Kraus operators and complete positivity

Assume a fixed environment state initially independent of the system, and a joint unitary. For a pure environment preparation, insert an orthonormal environment basis into the final partial trace. Each matrix element of the joint unitary between the initial environment vector and an output basis vector is an operator acting on the system alone. These are Kraus operators. A mixed environment can be purified, or its initial classical weights can be absorbed into an enlarged list of operators.

The construction relies on one environment preparation for every allowed input. If preparation procedures correlate system and environment differently for different inputs, a single channel on the entire system state space need not describe them. Conversely, every finite-dimensional completely positive, trace-preserving map has a dilation with a sufficiently large initially uncorrelated environment. The dilation is a mathematical representation, not a unique microscopic reconstruction: distinct environments, bases, and interactions can produce identical system input–output statistics.

Each term in a Kraus sum is positive when the input is positive: its expectation in any vector is the input expectation in a transformed vector. The same argument works after tensoring every Kraus operator with the identity of an arbitrary reference system. This is complete positivity. Ordinary positivity tests only the system, whereas complete positivity also protects probabilities when that system is one member of an entangled pair. Trace preservation alone says nothing about either positivity test.

Transposition illustrates the distinction. It preserves trace and the eigenvalues of each individual density matrix, so it is positive. Yet transposing only one half of a Bell-state projector produces a matrix with a negative eigenvalue. Therefore transposition cannot be a channel acting independently on an unknown member of an arbitrary entangled pair. This obstruction is stronger than saying the map is unfamiliar experimentally; it is a direct inconsistency with nonnegative joint measurement probabilities.

For a deterministic operation, cyclicity of trace turns output normalization into the operator identity in the first display. Reversing the order of the operators would test unitality instead, the preservation of the identity operator. Amplitude damping is trace preserving but not unital, so these conditions cannot be exchanged. A selected measurement outcome may instead have a trace-decreasing completely positive map; its output trace is the probability of that outcome before the conditional state is normalized.

Kraus decompositions are not unique. Unitary mixing of a fixed list leaves its summed channel unchanged, although a chosen interpretation of individual environment records can change. A random phase-flip channel is a simple example: apply the identity with one probability and a Pauli phase flip with the complementary probability. Each branch is unitary, but forgetting which branch occurred generally produces a mixed state. No single pure-state Bloch trajectory represents that averaged output for arbitrary input states.