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Let's consider a two-mode quantum state described in Fock state. A N00N state can be written as $|\psi _{{\text{NOON}}}\rangle ={\frac {|N\rangle _{a}|0\rangle _{b}+e^{{iN\theta }}|{0}\rangle _{a}|{N}\rangle _{b}}{{\sqrt {2}}}},\,$ where the subscript $a$ and $b$ denote two mode. This N00N state is an entangled state defined in Wikipedia, from where I feel the definition of entangled state should be the same as the conventional one, i.e., for the pure state, it cannot be written as the direct product form as $|n_1\rangle_a|n_2\rangle_b$. Then the weird thing happened to me: when we consider $N=1$ case, we have $|\psi\rangle=\frac{|1\rangle _a|0\rangle _b+|0\rangle _a|1\rangle _b}{\sqrt{2}}$, which should be entangled state as the definition I inferred. But this state only has one photon, while entanglement is a correlation between several particles. And the physical meaning of this state is that it has half probability in mode $a$ and has half probability in mode $b$ which sounds more like a coherent state to me.

So my problem is, does the entanglement has the conventional meaning in optical interferometer?

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Entanglement is entanglement, there's just one definition of it (well, there's more complex variations like genuine multipartite entanglement etc, but that's not relevant here). However, (bipartite) entanglement is defined with respect to a given bipartition of the state. See e.g. also the discussions on physics.SE here and here. Also very relevant is Van Enk's 2005 paper, which argues that the single-photon state $|10\rangle+|01\rangle$ should be considered as entangled.

Specifically about the single-photon state under consideration here, the point is that it's entangled with respect to the different Fock states. Or better said, there is entanglement with respect to the bipartition $\mathcal H_1\otimes\mathcal H_2$, with $\mathcal H_i$ the (infinite-dimensional) Hilbert space of the $i$-th photonic mode (be it spatial modes or whatever else). This means that you will observe correlations compatible with a maximally entangled two-qubit state, if you perform measures on the two modes which potentially probe coherences between vacuum state and excited states. These would be nonlinear measurements, so possibly nontrivial to implement experimentally, but nonetheless completely valid operations in principle.

On the other hand, you will observe no entanglement by simply measuring the position degree of freedom of the photon, as in that case the state is effectively something of the form $|1\rangle+|2\rangle$, and there isn't any bipartite structure to consider.

And just to conclude, let me stress that entanglement is not "correlation between different particles". It's a property of different degrees of freedom of a state (which produces correlations of a specific kind). But you don't need multiple particles to have entanglement.

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