Figure 1.

(a) Phase-space representations of various coherent communication schemes. QPSK, quadrature phase-shift keying. (b) Prepare-and-measure (PM) scheme of STCQ protocol. STCQ, simultaneous two-way classical and quantum; Hom1x(2p), homodyne detection of measuring the $X\left(P\right)$ quadrature; BS: 50:50 beam splitter; QM, quantum memory; $\alpha ,$ modulated signal amplitude of two-way classical communication."

Figure 2.

(a) Scenario of STCQ-based underwater submarine communication; one of the motherships releases an unmanned undersea vehicle as the repeater without exposing the risk. (b) Entanglement-based (EB) scheme of STCQ protocol. TMS, two-mode squeezed state; Het, heterodyne detection; Dis, displacement operation."

Figure 3.

Nine different measurement results are mapped in the phase space and classified by order (I), (II), and (IV). The number of times each color's measurement result appears in (a) is represented by these orders. In terms of the measurement findings, the transmitter and receiver are symmetric."

Figure 4.

Combination of the results based on the Bell-state measurement."

Figure 5.

Probability that Eve correctly deduces the coding sequence with length $n.$ For comparison, we assume that the user transmits the easiest and hardest guessable bits all the time in the most unfavorable and the most advantageous case, respectively."

Figure 6.

Secret key rate (SKR) (main graph) and amplitude of the classical signal (inset graph) as a function of the modulation variance in the symmetric scenario where the BER is less than 10−9 on oceans S2 for (a) and S5 for (b) and in the asymmetric scenario where the BER is less than 10−9 on oceans S2 for (c) and S5 for (d). Other simulation parameters are set as follows: the extra noise in two channels that is independent of the amplitude of the classical signal ${\varepsilon }_{{\rm{0A}}}={\varepsilon }_{{\rm{0B}}}=0.002,$ the detector efficiency ${\eta }_{{\rm{\hom }}}=0.98,$ the electronic noise ${\nu }_{{\rm{el}}}=0.01,$ the phase-noise variance ${\sigma }_{\phi }={10}^{-6},$ and the reconciliation efficiency $\beta =0.98.$"

Figure 7.

SKR as a function of the transmission distance in the symmetric scenario at oceans S2 for (a) and S5 for (b), and in the extremely asymmetric scenario at oceans S2 for (c) and S5 for (d), where the BER is less than 10−9. For comparison, the SKR following Beer's law (dotted line), considered the nonturbulence channel, is computed based on the same parameters of the turbulence channel (solid line). Other simulation parameters are set as follows: the extra noise in two channels that is independent of the amplitude of the classical signal ${\varepsilon }_{{\rm{0A}}}={\varepsilon }_{{\rm{0B}}}=0.002,$ the detector efficiency ${\eta }_{{\rm{\hom }}}=0.98,$ the electronic noise ${\nu }_{{\rm{el}}}=0.01,$ the phase-noise variance ${\sigma }_{\phi }={10}^{-6},$ and the reconciliation efficiency $\beta =0.98.$"

Figure 8.

Finite-size key rates of the STCQ protocol of the asymmetric scenario in ocean S2 for (a) and S5 for (b). Other simulation parameters are set the same as in figure 7."

Table A1.

Parameter values for S1–S6 oceans [36]."

Figure A1.

Extinction coefficient of extinction as a function of submarine depth including absorption and scattering in various ocean types. The S1–S6 oceanic parameters are given in table A1."

Figure B1.

Schematic of an ellipse model. In this model, the beam source is deformed from the transmission distance of $L$ with ellipse semi-axes ${w}_{1}$ and ${w}_{2},$ where $a$ and $r$ represent the aperture radius and the beam-deflection distance, respectively."

Figure B2.

Mean transmittance as a function of submarine depth and transmission distance, including the effects of extinction and turbulence on ocean S2 for (a) and ocean S5 for (b). Other simulation parameters are set as follows: the dissipation rate of temperature or salinity variance $\zeta ={10}^{-11},$ the kinetic energy dissipation rate $\omega ={10}^{-3},$ the receiver telescope radius $a=0.25{\rm{m}},$ and the initial beam radius ${w}_{0}=80{\rm{mm}}.$"