Entanglement activation from quantum coherence and superposition

Quantum entanglement and coherence are two fundamental features of nature, arising from the superposition principle of quantum mechanics. While considered as puzzling phenomena in the early days of quantum theory, it is only very recently that entanglement and coherence have been recognized as resources for the emerging quantum technologies, including quantum metrology, quantum communication, and quantum computing. In this work we study the limitations for the interconversion between coherence and entanglement. We prove a fundamental no-go theorem, stating that a general resource theory of superposition does not allow for entanglement activation. By constructing a quantum controlled-not gate as a free operation, we experimentally show that such activation is possible within the more constrained framework of quantum coherence. By using recent results from coherence theory, we further show that the trace norm entanglement is not a strong entanglement monotone.

Figure 1

The conceptual graph of the conversion process. Two individual quantum states are generated as the system state and the ancilla state, respectively. The system state $\rho$ is prepared with a nonzero amount of coherence $C\left(\rho \right)$, while the ancilla is initialized in an incoherent state ${\sigma }_i$. After an incoherent operation ${\mathrm{\Lambda }}_i$ acting on the system and ancilla, the two-qubit state is either entangled or separable, depending on whether the initial system state $\rho$ has coherence or not. Here, we choose the cnot gate as the optimal incoherent operation, which is decomposed into one controlled phase gate and two Hadamard gates.

Figure 2

Experimental setup. Pairs of identical photons are generated via a type-II spontaneous parametric down-conversion process in a BBO crystal by a 390-nm UV laser up-converted from a mode-lock Ti: sapphire oscillator. After passing the 3-nm bandpass filter (BPF), the photon pairs are coupled into the single-mode fibers and launched to the incoherent operation section. A quarter-wave plate (QWP) and a half-wave plate (HWP) are used for polarization compensation. A combination of HWPs and a partial polarizing beam splitter (PPBS) acts as the incoherent operation. The system states are prepared with different amounts of coherence by rotating a HWP following of the PBS. The two-qubit states and an additional copy of the system states are analyzed by quantum state tomography.

Figure 3

Experimental results of two-qubit tomography. The density matrices with different system states $cos\left(\vartheta \right)|H\rangle +sin(\vartheta \left)\right|V\rangle$ as the input state by scanning different polarizations (a) $\vartheta =0^{\circ }$, (b) $\vartheta ={15}^{\circ }$, (c) $\vartheta ={30}^{\circ }$, (d) $\vartheta ={45}^{\circ }$, (e) $\vartheta ={60}^{\circ }$, (f) $\vartheta ={75}^{\circ }$, and (g) $\vartheta ={90}^{\circ }$. From (a) to (d), it is obvious that the generated entangled states vary from separable states to maximal entangled state, while from (e) to (g), the entanglement gradually decreases due to the decline of the coherence.

Figure 4

Activation of entanglement from coherence. The blue bars represent the measured coherence of the system qubit as quantified by the ${\ell }_1$ norm of coherence in Eq. (8). The red bars represent the measured entanglement in the two-qubit state after the incoherent operation, quantified by the concurrence in Eq. (9). The outside frames are the theoretical prediction for coherence and entanglement. The experimental results show the same tendency as we vary the parameter $\vartheta$. All error bars are estimated by the Monte Carlo simulation with 1000 rounds by assuming the Poissonian distribution of the photon statistics.

Figure 5

Violation of strong monotonicity of trace norm entanglement for the state $\rho$ given in Eq. (C2). Solid line shows an upper bound on the trace norm entanglement of $\rho$. Dashed line shows the average entanglement $q_1{E}_{\mathrm{t}}\left({\sigma }_1\right)+q_2{E}_{\mathrm{t}}\left({\sigma }_2\right)$ after a suitable local measurement. Violation of strong monotonicity is obtained in the range $0.4<p<1$.