$ amplified by the weak value $A_w\approx i/\epsilon$. The probability for a single success after $n$ attempts, $P_s^{(n)}=1-(1-P_s{)}^n\approx n{\epsilon }^2$, is approximately linear in $n$.

Figure 2

Quantum circuit that simulates the entanglement-assisted WVA of a small parameter $\phi$. As in Fig. 1, a meter qubit is prepared in the state $|+⟩$, while $n$ ancilla qubits are prepared in a entangled state $|{\mathrm{\Psi }}_i⟩$. Each ancilla is then used as a control for a $Z$ rotation $R_z(2\phi )$ of the meter, simulating the unitary $\widehat{U}=\exp (-i\phi \widehat{A}\otimes {\widehat{\sigma }}_z/2)$ with $\widehat{A}$ being the sum of ancilla observables ${\widehat{\sigma }}_z$. The ancillas are then postselected in an entangled state $|{\mathrm{\Psi }}_f⟩$, and the meter qubit is measured in the $Z$ basis, yielding the linear response $⟨{\widehat{\sigma }}_z{⟩}_{{+}^{\prime }}\approx \phi \mathrm{Im}{A}_w$ amplified by a joint WV $A_w$.

Figure 3

Quantum circuit to prepare the optimal entangled preparation for $n$ ancilla qubits. The state $|{\mathrm{\Psi }}_i⟩=(|0{⟩}^{\otimes n}+|1{⟩}^{\otimes n})/\sqrt{2}$ is prepared from a single $|+⟩$ state by a sequence of CNOT gates. Because of this construction, we note that the ordering of the two-qubit gates in Figs. 2, 3, and 4 can be further optimized to pre- and postselect ($n-1$) of the ancilla qubits sequentially, which allows the $n$-qubit entangled ancilla to be practically simulated using only three physical qubits.

Figure 4

Quantum circuits for attaining optimal postselections, using the preparation in Fig. 3. (a) Keeping $A_w\approx \mathrm{i}/\epsilon$ fixed and maximizing $P_s$ produces the entangled postselection $⟨{\mathrm{\Psi }}_f|=⟨0{|}^{\otimes n}{e}^{\mathrm{i}n\epsilon }-⟨1{|}^{\otimes n}{e}^{-\mathrm{i}n\epsilon }$ with $P_s\approx n^2{\epsilon }^2$, which is a factor of $n$ larger than the single ancilla $P_s^{(n)}$ in Fig. 1. This postselection can be implemented as a sequence of CNOT gates and a rotation of the last qubit by $R_z^{†}(n2\epsilon )$ and $R_y^{†}(-\pi /2)$ before measuring all qubits in the $Z$ basis and keeping only $⟨0|$ events. For small $\epsilon$ this state is equivalent to Eq. (12). (b) Keeping $P_s=P_s^{(n)}\approx n{\epsilon }^2$ and maximizing $A_w$ produces a similar state $⟨{\mathrm{\Psi }}_f|=⟨0{|}^{\otimes n}{e}^{i\sqrt{n}\epsilon }-⟨1{|}^{\otimes n}{e}^{-i\sqrt{n}\epsilon }$ with $A_w\approx i\sqrt{n}/\epsilon$, which is a factor of $\sqrt{n}$ larger than $A_w$ in Fig. 1.