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Operational Resource Theory of Imaginarity
Kang-Da Wu, Tulja Varun Kondra, Swapan Rana, Carlo Maria Scandolo, Guo-Yong Xiang, Chuan-Feng Li, Guang-Can Guo, Alexander Streltsov
TL;DR
The paper examines whether complex numbers are essential to quantum mechanics and studies imaginarity using quantum resource-theory methods. It characterizes state conversion and approximate imaginarity distillation, then shows theoretically and experimentally that some real bipartite states can be perfectly distinguished by LOCC with general local measurements but not with real local measurements.
Problem
The paper asks whether complex numbers are essential to quantum mechanics and whether the theory can instead be formulated using real numbers only.
Source: p. 1Method
The authors use the resource theory of imaginarity to characterize state conversion and approximate distillation, and test local state discrimination experimentally with entangled photons and projective measurements.
Source: pp. 2, 4Results
Some real bipartite states are perfectly distinguishable by LOCC with general local measurements but cannot be distinguished with any nonzero probability using only real measurements.
Source: pp. 1, 4Takeaways & Limitations
The results show that complex numbers are an essential ingredient of quantum mechanics within the state-discrimination setting studied.
Source: pp. 2, 4
Abstract
from arXiv · showhide
Wave-particle duality is one of the basic features of quantum mechanics, giving rise to the use of complex numbers in describing states of quantum systems, their dynamics, and interaction. Since the inception of quantum theory, it has been debated whether complex numbers are actually essential, or whether an alternative consistent formulation is possible using real numbers only. Here, we attack this long-standing problem both theoretically and experimentally, using the powerful tools of quantum resource theories. We show that - under reasonable assumptions - quantum states are easier to create and manipulate if they only have real elements. This gives an operational meaning to the resource theory of imaginarity. We identify and answer several important questions which include the state-conversion problem for all qubit states and all pure states of any dimension, and the approximate imaginarity distillation for all quantum states. As an application, we show that imaginarity plays a crucial role for state discrimination: there exist real quantum states which can be perfectly distinguished via local operations and classical communication, but which cannot be distinguished with any nonzero probability if one of the parties has no access to imaginarity. We confirm this phenomenon experimentally with linear optics, performing discrimination of different two-photon quantum states by local projective measurements. These results prove that complex numbers are an indispensable part of quantum mechanics.
SUPPLEMENTAL MATERIAL
The supplement describes two-photon state preparation and local projective discrimination, including the restriction to real measurements and the conditions under which general measurements offer no advantage.
Experimental state preparation and discrimination
- The measurement module performs projective qubit measurements on each photon and records coincident outcomes for the local discrimination experiments.The experimental analysis considers four possible outcomes for the two local measurements.Source: SUPPLEMENTAL MATERIAL · p. 7
Local state discrimination
- The ideal success probability is expressed using the measurement directions α and β; real measurements impose α_y = β_y = 0, with the maximum then determined by a maximum singular value.In the experiments, δ_a = δ_b = 0 theoretically, giving p_succ = [1 + |β^T · (ΔE^T α)|]/2.Source: SUPPLEMENTAL MATERIAL · pp. 7–8
- A nonzero amount of imaginarity in local measurements gives an advantage over all real measurements.For one state family, however, general and real projective measurements have the same maximum guessing probability when p < 0.173.Source: SUPPLEMENTAL MATERIAL · p. 8
Experimental state preparation and discrimination
- The experiment measured maximum success probabilities for four classes of mixed states, using equal preparation probabilities w = w̃ = 0.5.The supplement describes preparing Bell, Werner, mixed two-qubit, and maximally correlated states; adjustable quartz plates prepare mixed and maximally correlated states.Source: SUPPLEMENTAL MATERIAL · pp. 7–8
Local state discrimination
- For the state family with threshold (m̃ − 1)/8 ≈ 0.173, general projective measurements provide no advantage over real projective measurements when p is below that threshold.The threshold is given as (m̃ − 1)/8, with m̃ = √(5 + 4 sin θ₀).Source: SUPPLEMENTAL MATERIAL · p. 8