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.
Imaginarity as a Resource within Quantum Coherence: Geometric Decomposition and Operational Conversion
School of Science, East China University of Technology（华东交通大学理学院）；School of Computer and Computing Science, Hangzhou City University（杭州城市大学计算机与计算科学学院）；School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）
Complex numbers are widely used in both classical and quantum physics, and are indispensable components for describing quantum systems and their dynamical behavior. Recently, the resource theory of imaginarity has been introduced, allowing for a systematic study of complex numbers in quantum mechanics and quantum information theory. In this work we develop theoretical methods for the resource theory of imaginarity, motivated by recent progress within theories of entanglement and coherence. We investigate imaginarity quantification, focusing on the geometric imaginarity and the robustness of im…
The use of imaginary numbers in modelling quantum mechanical systems encompasses the wave-like nature of quantum states. Here we introduce a resource theoretic framework for imaginarity, where the free states are taken to be those with density matrices that are real with respect to a fixed basis. This theory is closely related to the resource theory of coherence, as it is basis dependent, and the imaginary numbers appear in the off-diagonal elements of the density matrix. Unlike coherence however, the set of physically realizable free operations is identical to both completely resource non-gen…
Imaginarity Resource Theory of Gaussian Quantum Channels
School of Mathematical Science, Shanxi University（山西大学数学科学学院）；College of Mathematics, Taiyuan University of Technology（太原理工大学数学学院）；Key Laboratory of Complex Systems and Data Science of Shanxi University（山西大学复杂系统与数据科学重点实验室）
Uniqueness of imaginarity-assisted exact transformation from real orthogonal operations to arbitrary unitary operations
NTT Communication Science Laboratories, NTT Inc.（NTT通信科学实验室）；Mitsubishi Electric Corporation（三菱电机株式会社）；NTT Research Center for Theoretical Quantum Information, NTT Inc.（NTT理论量子信息研究中心）
Nonlocal advantage of quantum imaginarity in Schwarzchild spacetime
School of Mathematics and Systems Science, Guangdong Polytechnic Normal University（广东技术师范大学数学与系统科学学院）；School of Mathematics and Statistics, Central China Normal University（华中师范大学数学与统计学院）；School of Artificial Intelligence, Jianghan University（江汉大学人工智能学院）；School of Computer Science and Technology, Dongguan University of Technology（东莞理工学院计算机科学与技术学院）；School of Mathematics and Statistics, Hainan Normal University（海南师范大学数学与统计学院）
Indian Institute of Technology Jodhpur（印度理工学院乔德普尔分校）；Institute of High Energy Physics, Chinese Academy of Sciences（中国科学院高能物理研究所）；Kaiping Neutrino Research Center（开平中微子研究中心）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）
Experimental certification of the nonlocal advantages of quantum imaginarity
College of Physics and Optoelectronic Engineering, Ocean University of China（中国海洋大学物理与光电工程学院）；China Mobile (Suzhou) Software Technology Company Limited（中国移动（苏州）软件技术有限公司）；Engineering Research Center of Advanced Marine Physical Instruments and Equipment (Ministry of Education), Ocean University of China（中国海洋大学先进海工高技术装备教育部工程研究中心）；Qingdao Key Laboratory of Advanced Optoelectronics, Ocean University of China（中国海洋大学青岛先进光电子学重点实验室）
One of the central problems in the study of quantum resource theories is to provide a given resource with an operational meaning, characterizing physical tasks in which the resource can give an explicit advantage over all resourceless states. We show that this can always be accomplished for all convex resource theories. We establish in particular that any resource state enables an advantage in a channel discrimination task, allowing for a strictly greater success probability than any state without the given resource. Furthermore, we find that the generalized robustness measure serves as an exa…
We introduce the resource quantifier of weight of resource for convex quantum resource theories of states with arbitrary resources. We show that it captures the advantage that a resourceful state offers over all possible free states, in the operational task of exclusion of subchannels. Furthermore, we introduce information-theoretic quantities related to exclusion and find a connection between the weight of resource of a state, and the exclusion-type information of ensembles it can generate. These results provide support to a recent conjecture made in the context of convex quantum resource the…
The prevalent modus operandi within the framework of quantum resource theories has been to characterise and harness the resources within single objects, in what we can call \emph{single-object} quantum resource theories. One can wonder however, whether the resources contained within multiple different types of objects, now in a \emph{multi-object} quantum resource theory, can simultaneously be exploited for the benefit of an operational task. In this work, we introduce examples of such multi-object operational tasks in the form of subchannel discrimination and subchannel exclusion games, in wh…
Quantum resource theories have been widely studied to systematically characterize the non-classicality of quantum systems. Most resource theories focus on quantum states and study their interconversions. Although quantum channels are generally used as the tool for state manipulation, such a manipulation capability can be naturally regarded as a generalized quantum resource, leading to an open research direction in the resource theories of quantum channels. Various resource-theoretic properties of channels have been investigated, however, without treating channels themselves as operational reso…
Re-examining the Role of State Texture in Gate Identification and Fixed-Point Resource Theories
University of Toronto（多伦多大学）；Purdue University（普渡大学）；Purdue Quantum Science and Engineering Institute（普渡量子科学与工程研究所）；Oak Ridge National Laboratory（橡树岭国家实验室）；DEVCOM Army Research Laboratory（德弗科姆陆军研究实验室）；Tulane University（杜兰大学）
A fundamental approach for the characterization and quantification of all kinds of resources is to study the conversion between different resource objects under certain constraints. Here we analyze, from a resource-non-specific standpoint, the optimal efficiency of resource formation and distillation tasks with only a single copy of the given quantum state, thereby establishing a unified framework of one-shot quantum resource manipulation. We find general bounds on the optimal rates characterized by resource measures based on the smooth max/min-relative entropies and hypothesis testing relativ…
We establish a set of resource conversion relationships between quantum coherence and multipartite entanglement in many-body systems, where the operational measures of resource cost and distillation are focused. Under the multipartite incoherent operation, the initial coherence of single-party system bounds the maximal amounts of corresponding operational entanglement in an arbitrary bipartite partition as well as the genuine multipartite entanglement in many-body systems. Furthermore, the converted multipartite entanglement can be transferred to its subsystems and restored to coherence of a s…
The diverse range of resources which underlie the utility of quantum states in practical tasks motivates the development of universally applicable methods to measure and compare resources of different types. However, many of such approaches were hitherto limited to the finite-dimensional setting or were not connected with operational tasks. We overcome this by introducing a general method of quantifying resources for continuous-variable quantum systems based on the robustness measure, applicable to a plethora of physically relevant resources such as optical nonclassicality, entanglement, genui…
In order to analyze non-Markovianity of tripartite quantum states from a resource theoretical viewpoint, we introduce a class of quantum operations performed by three distant parties, and investigate an operational resource theory (ORT) induced it. A tripartite state is a free state if and only if it is a quantum Markov chain. We prove monotonicity of functions such as the conditional quantum mutual information, intrinsic information, squashed entanglement, a generalization of entanglement of purification, and the relative entropy of recovery. The ORT has five bound sets, each of which corresp…
Genuinely quantum states of a harmonic oscillator may be distinguished from their classical counterparts by the Glauber-Sudarshan P-representation -- a state lacking a positive P-function is said to be nonclassical. In this paper, we propose a general operational framework for studying nonclassicality as a resource in networks of passive linear elements and measurements with feed-forward. Within this setting, we define new measures of nonclassicality based on the quantum fluctuations of quadratures, as well as the quantum Fisher information of quadrature displacements. These lead to fundamenta…
Robustness- and weight-based resource measures without convexity restriction: Multicopy witness and operational advantage in static and dynamical quantum resource theories
