  The classic example of the destruction of interference fringes in a
``which-way'' experiment, caused by an environmental interaction, may be viewed
as the destruction of first-order coherence as defined by Glauber many years
ago (Glauber). However, the influence of an environment can also destroy the
$n$th-order quantum coherence in a quantum system, where this high order
coherence is captured. We refer to this phenomenon as the $n$% th-order
decoherence. In this paper we show that, just as the first-order coherence can
be understood as the interference of the amplitudes for two distinct paths, the
higher order coherence may be understood as the interference of multiple
amplitudes corresponding to multiple paths. To see this, we introduce the
concept of $n$th-order ``multi-particle wave amplitude''. It turns out that the
$n$th-order correlation function can be expressed as the square norm of some
``multi-particle wave amplitude'' for the closed system or as the sum of such
square norms for the open system. We also examine, as a specific example, how
an environment can destroy the second order coherence by eliminating the
interference between various multiple paths.
