Tracepreserving

Trace-preserving is a property of linear maps on operators, particularly in quantum information theory, where it means that the map does not alter the trace of an operator. If a map E acts on density operators, trace preservation ensures that Tr(E(ρ)) = Tr(ρ) for all ρ. Since density matrices have trace equal to one, trace-preserving maps send valid states to valid states without changing the total probability.

In finite dimensions, a common representation is the Kraus form. A completely positive trace-preserving (CPTP) map E can be written as E(ρ) = ∑_i K_i ρ K_i†, where the Kraus operators K_i satisfy the completeness relation ∑_i K_i† K_i = I. This condition guarantees Tr(E(ρ)) = Tr(ρ) for all ρ. Unitary channels, with a single Kraus operator U, are trace-preserving since U†U = I.

Trace-preserving maps are central to the description of quantum channels, which represent physical evolutions of quantum states. They guarantee that probabilities across all outcomes sum to one after the operation. Not all trace-preserving maps are physically implementable; a physically valid quantum operation must be completely positive in addition to trace-preserving. When the map is trace-decreasing instead, it corresponds to operations with post-selection or probabilistic filtering, where some outcomes are discarded and the resulting state is conditioned on a successful event.

Overall, trace preservation is a necessary attribute for quantum state evolution to conserve total probability, and it plays a foundational role alongside complete positivity in characterizing allowable quantum operations.