Table 5 Selection of related parameters to directly obtain the real and imaginary parts of the process matrix element \({\chi }_{J,K}^{M,N}(M\, > \,N,J\,\ne \,{\rm{K}})\)
From: Efficient non-destructive direct characterization of arbitrary many-body quantum channels
\({\chi }_{J,K}^{M,N}(M\, > \,N,J\,\ne \,K)\) | ||||||||
|---|---|---|---|---|---|---|---|---|
\(({j}_{{q}}\,\ne \,{k}_{{q}},{j}_{1}={k}_{1},\cdots {j}_{N}={k}_{N})\) | \(\left(\begin{array}{l}{j}_{{q}}\,\ne \,{k}_{{q}},\cdots {j}_{{N}}\,\ne \,{k}_{{N}},\\ {j}_{1}={k}_{1},\cdots {j}_{q-1}={k}_{q-1}\end{array}\right)\) | \(\left({j}_{{1}}\,\ne \,{k}_{{1}},\cdots {j}_{{N}}\,\ne \,{k}_{{N}}\right)\) | ||||||
\(\left(\begin{array}{l}{m}_{q}\,\ne \,{n}_{q},\\ {m}_{1}={n}_{1},\\ \cdots \\ {m}_{N}={n}_{N}\end{array}\right)\) | \(\left(\begin{array}{l}{m}_{{q}}\,\ne \,{n}_{{q}},\\ \cdots \\ {m}_{{N}}\,\ne \,{n}_{{N}},\\ {m}_{1}={n}_{1},\\ \cdots \\ {m}_{q-1}={n}_{q-1}\end{array}\right)\) | \(\left(\begin{array}{l}{m}_{{1}}\,\ne \,{n}_{{1}},\\ \cdots \\ {m}_{{N}}\,\ne \,{n}_{{N}}\end{array}\right)\) | \(\left(\begin{array}{l}{m}_{q}\,\ne \,{n}_{q},\\ {m}_{1}={n}_{1},\\ \cdots \\ {m}_{N}={n}_{N}\end{array}\right)\) | \(\left(\begin{array}{l}{m}_{q}\,\ne \,{n}_{q},\\ \cdots \\ {m}_{N}\,\ne \,{n}_{N},\\ {m}_{1}={n}_{1},\\ \cdots \\ {m}_{q-1}={n}_{q-1}\end{array}\right)\) | \(\left(\begin{array}{l}{m}_{1}\,\ne \,{n}_{1},\\ \cdots \\ {m}_{N}\,\ne \,{n}_{N}\end{array}\right)\) | \(\left(\begin{array}{l}{m}_{q}\,\ne \,{n}_{q},\\ {m}_{1}={n}_{1},\\ \cdots \\ {m}_{N}={n}_{N}\end{array}\right)\) | \(\left(\begin{array}{l}{m}_{q}\,\ne \,{n}_{q},\\ \cdots \\ {m}_{N}\,\ne \,{n}_{N},\\ {m}_{1}={n}_{1},\\ \cdots \\ {m}_{q-1}={n}_{q-1}\end{array}\right)\) | \(\left(\begin{array}{l}{m}_{1}\,\ne \,{n}_{1},\\ \cdots \\ {m}_{N}\,\ne \,{n}_{N}\end{array}\right)\) |
μ = 1, v = 1. | μ = 2, v = 1. | μ = 3, v = 1. | μ = 1, v = 2. | μ = 2, v = 2. | μ = 3, v = 2. | μ = 1, v = 3. | μ = 2, v = 3. | μ = 3, v = 3. |
\(\begin{array}{l}{\text{Re}}{\chi }_{J,K}^{M,N}=\frac{1}{4}(2{P}_{\mu ,\nu }^{6}-2{P}_{\mu ,\nu }^{10}+2{P}_{\mu ,\nu }^{15}-2{P}_{\mu ,\nu }^{19}-{P}_{\mu ,\nu }^{5}+{P}_{\mu ,\nu }^{9}-{P}_{\mu ,\nu }^{13}+{P}_{\mu ,\nu }^{17}-{P}_{\mu ,\nu }^{8}+{P}_{\mu ,\nu }^{12}-{P}_{\mu ,\nu }^{16}+{P}_{\mu ,\nu }^{20}),\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mu ,\nu \in \{1,2,3\}\\ {\text{Im}}{\chi }_{J,K}^{M,N}=\frac{1}{4}(2{P}_{\mu ,\nu }^{14}-2{P}_{\mu ,\nu }^{18}-2{P}_{\mu ,\nu }^{7}+2{P}_{\mu ,\nu }^{11}+{P}_{\mu ,\nu }^{5}-{P}_{\mu ,\nu }^{9}-{P}_{\mu ,\nu }^{13}+{P}_{\mu ,\nu }^{17}+{P}_{\mu ,\nu }^{8}-{P}_{\mu ,\nu }^{12}-{P}_{\mu ,\nu }^{16}+{P}_{\mu ,\nu }^{20}).\end{array}\) | ||||||||
