Complementarity relations of a delayed-choice quantum eraser in a quantum circuit

Abstract

We propose a quantum circuit that implements a delayed-choice quantum eraser via bipartite entanglement with the extension that the degree of entanglement between the two paired quantons is adjustable. This provides a broader setting to test complementarity relations between interference visibility and which-way distinguishability in the scenario that the which-way information is obtained through entanglement without direct contact with the quantum state for interference. The visibility-distinguishability relations are investigated from three perspectives that differ in how the which-way information is taken into consideration. These complementarity relations can be understood in terms of entropic uncertainty relations in the information-theoretic framework and the triality relation that incorporates single-particle and bipartite properties. We then perform experiments on the quantum computers provided by the IBM Quantum platform to verify the theoretical predictions. We also apply the delay gate to delay the measurement of the which-way information to affirm that the measurement can be made truly in the “delayed-choice” manner.

Appendices

Appendix A: An elementary account of (4.17)

It is notable that the complementarity relation (4.17) from the separate subensemble perspective is saturated to unity. Its meaning and significance can be understood in view of the triality relation (5.12) as discussed in Sec. 5.2. Here, we present a simple account for (4.17).

First, we express the state at slice 5 in Fig. 2 in the generic form

$$\begin{aligned} \vert {\psi _5}\rangle =a\vert {00}\rangle +b\vert {01}\rangle +c\vert {10}\rangle +d\vert {11}\rangle . \end{aligned}$$

(7.1)

The state at slice 6 is then given by

$$\begin{aligned} \vert {\psi _6}\rangle= & {} H\otimes \mathbb {1} \vert {\psi _5}\rangle \nonumber \\= & {} \frac{1}{\sqrt{2}}\big ((a+b)\vert {00}\rangle +(a-b)\vert {01}\rangle +(c+d)\vert {10}\rangle +(c-d)\vert {11}\rangle \big ). \end{aligned}$$

(7.2)

The contrast for the $0_d$ subensemble defined in (4.10) is computed from $\vert {\psi _6}\rangle$ as

$$\begin{aligned} {\mathcal {C}}_{0_d}= & {} p(0_i|0_d) - p(1_i|0_d) \nonumber \\= & {} \frac{|a+b|^2-|a-b|^2}{|a+b|^2+|a-b|^2} = \frac{a^*b+ab^*}{|a|^2+|b|^2}. \end{aligned}$$

(7.3)

Because the phase gate $P(\theta )$ acting on the interference qubit, we have $a^*b+ab^*=2|a||b|\cos \theta$. Consequently, the visibility ${\mathcal {V}}_{0_d}$ defined in (4.11) is given by

$$\begin{aligned} {\mathcal {V}}_{0_d}=\frac{2|a||b|}{|a|^2+|b|^2}. \end{aligned}$$

(7.4)

Meanwhile, the distinguishability for the $0_d$ subensemble defined in (4.16) is computed from $\vert {\psi _5}\rangle$ as

$$\begin{aligned} {\mathcal {D}}_{0_d}=2p(0_i|0_d)-1 =\frac{|a|^2-|b|^2}{|a|^2+|b|^2}. \end{aligned}$$

(7.5)

It follows that ${\mathcal {D}}_{0_d}^2+{\mathcal {V}}_{0_d}^2=1$. In the subensemble perspective, the complimentary relation always saturates.

Appendix B: System calibration data

Fig. 14

The qubit layout of the IBM Q machines ibm_auckland and ibmq_toronto. The numbers index the physical qubits and the edges denote the connections that allow CNOT operations

Table 1 The calibration data of the selected physical qubits for the pairs we have chosen. The data were retrieved directly from the IBM Quantum website on the same days of the experiments. $T_1$ is the longitudinal (thermal) relaxation time, and $T_2$ is the transverse (dephasing) relaxation time

The experimental results presented in the main text are performed on ibm_auckland, and more experimental results presented in Appendix C are performed on ibmq_toronto. Both ibm_auckland and ibmq_toronto share the same qubit layout as shown in Fig. 14.²¹

Fig. 15

The interference visibility and path distinguishability in the total-ensemble perspective on ibmq_toronto with $t_\text {delay}=0$ (a–c) and $t_\text {delay}=100\,\text {dt}$ (d–f). From the left to right columns, $\phi ^{\prime }=0$, $0.25\pi$, and $0.5\pi$ The calibration data of the selected physical qubits for the pairs we have chosen. The data were retrieved directly from the IBM Quantum website on the same days of the experiments. $T_1$ is the longitudinal (thermal) relaxation time, and $T_2$ is the transverse (dephasing) relaxation time

Fig. 16

The interference visibility and path distinguishability in the average perspective on ibmq_toronto with $t_\text {delay}=0$ (a–c) and $t_\text {delay}=100\,\text {dt}$ (d–f)

Fig. 17

The interference visibility and path distinguishability in the $1_d$ perspective on ibmq_toronto with $t_\text {delay}=0$ (a–c) and $t_\text {delay}=100\,\text {dt}$ (d–f)

We select three different pairs for our “i” and “d” qubits: (i) physical qubits 4 and 7 on ibm_auckland, (ii) physical qubits 1 and 4 on ibm_auckland, and (iii) physical qubits 1 and 4 on ibmq_toronto. The calibration data of these pairs were retrieved directly from the IBM Q website on the same days of the experiments and are reported in Table 1. Pair (i) is used for Fig. 4, 5, 6. Pair (ii) is used for Fig. 3 and Figs. 7, 8,  9,  10,  11,  12 , 13. Pair (iii) is used for the figures in Appendix C.

More experimental data

In order to know more about the nature of noises in the real devices of IBM Quantum, in addition to experiments performed on ibm_auckland, we also perform experiments on ibmq_toronto for comparison.

The experimental results on ibmq_toronto from the total ensemble perspective, the average perspective, and the subensemble perspective are presented in Figs. 15, 16, and 17, respectively. The top panels (a–c) show the results with the delay time $t_{\text {delay}}=0\,\text {dt}$ in the delay gate, while the bottom panels (d–f) show the results with $t_{\text {delay}}=100\,\text {dt}$. Compared with the experimental results performed on ibm_auckland as presented in the main text, the visibility and distinguishability are much noisier. Moreover, the time delay has stronger effect on ibmq_toronto than on ibm_auckland: the effect is appreciable with $t_{\text {delay}}\sim 100\,\text {dt}$ on the former, but it is not appreciable until $t_{\text {delay}}\sim 10^5\,\text {dt}$ on the latter. The fact that ibmq_toronto is much noisier than ibm_auckland is in agreement with the fact that the former has shorter relaxation times and a larger CNOT error rate, according to the daily calibration data provided by IBM Quantum as shown in Table 1.