Quantum operations that cannot be implemented using a small mixed environment

Abstract

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^2-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.

LATEX, 8 pages

References in corpus (1)

• An Analysis of Completely-Positive Trace-Preserving Maps on 2x2 Matrices

Cited by in corpus (8)

• The resource theory of informational nonequilibrium in thermodynamics
• Atemporal diagrams for quantum circuits
• Channel kets, entangled states, and the location of quantum information
• Simulating a single qubit channel using a mixed state environment
• Quantum and classical resources for unitary design of open-system evolutions
• Restrictions on the Schmidt rank of bipartite unitary operators beyond dimension two
• How two spins can thermalize a third spin
• Some Bipartite States do not Arise from Channels