The partial trace over a finite-dimensional Hilbert-space factor 
maps an operator on 
to an operator on 
by tracing only the 
indices, where 
is also finite-dimensional. In finite dimensions, for an orthonormal
basis 
of 
, it is
 |
This definition is independent of the chosen orthonormal basis. For a joint density matrix, the partial
trace is the reduced density matrix of subsystem
. It preserves the trace
and positivity (Watrous 2011, Example 2.2). For 
, it gives 
. Reduced density matrices
define entanglement entropy and one-particle occupation
numbers.
