Hardy’s Paradox and Violation of a State-Independent Bell Inequality in Time

Tests such as Bell’s inequality and Hardy’s paradox show that joint probabilities and correlations between distant particles in quantum mechanics are inconsistent with local realistic theories. Here we experimentally demonstrate these concepts in the time domain, using a photonic entangling gate to perform nondestructive measurements on a single photon at different times. We show that Hardy’s paradox is much stronger in time and demonstrate the violation of a temporal Bell inequality independent of the quantum state, including for fully mixed states.

Figure 1

Thought experiment for the violation of local realistic theories. (a) Spatial scenario: A source $S$ emits two (entangled) qubits, which are sent to two remote observers $A$ and $B$. Each subsystem is subject to two measurements $A_k$ and $B_l$, where $k$ and $l$ denote the measurement settings at different sites. The outcomes of individual measurements are labeled $r$ and $s$. (b) Temporal scenario for the violation of noninvasive, realistic theories. A single system is subjected to two measurements $A_k$ and $B_l$, in this case occurring at different times $t_B>t_A$.

Figure 2

Experimental scheme. (a) Temporal measurements. The signal and meter qubit are encoded in orthogonal polarization states of two single photons, which are created via spontaneous parametric down-conversion in a nonlinear crystal, pumped by a pulsed (76 MHz, 200 fs), frequency-doubled Ti:sapphire laser at $\lambda =820\mathrm{nm}$. States are prepared with polarizing beam splitters (PBS), a quarter- (QWP) and a half-wave plate (HWP). The signal photon passes a controlled-phase gate (cz), where it acts as the control qubit, with the meter photon being the target. Behind the gate, we analyze the meter photon polarization and detect it with a single-photon avalanche photodiode (APD), implementing the first measurement $A_k$. Two HWPs (one incorporated into the preparation stage) set the basis for this nondestructive measurement. The signal is stored in a 50 m long fiber spool and, after $A_k$ is concluded, measured projectively, implementing $B_l$. A fiber polarization controller and a combination of wave plates compensate for polarization rotation in the fiber. A coincidence logic analyzes detection events within a time window of 4.4 ns. (b) The cz gate in detail, here shown in dual-rail representation. We realize it with a single partially polarizing beam splitter (PPBS), with transmittivities ${\eta }_H=1/3$ (${\eta }_V=1$) for the $H$ ($V$) polarization [31, 32, 33]. Quantum interference results in a relative $\pi$ phase shift of the vertical polarization components $|V{⟩}_s|V{⟩}_m$. The correct functioning is heralded by a coincidence count between the two output arms of the PPBS, which occurs with probability $1/9$.

Figure 3

Experimental violation of the state-independent temporal Bell inequality (7). The classical limit is indicated by CL and the maximal achievable quantum value by QM. The first six bars correspond to pure signal states, the remaining two to mixed inputs, as explained in the main text. The latter were obtained by switching the signal state between states $|D⟩$ and $|A⟩$ while measurements were performed. The relative integration for $|D⟩$ and $|A⟩$ were chosen according to the target purity of 0.5 for ${\rho }_1$ and 0.75 for ${\rho }_2$. The mixed states were verified via single-qubit tomography.