Quantum operations that cannot be implemented using a small mixed environment

To implement any quantum operation (a.k.a. “superoperator” or “CP map”) on a d-dimensional quantum system, it is enough to apply a suitable overall unitary transformation to the system and a d-dimensional environment which is initialized in a fixed pure state. It has been suggested that a d-dimensional environment might be enough if we could initialize the environment in a mixed state of our choosing. In this note we show with elementary means that certain explicit quantum operations cannot be realized in this way. Our counterexamples map some pure states to pure states, giving strong and easily manageable conditions on the overall unitary transformation. Everything works in the more general setting of quantum operations from d-dimensional to d′-dimensional spaces, so we place our counterexamples within this more general framework. 1 Quantum operations Quantum operations (see e.g. [3]) are also known as “superoperators”, “superscattering operators” or “completely positive maps” (“CP maps”). They can be viewed as a generalization of unitary transformations and are the most general transformations that can be applied to a quantum system in an unknown (possibly mixed) state. More precisely, quantum operations are the most general transformations that can be implemented deterministically, thus excluding operations which only succeed with a certain probability, like those depending on a measurement outcome. Under a quantum operation pure states are frequently mapped to mixed states. All quantum operations on a d-dimensional system can be implemented as the partial trace of a unitary operator acting on the system together with an auxiliary system (the “environment”). The question is how small an environment suffices to implement all possible quantum operations on a d-dimensional ∗Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1, e-mail: [email protected] †FX Palo Alto Laboratory, 3400 Hillview Avenue, Palo Alto, CA 94304, USA, e-mail: [email protected]