In quantum computing, the partial trace operation is used to calculate the reduced density matrix of a subsystem in a composite quantum system. The reduced density matrix represents the state of the subsystem when the rest of the system is ignored or traced out.
To understand why the partial trace operation is equivalent to measuring and discarding, let's consider a simple scenario involving two quantum systems, A and B, that are entangled. The joint state of the composite system is represented by the density matrix ρAB.
When we perform a partial trace over system B, we effectively "trace out" or discard the degrees of freedom associated with system B, focusing solely on system A. The resulting reduced density matrix, denoted as ρA, describes the state of system A.
Now, let's compare this to the process of measuring and discarding. When we measure system B and obtain a measurement outcome, it collapses the joint state ρAB into a specific state associated with the measurement outcome. This process is known as a projective measurement. However, since we are only interested in system A, we disregard the measurement outcome and only consider the state of system A after the measurement. This resulting state of system A, after discarding the measurement outcome associated with system B, should be equivalent to the reduced density matrix obtained by performing the partial trace.
Mathematically, the equivalence between partial trace and measuring and discarding can be shown as follows:
Consider the joint state of systems A and B: ρAB.
Perform a partial trace over system B to obtain the reduced density matrix of system A: ρA = TrB(ρAB).
Perform a projective measurement on system B and obtain a measurement outcome. This collapses the joint state ρAB into a specific state associated with the measurement outcome.
Discard the measurement outcome and consider only the resulting state of system A.
The key point is that the reduced density matrix ρA obtained by performing the partial trace and the resulting state of system A after discarding the measurement outcome should be equivalent. This equivalence arises due to the properties of entanglement and the way composite quantum systems behave when subsystems are traced out or measurements are performed.
It's important to note that the equivalence between partial trace and measuring and discarding is a fundamental concept in quantum mechanics, supported by mathematical reasoning and consistent with experimental observations. It has been extensively studied and verified in the field of quantum information theory.