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arXiv:0802.4248 · Physical Review A

Coexistence of qubit effects

Peter Stano, Daniel Reitzner, Teiko Heinosaari

Published Jul 10, 2008• 60 citations
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Abstract

We characterize all coexistent pairs of qubit effects. This gives an exhaustive description of all pairs of events allowed, in principle, to occur in a single qubit measurement. The characterization consists of three disjoint conditions, which are easy to check for a given pair of effects. Known special cases are shown to follow from our general characterization theorem.

DOI
10.1103/physreva.78.012315

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We characterize all coexistent pairs of qubit effects. This gives an exhaustive description of all pairs of events allowed, in principle, to occur in a single qubit measurement. The characterization consists of three disjoint conditions, which are easy to check for a given pair of effects. Known special cases are shown to follow from our general characterization theorem.
The coexistence relation of quantum effects is a fundamental structure, describing those pairs of experimental events that can be implemented in a single setup. Only in the simplest case of qubit effects is an analytic characterization of coexistent pairs known. We generalize the qubit coexistence characterization to all pairs of effects in arbitrary dimensions that belong to the von Neumann algebra generated by two projections. We demonstrate the presented mathematical machinery by several examples, and show that it covers physically relevant classes of effect pairs.
Abstract. The coexistence relation of quantum effects is a fundamental struc-ture, describing those pairs of experimental events that can be implemented in a single setup. Only in the simplest case of qubit effects an analytic charac-terization of coexistent pairs is known. We generalize the qubit coexistence characterization to all pairs of effects in arbitrary dimension that belong to the von Neumann algebra generated by two projections. We demonstrate the pre-sented mathematical machinery by several examples, and show that it covers physically relevant classes of effect pairs. 1.
One of the hallmarks of quantum theory is the realization that distinct measurements cannot in general be performed simultaneously, in stark contrast to classical physics. In this context the notions of coexistence and joint measurability are employed to analyze the possibility of measuring together two general quantum observables, characterizing different degrees of compatibility between measurements. It is known that two jointly measurable observables are always coexistent, and that the converse holds for various classes of observables, including the case of observables with two outcomes. Here we resolve, in the negative, the open question of whether this equivalence holds in general. Our resolution strengthens the notions of coexistence and joint measurability by showing that both are robust against small imperfections in the measurement setups.
I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii It is known that almost all 2-qubit gates are universal for quantum computing (Lloyd 1995; Deutsch, Barenco, Eckert 1995). However, an explicit characterization of non-universal 2-qubit gates is not known. We consider a closely related problem of characterizing the set of non-universal 2-qubit Hamiltonians. We call a 2-qubit Hamiltonian n-universal if, when applied on different pairs of qubits, it can be used to approximate any unitary operation on n qubits. It follows directly from the results of Lloyd and Deutsch, Barenco, Eckert, that almost any 2-qubit Hamiltonian is 2-universal. Our main result is a complete characterization of 2-non-universal 2-qubit Hamiltonians. There are three cases when a 2-qubit Hamiltonian H is not universal: (1) H shares an eigenvector with the gate that swaps two qubits; (2) H
A theory of quantum relations is proposed in which physical meaning is not exhausted by the structure visible on an instantaneous time slice. The central principle is that a quantum relation remains physically open while an admissible future operation can still complete, distinguish, or reroute that relation. Recorded local events remain fixed; relational topology can remain unresolved. The causal-relational closure principle is inspired by conceptual patterns from the Quantum Entanglement Spacetime Theory research programme, but no constructive derivation from Quantum Entanglement Spacetime Theory is assumed. Several established experiments become manifestations of one physical idea. Quantum erasure shows that later access to path information controls which interference relation becomes visible. Delayed-choice entanglement swapping shows that a later measurement can select entangled or separable conditional relations among already registered events. Entanglement swapping between photons that never coexisted shows that a quantum relation need not correspond to simultaneously existing endpoint systems. Localizable entanglement and measurement-based graph-state experiments show that weak pairwise signatures can coexist with strong future-realizable entanglement. These results provide retrospective empirical consistency tests, not exclusive confirmation over standard quantum mechanics. A new composite prediction follows: after endpoint events have been recorded, later Bell measurements on retained mediator systems can program which Bell-correlation graph is realized among those records, including pairings of endpoints that never coexisted. For 2m endpoints, (2m − 1)!! perfect-matching topologies are selectable in the ideal pairwise construction. The state vector remains a valid predictive representation, while instantaneous state structure need not coincide with final physically meaningful relational structure.