We give analytical solutions for the time-optimal synthesis of entangling gates between indirectly coupled qubits 1 and 3 in a linear spin chain of three qubits subject to an Ising Hamiltonian interaction with equal coupling $J$ plus a local magnetic
field acting on the intermediate qubit. The energy available is fixed, but we relax the standard assumption of instantaneous unitary operations acting on single qubits. The time required for performing an entangling gate which is equivalent, modulo local unitary operations, to the $mathrm{CNOT}(1, 3)$ between the indirectly coupled qubits 1 and 3 is $T=sqrt{3/2} J^{-1}$, i.e. faster than a previous estimate based on a similar Hamiltonian and the assumption of local unitaries with zero time cost. Furthermore, performing a simple Walsh-Hadamard rotation in the Hlibert space of qubit 3 shows that the time-optimal synthesis of the $mathrm{CNOT}^{pm}(1, 3)$ (which acts as the identity when the control qubit 1 is in the state $ket{0}$, while if the control qubit is in the state $ket{1}$ the target qubit 3 is flipped as $ket{pm}rightarrow ket{mp}$) also requires the same time $T$.
The formation of a deeply-bound $K^- pp$ state with $I=1/2$, $J^pi=0^-$ by the $^3$He(in-flight $K^-$, $n$) reaction is theoretically investigated in a distorted-wave impulse approximation using the Greens function method. The expected inclusive and
semi-exclusive spectra at $p_{K^-} = 1.0$ GeV/c and $theta_{rm lab} = 0^{circ}$ are calculated for the forthcoming J-PARC E15 experiment. We demonstrate these spectra with several types of phenomenological $K^-$-``$pp$ optical potentials $U^{rm opt}(E)$ which have an energy-dependent imaginary part multiplied by a phase space suppression factor, fitting to recent theoretical predictions or experimental candidates of the $K^-pp$ bound state. The results show that a cusp-like peak at the $pi Sigma N$ threshold is an unique signal for the $K^-pp$ bound state in the spectrum including the [$K^-pp$] $to$ $Y + N$ decay process from the two-nucleon $K^-$ absorption, as well as a distinct peak of the $K^-pp$ bound state. The shape of the spectrum is explained by a trajectory of a moving pole of the $K^-pp$ bound state in the complex energy plane. The importance of the spectrum with [$K^-pp$] $to$ $Y + N$ from the two-nucleon $K^-$ absorption is emphasized in order to extract clear evidence of the $K^-pp$ bound state.
The formation of deeply-bound antikaonic $K^-/bar{K}^0$ nuclear states by nuclear ($K^-$, $N$) reactions is investigated theoretically within a distorted-wave impulse approximation (DWIA), considering the isospin properties of the Fermi-averaged $K^-
+ N to N + bar{K}$ elementary amplitudes. We calculate the formation cross sections of the deeply-bound $bar{K}$ states by the ($K^-$, $N$) reactions on the nuclear targets, $^{12}$C and $^{28}$Si, at incident $K^-$ lab momentum $p_{K^-}$ = 1.0 GeV/c and $theta_{rm lab} = 0^{circ}$, introducing a complex effective nucleon number $N_{rm eff}$ for unstable bound states in the DWIA. The results show that the deeply-bound $bar{K}$ states can be populated dominantly by the ($K^-$, $n$) reaction via the total isoscalar $Delta T=0$ transition owing to the isospin nature of the $K^-+ N to N + bar{K}$ amplitudes, and that the cross sections described by ${rm Re}N_{rm eff}$ and ${rm Arg}N_{rm eff}$ enable to deduce the structure of the $bar{K}$ nuclear states; the calculated inclusive nucleon spectra for a deep $bar{K}$-nucleus potential do not show distinct peak structure in the bound region. The few-body $bar{K}otimes [NN]$ and $bar{K}otimes [NNN]$ states formed in ($K^-$, $N$) reactions on $s$-shell nuclear targets, $^3$He, $^3$H and $^4$He, are also discussed.
