[Paper Review] The generation of an entangled four-photon state from two pairs of entangled two-photon states by using linear optical elements

Xu‐Bo Zou, K. Pahlke+1 more|ArXiv.org|Nov 2, 2001

Quantum Information and Cryptography1 references3 citations

TL;DR

This paper proposes a linear optical scheme to generate a four-photon entangled state from two pairs of pre-prepared two-photon entangled states using beam splitters and post-selection. The resulting state is equivalent to two maximally entangled spin-1 particles and enables 1→2 optimal quantum cloning, with the method generalizable to 2N-photon states from N entangled pairs.

ABSTRACT

We present a scheme to produce an entangled four-photon state from two pairs of entangled two-photon states. Such entangled four-photon states are equivalent to the quantum state of two maximally entangled spin-1 particles. The scheme can also be generated to prepare an entangled 2N-photon state from N pairs of entangled two-photon states. Such multi-photon states play a crucial role in fundamental tests of quantum mechanics versus local realism and in many quantum information and quantum computation schemes.

Motivation & Objective

• To develop a deterministic method for generating high-dimensional multi-photon entanglement using only linear optics and post-selection.
• To demonstrate that the resulting four-photon state is equivalent to two maximally entangled spin-1 particles, enabling violation of Bell inequalities.
• To show that the generated state can serve as a resource for 1→2 optimal quantum cloning via telecloning protocols.
• To generalize the scheme to produce 2N-photon entangled states from N pairs of two-photon entangled states.

Proposed method

• Two pairs of polarization-entangled two-photon states are prepared in the form $\frac{1}{\sqrt{2}}(|H\rangle_i|V\rangle_j + |V\rangle_i|H\rangle_j)$ for photons 1–2 and 3–4.
• The photons pass through four symmetric beam splitters, with the beam splitter transmission coefficient $\theta$ tuned to maximize success probability.
• Post-selection is applied by detecting no photons in the auxiliary output modes ($H_3', V_3', H_4', V_4'$), projecting the system into the target four-photon entangled state.
• The resulting state is $\Psi_2 = \frac{1}{2\sqrt{3}}(a_{H_1'}^\dagger a_{V_2'}^\dagger + a_{V_1'}^\dagger a_{H_2'}^\dagger)^2|0\rangle$, equivalent to a symmetric superposition of Fock states.
• The scheme generalizes to $2N$-photon states by using $N$-port symmetric beam splitters and post-selecting on vacuum in all but the first output mode.
• The protocol enables quantum telecloning by projecting the state into a form suitable for 1→2 optimal cloning using symmetric beam splitters and coincidence detection.

Experimental results

Research questions

• RQ1 Can a four-photon entangled state equivalent to two maximally entangled spin-1 particles be generated using only linear optical elements and post-selection?
• RQ2 What is the maximum success probability of generating such a state using symmetric beam splitters?
• RQ3 Can the generated four-photon state serve as a resource for 1→2 optimal quantum cloning via telecloning?
• RQ4 How can the scheme be generalized to produce $2N$-photon entangled states from $N$ pairs of two-photon entangled states?
• RQ5 Does the resulting multi-photon state violate Bell-type inequalities, as predicted by Drummond?

Key findings

• The success probability of generating the four-photon entangled state reaches a maximum of $3/16$ when symmetric beam splitters ($\theta = \pi/4$) are used.
• The final state $\Psi_2$ is equivalent to $\frac{1}{\sqrt{3}}(|2H\rangle_1|2V\rangle_2 + |H\rangle_1|V\rangle_1|H\rangle_2|V\rangle_2 + |2V\rangle_1|2H\rangle_2)$, confirming its spin-1 equivalence.
• Post-selection on vacuum in the auxiliary output modes successfully projects the system into the desired entangled four-photon state with high fidelity.
• The scheme enables 1→2 optimal quantum cloning by generating a telecloning state via symmetric beam splitters and four-photon coincidence detection.
• The method generalizes to $2N$-photon states, producing a state equivalent to two maximally entangled spin-$N/2$ particles.
• The resulting $2N$-photon state is of the form $\Psi_6 = \frac{1}{\sqrt{N}}\sum_{m=0}^{N} |mH\rangle_1|(N-m)V\rangle_1|(N-m)H\rangle_2|mV\rangle_2$, confirming its symmetric, entangled structure.