Coherence and complementarity in state-channel interaction

While Bohr's complementarity principle constitutes a bedrock of quantum mechanics with profound implications, coherence, as a defining feature of the quantum realm originating from the superposition principle, pervades almost every quantum consideration. By exploiting the algebraic and geometric structure of state-channel interaction, we show that an information-theoretic measure of coherence and a quantitative symmetry-asymmetry complementarity emerge naturally from the formalism of quantum mechanics. This is achieved by decomposing the state-channel interaction into a symmetric part and an asymmetric part, which satisfy a conservation relation. The symmetric part is represented by the symmetric Jordan product, and the asymmetric part is synthesized by the skew-symmetric Lie product. The latter further leads to a significant extension of the celebrated Wigner–Yanase skew information, and has an operational interpretation as quantum coherence of a state with respect to a channel. This not only presents a basic and alternative framework for addressing complementarity, but also puts the study of coherence in a broad context involving channels. Fundamental properties of the symmetry-asymmetry complementarity are revealed, and applications and implications are illustrated via several prototypical channels as well as the Mach–Zehnder interferometry, in which the fringe visibility is linked to symmetry and the which-path is linked to asymmetry.

Figure 1

Mach–Zehnder interferometry (referred to party $a$) with path detector operation $V$ (referred to party $b$). A quantum particle with an external state $\rho$ and an internal state $\tau$ traverses the path. Its external state subjects to the action of the beam splitter $U_{\mathrm{B}},$ the phase shift $U_{\alpha }$, the mirror $U_{\mathrm{M}}$, and finally the beam splitter $U_{\mathrm{B}}$ again. In the course, when the particle traverses path $|1\rangle$ immediately after the first beam splitter, its internal state $\tau$ is subject to the action of $V$. The overall initial state is the bipartite state $\rho \otimes \tau ,$ and the final external state is $\mathrm{\Phi }\left(\rho \right)={\mathrm{tr}}_b(U(\rho \otimes \tau )U^{†})$ with $U=U_{\mathrm{B}}^{ab}{U}_{\mathrm{M}}^{ab}{V}^{ab}{U}_{\mathrm{B}}^{ab},$ while $U_{\mathrm{B}}^{ab}=U_{\mathrm{B}}\otimes {\mathbf{1}}^b$, $U_{\mathrm{M}}^{ab}=U_{\mathrm{M}}\otimes {\mathbf{1}}^b$, $V^{ab}=e^{i\alpha }|0\rangle \langle 0|\otimes {\mathbf{1}}^b+|1\rangle \langle 1|\otimes V.$