Perfect Wave Transfer in Continuous Quantum Systems

Abstract

The transfer of information from one part of a quantum system to another is fundamental to the understanding and design of quantum information processing devices. In the realm of discrete systems, such as spin chains, inhomogeneous networks have been engineered that allow for the perfect transfer of qubits from one end to the other. Here, by contrast, we investigate perfect transfer of information in continuous systems, phrased in terms of wave propagation. A remarkable difference is found between systems that possess conformal invariance and those that do not. Systems in the first class enjoy perfect wave transfer (PWT), explicitly shown for one-particle excitations and expected in general. In the second class, those that exhibit PWT are characterized as solutions to an inverse spectral problem. As a concrete example, we demonstrate how to formulate and solve this problem for a prototypical class of bosonic theories, showing the importance of conformal invariance for these theories to enjoy PWT. Using bosonization, our continuum results extend to theories with interactions, broadening the scope of perfect information transfer to more general quantum systems.

Figures (2)

• Figure 1: (a) Illustration of PWT with Neumann BCs showing an initial wave that evolves to its reflection at a certain time $T$. (b) Its manifestation in an inhomogeneous CFT with a given profile $v(x)/v_{0}$ (inset), showing a localized initial wave propagating along curved light-cone trajectories that recombine at the reflected point at $t = T \equiv L/v_{0}$ given by Eq. \ref{['yx_v0_def']}. Specifically, the magnitude of boson two-point correlations \ref{['varphivarphi']} at the free-fermion point is plotted, which by bosonization encodes the same information as $\log F(t)$ in Eq. \ref{['Ft']} for delta-like waves $\xi_{1}$ centered at $-3L/8$ and $\xi_{2}$ at $x$ (setting $k_{\textrm{F}} = 0$ for simplicity and $t' = 0$).
• Figure 2: Plot of $C(x, t) \equiv \operatorname{Re} \langle\Omega| \varphi(x,t)\varphi(-3L/8,0) |\Omega\rangle$ in Eq. \ref{['varphivarphi']} for eigenfunctions in the form of (a) Chebyshev polynomials of the first kind, which exhibit PWT, contrasted to (b) Legendre polynomials and (c) Chebyshev polynomials of the second kind, which do not. The upper row contains intensity plots of the correlations as functions of $x/L \in [-1/2, 1/2]$ and $t/T \in [0,1]$, while the lower row compares their profiles at $t = 0$ and $T$. Note the asymmetry in (b) and (c). The velocity $v(x)$ (same in all cases) and the respective Luttinger parameter $K(x)$ are shown in (d).