The quantum switch is uniquely defined by its action on unitary operations

Abstract

The quantum switch is a quantum process that creates a coherent control between different unitary operations, which is often described as a quantum process which transforms a pair of unitary operations $(U_1,U_2)$ into a controlled unitary operation that coherently applies them in different orders as $|0⟩⟨0|\otimes U_1{U}_2+|1⟩⟨1|\otimes U_2{U}_1$. This description, however, does not directly define its action on non-unitary operations. The action of the quantum switch on non-unitary operations is then chosen to be a ``natural'' extension of its action on unitary operations. In general, the action of a process on non-unitary operations is not uniquely determined by its action on unitary operations. It may be that there could be a set of inequivalent extensions of the quantum switch for non-unitary operations. We prove, however, that the natural extension is the only possibility for the quantum switch for the 2-slot case. In other words, contrary to the general case, the action of the quantum switch on non-unitary operations (as a linear and completely CP preserving supermap) is completely determined by its action on unitary operations. We also discuss the general problem of when the complete description of a quantum process is uniquely determined by its action on unitary operations and identify a set of single-slot processes which are completely defined by their action on unitary operations.

arXiv comment24 pages, 2 figures. Typos corrected, published version

References in corpus (30)

• Quantum correlations with no causal order
• Quantum computations without definite causal structure
• Theoretical framework for quantum networks
• Quantum Circuits Architecture
• Bloch vectors for qudits
• Transforming quantum operations: quantum supermaps
• Computational advantage from quantum-controlled ordering of gates
• Enhanced communication with the assistance of indefinite causal order
• Perfect discrimination of no-signalling channels via quantum superposition of causal structures
• Exponential Communication Complexity Advantage from Quantum Superposition of the Direction of Communication
• Causal and causally separable processes
• Experimental transmission of quantum information using a superposition of causal orders
• Unital Quantum Channels - Convex Structure and Revivals of Birkhoff's Theorem
• Indefinite causal order enables perfect quantum communication with zero capacity channels
• Communicating via ignorance: Increasing communication capacity via superposition of order
• Reversing Unknown Quantum Transformations: Universal Quantum Circuit for Inverting General Unitary Operations
• On the existence of physical transformations between sets of quantum states
• Quantum circuits with classical versus quantum control of causal order
• A purification postulate for quantum mechanics with indefinite causal order
• Cyclic Quantum Causal Models
• Probabilistic exact universal quantum circuits for transforming unitary operations
• Extending quantum operations
• Quantum operations with indefinite time direction
• Deterministic Quantum State Transformations
• Complex conjugation supermap of unitary quantum maps and its universal implementation protocol
• When do pieces determine the whole? Extremal marginals of a completely positive map
• Success-or-Draw: A Strategy Allowing Repeat-Until-Success in Quantum Computation
• Consequences of preserving reversibility in quantum superchannels
• Uniqueness of the joint measurement and the structure of the set of compatible quantum measurements
• Implementing positive maps with multiple copies of an input state

Cited by in corpus (4)

• Experimental Aspects of Indefinite Causal Order in Quantum Mechanics
• Higher-order Process Matrix Tomography of a passively-stable Quantum SWITCH
• Emergent non-Markovianity and dynamical quantification of the quantum switch
• Simulating the quantum switch with quantum circuits is computationally hard