Abstract
Measurements in quantum theory exhibit incompatibility, i.e., they can fail to be jointly measurable. An intuitive way to represent the (in)compatibility relations among a set of measurements is via a hypergraph representing their joint measurability structure: its vertices represent measurements and its hyperedges represent (all and only) subsets of compatible measurements. Projective measurements in quantum theory realize (all and only) joint measurability structures that are graphs. On the other hand, general measurements represented by positive operator-valued measures (POVMs) can realize arbitrary joint measurability structures. Here we explore the scope of joint measurability structures realizable with qubit POVMs. We develop a technique that we term marginal surgery to obtain nontrivial joint measurability structures starting from a set of compatible measurements. We show explicit examples of marginal surgery on a special set of qubit POVMs to construct joint measurability structures such as N-cycle and N-Specker scenarios for any integer N≥3. We also show the realizability of various joint measurability structures with N∈{4,5,6} vertices. In particular, we show that all possible joint measurability structures with N=4 vertices are realizable. We conjecture that all joint measurability structures are realizable with qubit POVMs. This contrasts with the unbounded dimension required in R. Kunjwal et al. [Phys. Rev. A 89, 052126 (2014)]. Our results also render this previous construction maximally efficient in terms of the required Hilbert space dimension. We also obtain a sufficient condition for the joint measurability of any set of binary qubit POVMs which powers many of our results and should be of independent interest.25 MoreReceived 13 March 2020Revised 4 August 2020Accepted 5 October 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.043147Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum information processingQuantum Information
Highlights
A fundamental sense in which quantum theory departs from classical physics is the existence of incompatible measurements in the former
All joint measurability structures are realizable with qubit positive operatorvalued measures (POVMs)
Critical to the constructions in this paper is the method of marginal surgery that lets us use a joint POVM for some given set of compatible POVMs to construct nontrivial joint measurability structures
Summary
A fundamental sense in which quantum theory departs from classical physics is the existence of incompatible measurements in the former. [9], note that it requires a steadily growing Hilbert space dimension to realize joint measurability structures corresponding to ever larger sets of measurements, rendering it quite inefficient in this sense This raises the natural question of whether it is possible to realize joint measurability structures for arbitrarily large sets of measurements using the smallest possible Hilbert space dimension, i.e., using qubit POVMs. All of the considerations above motivate the present study as a first step towards addressing general features of joint measurability in quantum theory for systems of limited Hilbert space dimension.
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