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arXiv:quant-ph/0103030 · Physical Review Letters

Virtual Quantum Subsystems

Paolo Zanardi A

Published Jul 25, 2001• 309 citations
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Abstract

The physical resources available to access and manipulate the degrees of freedom of a quantum system define the set A of operationally relevant observables. The algebraic structure of A selects a preferred tensor product structure, i.e., a partition into subsystems. The notion of compoundness for quantum systems is accordingly relativized. Universal control over virtual subsystems can be achieved by using quantum noncommutative holonomies

DOI
10.1103/physrevlett.87.077901

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The physical resources available to access and manipulate the degrees of freedom of a quantum system define the set A of operationally relevant observables. The algebraic structure of A selects a preferred tensor product structure, i.e., a partition into subsystems. The notion of compoundness for quantum systems is accordingly relativized. Universal control over virtual subsystems can be achieved by using quantum noncommutative holonomies
Dividing the world into subsystems is an important component of the scientific method. The choice of subsystems, however, is not defined a priori. Typically, it is dictated by experimental capabilities, which may be different for different agents. Here, we propose a way to define subsystems in general physical theories, including theories beyond quantum and classical mechanics. Our construction associates every agent A with a subsystem S A , equipped with its set of states and its set of transformations. In quantum theory, this construction accommodates the notion of subsystems as factors of a tensor product, as well as the notion of subsystems associated with a subalgebra of operators. Classical systems can be interpreted as subsystems of quantum systems in different ways, by applying our construction to agents who have access to different sets of operations, including multiphase covariant channels and certain sets of free operations arising in the resource theory of quantum coherence. After illustrating the basic definitions, we restrict our attention to closed systems, that is, systems where all physical transformations act invertibly and where all states can be generated from a fixed initial state. For closed systems, we show that all the states of all subsystems admit a canonical purification. This result extends the purification principle to a broader setting, in which coherent superpositions can be interpreted as purifications of incoherent mixtures.
Using the quantum two-body system as a familiar model, this talk will describe how entanglement can be used to select preferred observables for interrogating a physical system. The symmetries and dynamics of the quantum two-body system provide a backdrop for testing the relativity of entanglement with respect to observable-induced tensor product structures. We believe this exploration leads us to a general statement: the physically-meaningful observable subalgebras are the ones that minimize entanglement in typical states.
We describe a broad dynamical-algebraic framework for analyzing the quantum control properties of a set of naturally available interactions. General conditions under which universal control is achieved over a set of subspaces/subsystems are found in terms of the representation theory of the group UA{\mathcal{U}}_{A}UA​ of allowed quantum evolutions. In some cases universal control can be achieved in all the state-space sectors carrying an irreducible action of UA.{\mathcal{U}}_{A}.UA​. All known physical examples of universal control on subspaces/systems are related to the framework developed here. Implications for quantum-information processing are discussed.
Quantum theory typically presupposes a fixed factorisation of the total Hilbert space into subsystems, or an equivalent fixed choice of commuting subalgebras. This convention hard-codes a classical notion of boundary between "System A" and "System B". Recent work has formalised subsystem ambiguity by defining partitions directly at the level of C*-subalgebras, including non-tensor-factor subsystems, and by studying representability and the role of centres. In parallel, work under the label "quantum mereology" studies how subsystems can emerge top-down from spectral structure or from changes of tensor product structure. The aim here is different. We define RCC-style region relations directly from unitary invariants of pairs of sharp region projectors, so that boundary indefiniteness is a relation-level observable rather than a chosen decomposition. A quantum mereotopology is developed in which "regions" are represented by projection operators and their topological relations are defined intrinsically, without background space and without a privileged tensor product structure. The central object is the commutator of region projectors, whose spectrum and operator norms quantify the extent to which a putative boundary is definite or indefinite. A quantum extension of region connection reasoning is proposed in which the commuting regime yields a classical Boolean core of region relations, while a noncommuting regime captures superposed and dynamically entangled partitions through an explicit interface sector. This yields a coordinate-free separability criterion expressed purely in terms of region relations, and it provides a framework in which boundary fuzziness becomes an observable notion tied to measurement compatibility.