Fluctuation of information content and the optimum capacity for bosonic transport

Abstract

We discuss the optimum communication capacity of bosonic information carriers propagating through a tunnel junction. Based on the multi-contour Keldysh Green function, we evaluate the information generating function, or the Rényi entanglement entropy, of the reduced density matrix subjected to the constraint of local heat quantity in the steady state. The Rényi entanglement entropy of order zero is the partition function, which exponentially depends on the optimum capacity. For the perfect transmission, the self-information and the local heat quantity, i.e., the energy of signals, are perfectly linearly correlated. The water-filling theorem is recovered in the wave-like regime.

Appendix A: Multi-contour Keldysh technique

A component of $2 M \times 2 M$ multi-contour Keldysh Green function matrix $\mathbf{g}_{A}$ connecting $C_{m^\prime ,s^\prime }$ and $C_{m,s}$ is [22,23,24,25,26,27,28,29],

$$\begin{aligned} g_{A}^{ms,m's'}(t,t')= & {} g_{A}(t_{ms},t'_{m's'}) \nonumber \\= & {} -i \sum _k \left\langle {\hat{T}}_C {\hat{a}}_{Ak}(t_{ms})_I {\hat{a}}_{Ak}^\dagger (t_{m's'}')_I \right\rangle _{M,\chi } . \nonumber \\ \end{aligned}$$

(31)

The $2 M \times 2M$ matrix $\mathbf{g}_{A}$ is a block circulant as,

$$\begin{aligned}&\mathbf{g}_{A}(\omega ) = \left[ \begin{array}{cccc} \mathbf{A}_0(\omega ) &{} \mathbf{A}_{M-1}(\omega ) &{} \cdots &{} \mathbf{A}_{1}(\omega ) \\ \mathbf{A}_1(\omega ) &{} \mathbf{A}_{0}(\omega ) &{} \cdots &{} \mathbf{A}_{2}(\omega ) \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \mathbf{A}_{M-1}(\omega ) &{} \mathbf{A}_{M-2}(\omega ) &{} \cdots &{} \mathbf{A}_{0}(\omega ) \end{array} \right] \nonumber ,\\ \,&\mathbf{A}_0(\omega ) = \epsilon _A(\omega ) {\tau }_z -i \pi \rho _A(\omega ) \nonumber \\&\quad \times \left[ \begin{array}{cc} n_{A, 0 }(\omega ) + n_{A, M-i \chi /\beta _A}(\omega ) &{} 2 n_{A, 1}(\omega ) \\ 2 n_{A, M(1-i \chi /(\beta _A M))-1}(\omega ) &{} n_{A, 0 }(\omega ) + n_{A, M-i \chi /\beta _A}(\omega ) \end{array} \right] , \nonumber \\&\mathbf{A}_m(\omega ) = -2 i \pi \rho _A(\omega ) \nonumber \\&\quad \times \left[ \begin{array}{cc} n_{A, m(1-i \chi /(\beta _A M))}(\omega ) &{} n_{A, m(1-i \chi /(\beta _A M))+1}(\omega ) \\ n_{A, m(1-i \chi /(\beta _A M))-1}(\omega ) &{} n_{A, m(1-i \chi /(\beta _A M))}(\omega ) \end{array} \right] \nonumber \\&\quad \times (m \ne 0) ,\nonumber \\ \end{aligned}$$

(32)

where $n_{A , m}(\omega ) = \mathrm{e}^{-\beta _{A} \omega m} /\left( 1- \mathrm{e}^{ -\beta _A \omega (M-i \chi )} \right)$. The block circulant matrix is block-diagonalized by the discrete Fourier transform, $\sum _{m=0}^{M-1} \mathbf{A}_{m} \mathrm{e}^{i 2 \pi \ell m/M } = \mathbf{g}_{A}^{\lambda _\ell }$, where $\lambda _\ell = (2 \pi \ell + \omega \chi )/M$ and

$$\begin{aligned}&\mathbf{g}_{A}^{\lambda }(\omega ) = \epsilon _A(\omega ) {\tau }_z - 2 i \pi \rho _A(\omega ) \nonumber \\&\quad \times \left[ \begin{array}{cc} n_{A}^{\lambda ,-}(\omega ) + n_{A}^{\lambda ,+}(\omega ) &{} n_{A}^{\lambda ,+}(\omega ) \mathrm{e}^{-i \lambda } \\ n_{A}^{\lambda ,-}(\omega ) \mathrm{e}^{i \lambda } &{} n_{A}^{\lambda ,-}(\epsilon _A) + n_{A}^{\lambda ,+}(\omega ) \end{array} \right] . \end{aligned}$$

(33)

A multi-contour Keldysh Green function of subsystem B is nonzero when ${\hat{a}}_{B,m}$ and ${\hat{a}}_{B,m'}^\dagger$ are on the same (standard) Keldysh contour $m=m'$. It is given as,

$$\begin{aligned}&\mathbf{g}_{B}(\omega ) = \mathbf{1}_{M} \otimes \nonumber \\&\quad \times \left( - 2 \pi i \rho _B(\omega ) \left[ \begin{array}{cc} n_B^-(\omega ) + n_B^+ (\omega ) &{} n_B^+ (\omega ) \\ n_B^- (\omega ) &{} n_B^-(\omega ) + n_B^+ (\omega ) \end{array} \right] \right) ,\nonumber \\ \end{aligned}$$

(34)

where $\mathbf{1}_M$ is the $M \times M$ identity matrix.

The Keldysh partition function (21) can be calculated by exploiting standard Keldysh techniques, see e.g. [32,33,34,35,36,37,38]. In the limit of long measurement time, the leading contribution is proportional to $\tau$ as,

$$\begin{aligned}&\ln \frac{S_{M}( \chi )}{s_{M}( \chi )} \approx - \sum _{\ell =0}^{M-1} \tau \int \frac{\mathrm{d} \omega }{2 \pi } \ln \mathrm{det}\nonumber \\&\quad \times \left[ \begin{array}{cc} \mathbf{1}_{2} &{} -J \mathbf{g}_A^{\lambda _\ell }(\omega ) {\tau }_z \\ -J \mathbf{g}_B(\omega ) {\tau }_z &{} \mathbf{1}_{2} \end{array} \right] = \sum _{\ell =0}^{M-1} \ln {{\mathcal {Z}}}_\tau (\mathrm{e}^{i \lambda _\ell }) , \end{aligned}$$

where $\ln {{\mathcal {Z}}}_\tau$ is the current cummulant generating function of full counting statistics in the single-time measurement protocol,

$$\begin{aligned}&\ln {{\mathcal {Z}}}_\tau (\mathrm{e}^{i \lambda }) = - \tau \int \frac{\mathrm{d} \omega }{2 \pi } \ln \left( \frac{{\tilde{n}}_A^-(\omega ) - {\tilde{n}}_A^+(\omega ) \mathrm{e}^{i \lambda }}{{n}_A^-(\omega ) - {n}_A^+(\omega ) \mathrm{e}^{i \lambda }} \right) . \end{aligned}$$

(35)

We rewrite the summation over $\ell$ as the contour integral (Fig. 3a) as [39],

$$\begin{aligned} \ln \frac{S_{M}( \chi )}{s_{M}( \chi )} = \int _{C_{\mathrm{even}}} \frac{\mathrm{d}u}{2 \pi } \sum _{\ell =0}^{M-1} \frac{\ln {{\mathcal {Z}}}_\tau (u \mathrm{e}^{i \omega \chi /M})}{u-\mathrm{e}^{i 2 \pi \ell /M}} . \end{aligned}$$

(36)

If $i \chi <0$ and $\beta _B<\beta _A$, branch points of the integrand $\ln {{\mathcal {Z}}}_\tau$, $u_+(\omega )=n_A^-(\omega )/(n_A^+(\omega )\mathrm{e}^{i \omega \chi /M})$ and $u_-(\omega )={\tilde{n}}_A^-(\omega )/({\tilde{n}}_A^+(\omega )\mathrm{e}^{i \omega \chi /M})$ are on the real axis and satisfy $1<u_-(\omega )<u_+(\omega )$. By changing the contour as Fig. 3b and by utilizing $\sum _{\ell =0}^{M-1} (u-\mathrm{e}^{i 2 \pi \ell /M})^{-1} = \partial _u \ln (u^M-1)$, we obtain Eq. (22).