We examine multiphoton interference with a symmetric $\mathrm{SU}\left(N\right)$ beam splitter $S_N$, an extension of features of the $\mathrm{SU}\left(2\right)$ 50/50 beam splitter extended Hong-Ou-Mandel (eHOM) effect, whereby one obtains a zero amplitude (probability) for the output coincidence state (defined by equal number of photons $n/N$ in each output port), when a total number $n$ of photons impinges on the $N$-port device. These are transitions of the form $|n_1,n_2,...,n_N\rangle \stackrel{S_N}{\to }{|n/N\rangle }^{\otimes N}$, where $n={\sum }_{i=1}^N{n}_i$, which generalize the Hong-Ou-Mandel effect $|1,1\rangle \stackrel{S_2}{\to }|1,1\rangle$, the eHOM effect $|n_1,n_2\rangle \stackrel{S_2}{\to }|\frac{n_1+n_2}{2},\frac{n_1+n_2}{2}\rangle$, and the generalized HOM effect ${|1\rangle }^{\otimes N}\stackrel{S_N}{\to }{|1\rangle }^{\otimes N}$, which have previously been studied in the literature. The emphasis of this work is on illuminating how the overall destructive interference occurs in separate groups of destructive interferences of subamplitudes of the total zero amplitude. We also consider the more general case for zero-coincidences for the symmetric $\mathrm{SU}\left(N\right)$ beam splitter transformations on multiphoton Fock input states $|n_1,n_2,...,n_N\rangle \stackrel{S}{\to }|m_1,m_2,...,m_N\rangle$ such that ${\sum }_{i=1}^N{n}_i={\sum }_{i=1}^N{m}_i\stackrel{\mathrm{def}}{=}n$. We relate these zero-coincidences to the symmetry properties of $S_N$, beyond the well-known condition that zero-coincidence implies zero-permanent Perm $\left(S_N\right)$=0, which governs the transformation ${|1\rangle }^{\otimes N}\stackrel{S_N}{\to }{|1\rangle }^{\otimes N}$ for arbitrary $N\in \mathrm{even}$. We extend these symmetry properties to the case of the generalized eHOM effect (geHOM) $|n_1,n_2,...,n_N\rangle \stackrel{S_N}{\to }{|n/N\rangle }^{\otimes N}$ involving a zero amplitude governed by $\text{Perm}\left(\mathrm{\Lambda }\right)=0$, for an appropriately constructed matrix $\mathrm{\Lambda }\left(S_N\right)$ built from the matrix elements of $S_N$. We develop an analytical constraint equation for $\text{Perm}\left(\mathrm{\Lambda }\right)$ for arbitrary $N$ that allows us to determine when it is zero, implying complete destructive interference on the geHOM output coincident state $|\frac{n}{N}{\rangle }^{\otimes N}$. We generalize the SU(2) beam splitter feature of the central nodal line, which has a zero diagonal along the output probability distribution when one of the input states is of odd parity (containing only odd number of photons), to the general case of $N=2\times N^{\prime }$ where $N^{\prime }\in \mathrm{odd}$.

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• Quantum information processing
• Quantum measurements
• Quantum metrology