Revealing Hidden Coherence in Partially Coherent Light

Coherence and correlations represent two related properties of a compound system. The system can be, for instance, the polarization of a photon, which forms part of a polarization-entangled two-photon state, or the spatial shape of a coherent beam, where each spatial mode bears different polarizations. Whereas a local unitary transformation of the system does not affect its coherence, global unitary transformations modifying both the system and its surroundings can enhance its coherence, transforming mutual correlations into coherence. The question naturally arises of what is the best measure that quantifies the correlations that can be turned into coherence, and how much coherence can be extracted. We answer both questions, and illustrate its application for some typical simple systems, with the aim at illuminating the general concept of enhancing coherence by modifying correlations.

Figure 1

Coherence ($D^2$) and maximal violation of the CHSH inequality ($B_{\max }$) for (a) and (b): example I (${\widehat{\rho }}_{\mathrm{MNMS}}$), and (c) and (d): example II (${\widehat{\rho }}_{\mathrm{MEMS}}$). Green lines depict the values of the original state, prior to any unitary transformation. The maximal coherence and minimal violation of the CHSH inequality are marked by dashed-red lines, and the minimal coherence and maximal violation of the CHSH inequality are marked by dotted-blue lines. The black dashed-dotted line represents the classical correlation limit, $B_{\max }=2$. Grey areas correspond to all admissible values achievable by all unitary operations.

Figure 2

Coherence ($D^2$) and maximal violation of the CHSH inequality ($B_{\max }$) for (a) and (b): example III (${\widehat{\rho }}_{\mathrm{E}\mathrm{X}\mathrm{C}}$), and (c) and (d): example IV (${\widehat{\rho }}_W$). Green lines depict the values of the original state, prior to any unitary transformation. The maximal coherence and minimal violation of the CHSH inequality are marked by dashed-red lines, and the minimal coherence and maximal violation of the CHSH inequality are marked by dotted-blue lines. The black dashed-dotted line represents the classical correlation limit $B_{\max }=2$. Grey areas correspond to all admissible values achievable by all unitary operations.