Quantum Entanglement and Cryptography for Automation and Control of Dynamic Systems
Farbod Khoshnoud, Ibrahim I. Esat, Shayan Javaherian, Behnam Bahr
Quantum Entanglement and Cryptography for Automation and Control of Dynamic Systems
Farbod K HOSHNOUD , Ibrahim I. E
SAT , Shayan J AVAHERIAN , Behnam B AHR Department of Electromechanical Engineering Technology, College of Engineering, California State Polytechnic University, Pomona, CA 91768, USA Center for Autonomous Systems and Technologies, Department of Aerospace Engineering, California Institute of Technology, Pasadena, CA 91125, USA Department of Mechanical and Aerospace Engineering, Brunel University London, Uxbridge UB8 3PH, United Kingdom Department of Mechanical Engineering, College of Engineering, California State Polytechnic University, Pomona, CA 91768, USA
Abstract:
This paper addresses the application of quantum entanglement and cryptography for automation and control of dynamic systems. A dynamic system is a system where the rates of changes of its state variables are not negligible. Quantum entanglement is realized by the
Spontaneous Parametric Down-conversion process. Two entangled autonomous systems exhibit correlated behavior without any classical communication in between them due to the quantum entanglement phenomenon. Specifically, the behavior of a system, Bob, at a distance, is correlated with a corresponding system, Alice. In an automation scenario, the “Bob Robot” is entangled with the “Alice Robot” in performing autonomous tasks without any classical connection between them. Quantum cryptography is a capability that allows guaranteed security. Such capabilities can be implemented in control of autonomous mechanical systems where, for instance, an “Alice Autonomous System” can control a “Bob Autonomous System” for applications of automation and robotics. The applications of quantum technologies to mechanical systems, at a scale larger than the atomistic scale, for control and automation, is a novel contribution of this paper. Notably, the feedback control transfer function of an integrated classical dynamic system and a quantum state is proposed.
Keywords:
Quantum entanglement, Quantum multibody dynamics, Quantum cooperative robotics, Quantum cryptography, Quantum autonomy.
In this Quantum era, quantum technologies are making the biggest and the most promising impacts in developing new technologies towards improving the quality of life. The applications of quantum technologies in our future everyday lives are inevitable due to their many unmatched advantages. By integrating the unmatched possibilities of quantum supremacy with engineering applications, such interdisciplinary quantum and engineering systems can push the engineering boundaries beyond any classical technique. However, currently, the most focus on the application of quantum technologies in engineering systems is on quantum computing and related areas (For example, see [1], [4]). Until recently, a Quantum Robot is referred to as a mobile autonomous platform that is equipped with a quantum computer, as its processing system (See for example, [3]). Also, the advantages of quantum technologies have not been realized and introduced at the macro scale. Very few references can be found on the application of quantum capabilities to actual mechanical systems. The applications of quantum technologies in advancing the performance of mechanical systems (at the macro-scale) are found in the literature merely on developing novel sensors and actuators ([5]-[15]), and quantum information processing systems, with optics-based distributed networks (See for example, [16]), optical switching, and a self-correcting quantum memory, for developing quantum devices that can control themselves [16]. The applications of quantum optics in controls [17]-[20], feedback systems [21]-[22], and programmable logic devices in quantum optics [23] have shown significant advances. An integration of experimental quantum interference with soft computing as an experimental quantum-enhanced stochastic simulation device can execute a simulation using less memory than possible by classical means [24]. Although there is rich literature in areas related to “quantum engineering” (e.g., [16]-[24]), the actual integration of such technologies with macro-scale mechanical systems, for instance in robotics, incorporation of autonomy, and autonomous systems (e.g., unmanned systems), has not been addressed, despite its importance and its capacity for potential technological breakthrough. Optics technologies are used in quantum mechanics, where single photons (or alternatively electrons, or sound energy levels [25]) are considered in the study of a quantum system. Similar, already well-developed technologies in optics and photonics areas are utilized in quantum mechanics. For instance, classical free-space optical communication tools in the autonomy of unmanned systems [26] (although not in quantum context), can be implemented towards the development of quantum engineering systems. The applications of the principles of quantum mechanics integrated with the laws of analytical mechanics in analyzing the motion of elementary particles (for example, see [27]), such as molecular dynamics topics, are readily available. However, the problem of multibody dynamical systems (at the macro-scale) integrated with quantum mechanics (for instance, where the quantum control of the system is of interest) has not been addressed yet. The research by the authors of this paper, and their earlier work (see for example, [48]-[50]) is the first effort towards the theoretical and experimental research and establishing the interdisciplinary field of Quantum Multibody Dynamics ([28], [48]-[50]). Recent advances in experimental and theoretical quantum systems allows: Secure quantum communication code, with no classical communication [29], quantum correlations over more than 10 km [30], Entanglement in noisy quantum channels [31], Entanglement-based quantum communication over 144 km [32], free-space entangled photon distributions in long distances [33], and Secured quantum key distribution with entangled photons [34]. A quantum-inspired approach has been proposed to solve problems of two robotic agents finding each other or pushing an object ([35], [36]), without any knowledge of each other. Furthermore, a research work on Psi Intelligent Control [37], inspired by precognition, has led to an approach in a quantum entanglement-based autonomy concept for autonomous vehicles [38]. Recent technological advances in quantum mechanics make such experiments feasible with considerable cost and size reductions [39]-[44]. On the other hand, reconfigurable Quantum Key Distribution networking [45] techniques allow free-space quantum communication over significant distances and overcomes signal degradation issues (e.g., due to weather events), where, unconditionally secure bit commitment by transmitting measurement outcomes is possible to attain perfect security ([46], [47]). Collectively, such technology resources and potential capacities allow us to apply quantum capabilities in engineering applications effectively, particularly in robotics and autonomous domains. Developing quantum capabilities in the mechanical system domain pushes the engineering boundaries beyond any existing classical technique, which gives advantageous and unmatched capabilities, such as the possibility of entangling the robotic agents in a distributed robotic system, quantum superposition, and guaranteed security. The present paper introduces the integration of quantum technologies with engineering systems (in a physical system domain), where the quantum advantages are applied to robots and autonomous systems in cooperative multi-agent robotic network scenarios. It demonstrates how experimental quantum entanglement and cryptography can be integrated with engineering systems (for example, how quantum can be used in control of a mechanical system). To the authors’ knowledge, this paper, and their recent work ([48]-[50]), are the first attempts in establishing the integration of experimental quantum capabilities with multibody dynamic systems (such as multi-agent robotic problems). A goal for such integrated quantum engineering approach is to demonstrate the interface of Quantum Technologies with Engineering Systems for research and educational purposes. The authors introduced the concept of experimental quantum cryptography and entanglement in robotics for the first time in the references [48]-[50]. The concepts of Quantum Multibody Dynamics, Robotics, Controls, and Autonomy are integrated with, and take advantage of, quantum technologies here. The paper is organized as follows. Section 2 presents an automation scenario, and Section 3 addresses the classical dynamics and controls aspects of such problems. Sections 4-6 give an introduction to quantum mechanics (to allow the readers to use the paper as a self-contained article). In Sections 7, the control problem for autonomy in the context of quantum states as the integrated quantum and classical feedback control transfer function for the autonomy of mechanical systems, at nonatomistic scales, is proposed, for the first time. Sections 8 and 9 discuss the quantum cryptography and quantum entanglement, respectively, and how they are implemented for automation and robotics applications. Automation
The aim of the present paper to introduce a technique to enhance autonomy through quantum technologies. An integrated automation and robotics system is shown in Fig. 1. This system consists of autonomous mobile platforms, robotic manipulators, and an automation facility equipped with actuators, motion detection sensors, and microcontrollers. The autonomy of such systems enhanced by quantum entanglement and quantum cryptography, is discussed in the following sections. Dynamics of the Autonomous Platforms
The equations of motion of an autonomous system containing robotic platforms and manipulators are presented now. A schematic representation of the system is shown in Fig. 2. The moment of inertia matrix of the vehicle in principal axes is given by, 𝐼 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 = [𝐼 𝑥𝑥 𝑦𝑦
00 0 𝐼 𝑧𝑧 ] The Euler angles for the sequence of rotations from the inertial frame, 𝐼 , to the body frame, 𝐵 , can be given by yaw, then pitch, and then roll, with yaw rotation 𝜃 𝑧 about 𝑧 𝐼 , in the inertial frame, or in mathematical form 𝒓 = 𝐇 𝐼1 𝒓 𝐼 , pitch rotation 𝜃 𝑦 about 𝑦 , in the intermediate frame, or 𝒓 = 𝐇 𝒓 , and roll rotation 𝜃 𝑥 about 𝑥 , in the body frame, as 𝒓 𝐵 = 𝐇 (𝜃 𝑥 )𝒓 . The three-angle rotation matrix is the product of 3 single-angle rotation matrices, as, 𝐇 𝐼𝐵 = 𝐇 (𝜃 𝑥 )𝐇 (𝜃 𝑦 )𝐇 𝐼1 (𝜃 𝑧 ) where 𝐇 𝐼𝐵 (𝜃 𝑥 , 𝜃 𝑦 , 𝜃 𝑧 ) is the rotation matrix of the inertial frame with respect to the body frame, and [𝐇 𝐼𝐵 ] −1 =[𝐇 𝐼𝐵 ] 𝑇 = 𝐇 𝐵𝐼 ; and 𝐇 𝐵𝐼 𝐇 𝐼𝐵 = 𝐇 𝐼𝐵 𝐇 𝐵𝐼 = 𝐈 . The relationship between the Euler-angle rates and body-axis rates is expressed as, [𝑝𝑞𝑟] = [1 0 −sin 𝜃 𝑦 𝑥 sin 𝜃 𝑥 cos 𝜃 𝑦 𝑥 cos 𝜃 𝑥 cos 𝜃 𝑦 ] [𝜃 𝑥 ̇𝜃 𝑦 ̇𝜃 𝑧 ̇ ] = 𝐋 𝑰𝑩 𝚯̇ Or [𝜃 𝑥 ̇𝜃 𝑦 ̇𝜃 𝑧 ̇ ] = 𝐋 𝑩𝑰 𝝎 𝐵 where 𝜃 𝑧 ̇ is measured in the inertial frame, 𝜃 𝑦 ̇ is measured in the intermediate frame, and 𝜃 𝑥 ̇ is measured in the body frame. Here, 𝚯̇ denotes the Euler angle rate vector, and 𝝎 𝐵 is the angular velocity vector in the body frame. Fig. 1 An integrated automation and robotic system.
Mobile Robot Mobile Robot Robotic manipulator An automation facility Conveyor belt Laser and a polarizer Manipulator Gimbal
Fig. 2 A generalized automation system and robotic platform.
The rate of change of the translational position is obtained by 𝐫̇ 𝐼 (𝑡) = 𝐇 𝑩𝑰 (𝑡)𝒗 𝑩 (𝑡) where 𝐫 𝐼 = [𝑥𝑦𝑧] 𝐼 is the translational position, and 𝒗 𝑩 =[𝑢𝑣𝑤] is the translational velocity in the body frame. The rate of change of angular position is 𝚯̇ (𝑡) = 𝐋 𝑩𝑰 (𝑡)𝝎 𝐵 (𝑡) The equaion of motion can be written in terms of the rate of change of translational velocity as, 𝒗̇ 𝑩 (𝑡) = 1𝑚(𝑡) 𝐅 𝐵 (𝑡) + 𝐇 𝐼𝐵 (𝑡)𝐠 𝐼 − 𝝎̃ 𝐵 (𝑡)𝒗 𝑩 (𝑡) (1) and the rate of change of angular velocity is, 𝝎̇ 𝐵 (𝑡) = 𝑰 𝐵−1 (𝑡)[𝐌 𝐵 (𝑡) − 𝝎̃ 𝐵 (𝑡)𝑰 𝐵 (𝑡)𝝎 𝐵 (𝑡)] (2) where 𝝎̃ is the cross product equivalent matrix of 𝝎 given by, 𝝎̃ = [ 0 −𝜔 𝑧 𝜔 𝑦 𝜔 𝑧 𝑥 −𝜔 𝑦 𝜔 𝑥 and 𝐇̇ 𝐼𝐵 = −𝝎̃ 𝐵 𝐇 𝐼𝐵 ; 𝐇̇ 𝐵𝐼 = 𝝎̃ 𝐼 𝐇 𝐵𝐼 . The external forces include the drag and propulsion forces in the body frame as, 𝐅 𝐵 = 𝐅 𝐵,𝑑𝑟𝑎𝑔 + 𝐅
𝐵,𝑝𝑟𝑜𝑝𝑢𝑙 = [𝑋 𝑑𝑟𝑎𝑔 + 𝑋 𝑝𝑟𝑜𝑝𝑢𝑙 𝑌 𝑑𝑟𝑎𝑔 + 𝑌 𝑝𝑟𝑜𝑝𝑢𝑙 𝑍 𝑑𝑟𝑎𝑔 + 𝑍 𝑝𝑟𝑜𝑝𝑢𝑙 ] 𝐵 (3) and the applied moments in body frame are 𝐌 𝐵 = 𝐌 𝐵,𝑑𝑟𝑎𝑔 + 𝐌
𝐵,𝑝𝑟𝑜𝑝𝑢𝑙 = [ 𝐿 𝑑𝑟𝑎𝑔 + 𝐿 𝑝𝑟𝑜𝑝𝑢𝑙 𝑀 𝑑𝑟𝑎𝑔 + 𝑀 𝑝𝑟𝑜𝑝𝑢𝑙 𝑁 𝑑𝑟𝑎𝑔 + 𝑁 𝑝𝑟𝑜𝑝𝑢𝑙 ] 𝐵 (4) The propulsion forces are generalized here, which can represent the driving force of a ground vehicle and the thrust of the propellers of an aerial vehicle [51], [52]. In the case of a mobile ground platform, as in Fig. 2, the vertical translations are due to the road roughness. The motion of the end effector in Fig. 2 with respect to the reference frame, 𝑅 (located at the base of the manipulator), can be obtained by, 𝑇 𝐸 𝑒𝑓𝑓 𝑅 = 𝑇 𝑇 𝑇 ⋯ 𝑇 𝑓𝑛𝑓𝑛−1 𝑇 𝐸 𝑒𝑓𝑓 𝑓𝑛 where 𝑇 𝐸𝑟𝑓 is the transformation matrix of frame 𝐸 relative to frame 𝑓𝑛 , and 𝑓𝑛 denotes the frame of link 𝑛 . If the position and orientation of the end effector are given by the vectors 𝑷 𝐸 and 𝑶 𝐸 relative to the reference frame 𝑅 , attached to the mobile vehicle, we have, [𝑷 𝐸 𝑶 𝐸 ] 𝐼 = [𝑷 𝐸 𝑶 𝐸 ] 𝑅 + [ 𝐫 𝐼 𝚯 𝐼 ] The control of the robotic platforms in a multi-agent 𝑥 𝐵 𝑧 𝐵 𝜷 𝜷 𝜷 𝑖 𝜔 𝑝1 𝜔 𝑝2 𝜔 𝑝𝑖 Hull boundary 𝑇 𝑇 𝑇 𝑖 𝑝, 𝜙 𝑟 Propellers 𝑦 𝐵 𝑞 𝜶 𝑓 𝜶 𝑓 𝜶 𝑖 𝑓 𝑖 𝑙 𝑙 Robotic arm
End effector
Mobile Ground platform
Manipulator reference
Automated manufacturing (Conveyers, sensors, actuators, etc.) networked scenario is discussed now. In a multi-agent robotic system scenario, various control tasks may be desirable, such as formations, swarm and cooperative physical tasks. In performing robotic tasks, a common problem is involved with the kinematic synchronization of the robotic platforms in a network of autonomous systems, in a dynamic environment. For instance, in a formation problem for a group of robotic platforms (or unmanned autonomous systems), maintaining the reference relative positions between the agents is required. Such formation may be defined by, lim 𝑡→∞ ‖(𝐫
𝐼,𝑖 (𝑡) − 𝒓 𝐿 (𝑡)) − 𝒅 𝑖 (𝑡)‖ = 0 where 𝐫 𝐼,𝑖 is the position of the robotic agent 𝑖 , 𝒓 𝐿 is the position of a reference agent, and 𝒅 𝑖 is the desired position vector of robot 𝑖 with respect to the reference or another agent. A controller 𝑔(𝑘 𝑝𝑖𝑑 ) can now be implemented to realize the required task, by controlling the actuation forces of the individual robots that drive each agent 𝑖 , with a strategy given by, ∑ 𝑎 𝑖𝑗𝑁+1𝑗=1 {𝑔(𝑘 𝑝𝑖𝑑 ) [( 𝐫 𝐼,𝑖 (𝑡) − 𝐫
𝐼,𝑗 (𝑡)) − 𝒅 𝑖 (𝑡)]} where 𝑎 𝑖𝑗 is equal to 1 if an agent 𝑖 is connected to 𝑗 , and it is equal to zero otherwise. The Quantum State Vector
The spin state of a particle is denoted by |+𝐳⟩ with the value of intrinsic angular momentum in the 𝑧 direction, 𝑆 𝑧 , equal to ℏ 2⁄ , where ℏ = ℎ 2𝜋⁄ = 1.055 × -27 erg ∙ s = 6.582 × -16 eV ∙ s, and ℎ is the Planck’s constant. In classical physics, a coordinate system with the basis vectors 𝐢 , 𝐣 , and 𝐤 is applied in representing vectors. The quantum representation of vectors is realized based on the assumption that if a (spin- ) particle is sent to a magnetic field in the z direction we obtain only the values ℏ 2⁄ and − ℏ 2⁄ , corresponding to the particle ending up in the state |+𝐳⟩ , and ending down in the state |−𝐳⟩ , respectively. These two states can be considered as the vectors in the quantum mechanical vector space. In this vector system, ⟨−𝐳|+𝐳⟩ = 0 implies that a particle in the state |+𝐳⟩ has an amplitude of zero to be in the state |−𝐳⟩ . This is analogous to the orthogonal property in classical physics (e.g., for an electric field, or a force field) where, for instance, 𝐢 ∙ 𝐣 = 0 . Similarly, the amplitude of a particle in the state |+𝐳⟩ to be in the state |+𝐳⟩ is equal to one, or ⟨+𝐳|+𝐳⟩ = 1 (the analogy is 𝐢 ∙ 𝐢 = 1 in classical Physics. If a particle initially in the state |+𝐱⟩ passes through a magnetic field in the z direction, it can attain the amplitudes 𝑐 + and 𝑐 − in the states |+𝐳⟩ and |−𝐳⟩ , respectively. Therefore, the state |+𝐱⟩ can be expressed as a linear combination of the states |+𝐳⟩ and |−𝐳⟩ by, |+𝐱⟩ = 𝑐 + |+𝐳⟩ + 𝑐 − |−𝐳⟩ (5) Assume an arbitrary spin state |𝜓⟩ , created by sending a beam of particles with the intrinsic spin- through an inhomogeneous magnetic field oriented in some arbitrary direction. If the measurement of the intrinsic angular momentum for an arbitrary state |𝜓⟩ in 𝑧 direction, 𝑆 𝑧 , is made, this state can be expressed by, |𝜓⟩ = 𝑐 + |+𝐳⟩ + 𝑐 − |−𝐳⟩ (6) where the arbitrary state |𝜓⟩ is in the superposition of the states |+𝐳⟩ and |−𝐳⟩ with amplitudes of 𝑐 + and 𝑐 − , respectively. The amplitudes 𝑐 + and 𝑐 − depend on the orientation of the magnetic field that the particles pass through. For a ket vector |𝜓⟩ , a corresponding bra vector can be defined as, ⟨𝜓| . The amplitude of a particle in the spin state |𝜓⟩ to be in the state |𝜑⟩ can be shown by ⟨𝜑|𝜓⟩ (which is an inner product representation). Thus, for example, we can write, ⟨+𝐳|𝜓⟩ = 𝑐 + ⟨+𝐳|+𝐳⟩ + 𝑐 − ⟨+𝐳|−𝐳⟩ = 𝑐 + (7) ⟨−𝐳|𝜓⟩ = 𝑐 + ⟨−𝐳|+𝐳⟩ + 𝑐 − ⟨−𝐳|−𝐳⟩ = 𝑐 − (8) Substituting 𝑐 + and 𝑐 − from Equations (7) and (8) into Equation (6) we obtain, |𝜓⟩ = ⟨+𝐳|𝜓⟩|+𝐳⟩ + ⟨−𝐳|𝜓⟩|−𝐳⟩ (9) By placing the amplitudes ⟨+𝐳|𝜓⟩ , and ⟨−𝐳|𝜓⟩ after the kets in Equation (9), we have, |𝜓⟩ = |+𝐳⟩⟨+𝐳|𝜓⟩ + |−𝐳⟩⟨−𝐳|𝜓⟩ (10) The amplitudes are complex numbers, and therefore placing them before or after the kets in Equations (9) and (10) does not make any difference. An analogy to presenting the amplitudes and the states of the quantum form in a classical mechanics vector form, such as a force vector 𝐅 in a two-dimensional problem, is the force 𝐅 =𝐹 𝑥 𝐢 + 𝐹 𝑦 𝐣 , which is the same as 𝐅 = 𝐢𝐹 𝑥 + 𝐣𝐹 𝑦 . By definition, there exists a bra vector for each ket . In order to complete the quantum notation for the states, the bra vector is also represented similarly to the ket representation of the amplitudes and the states as follows: ⟨𝜓| = 𝑐 +′ ⟨+𝐳| + 𝑐 −′ ⟨−𝐳| (11) If the same procedure for kets as in Equations (7) to (10), is carried out for the bra representation in Equation (11), we have, ⟨𝜓|+𝐳⟩ = 𝑐 +′ ⟨+𝐳|+𝐳⟩ + 𝑐 −′ ⟨+𝐳|−𝐳⟩ = 𝑐 +′ (12) ⟨𝜓|−𝐳⟩ = 𝑐 +′ ⟨+𝐳|−𝐳⟩ + 𝑐 −′ ⟨−𝐳|−𝐳⟩ = 𝑐 −′ (13) and substituting the results in (12) and (13) into (11) leads to, ⟨𝜓| = ⟨𝜓|+𝐳⟩⟨+𝐳| + ⟨𝜓|−𝐳⟩⟨−𝐳| (14) In order to evaluate ⟨𝜓|𝜓⟩ , Equations (9) and (14) are combined, noting that ⟨+𝐳|+𝐳⟩ = 1 , ⟨−𝐳|+𝐳⟩ = 0 , ⟨−𝐳|−𝐳⟩ = 1 , ⟨+𝐳|−𝐳⟩ = 0 ⟨𝜓|𝜓⟩ = ⟨𝜓|+𝐳⟩⟨+𝐳|𝜓⟩ + ⟨𝜓|−𝐳⟩⟨−𝐳|𝜓⟩ (15) The term, ⟨𝜓|𝜓⟩ , which represents the amplitude of a particle in the spin state 𝜓 to be in the state 𝜓 , is equal to one, by definition. For a case that ⟨𝜓|+𝐳⟩ implies ⟨+𝐳|𝜓⟩ , and ⟨𝜓|−𝐳⟩ implies ⟨−𝐳|𝜓⟩ , which in fact is true by definition, we can write Equation (15) as, ⟨𝜓|𝜓⟩ = |⟨+𝐳|𝜓⟩| + |⟨−𝐳|𝜓⟩| = 1 (16) The term |⟨+𝐳|𝜓⟩| is referred to the probability of a particle in the state |𝜓⟩ being in the state |+𝐳⟩ when 𝑆 𝑧 is measured. Similarly, the statement for |⟨+𝐳|𝜓⟩| is: the probability of a particle in the state |𝜓⟩ being in the state |−𝐳⟩ when 𝑆 𝑧 is measured. Equation (16) is interpreted as the sum of the probabilities of finding the particle in the state |+𝐳⟩ or |−𝐳⟩ is equal to one . Quantum physics, therefore, unlike any classical mechanics expectation, states that the probability amplitudes of ⟨+𝐳|𝜓⟩ and ⟨−𝐳|𝜓⟩ can be nonzero simultaneously, where a particle can be in the superposition of the states |+𝐳⟩ and |−𝐳⟩ . Thus, measuring intrinsic angular momentum can result in non-zero probabilities for both 𝑆 𝑧 = ℏ 2⁄ and 𝑆 𝑧 = − ℏ 2⁄ simultaneously. In classical mechanics, we only expect to obtain one value for the angular momentum of a particle at a time. Rotation of Basis States
A spin state |𝜓⟩ of a spin- particle is developed when sending particles through a magnetic field oriented in an arbitrary direction. The amplitudes ⟨+𝐳|𝜓⟩ and ⟨−𝐳|𝜓⟩ give the projected values of |𝜓⟩ onto the states |+𝐳⟩ and |−𝐳⟩ , respectively (Equation (9)). As in a classical case for force vector, for instance, the amplitudes and the projected values are used in representing the vector 𝐅 , in a quantum scenario the projected values or the amplitudes are complex numbers representing a state |𝜓⟩ . In classical mechanics, a vector, for instance, force 𝐅 is shown as 𝐅 = 𝐹 𝑥 𝐢 + 𝐹 𝑦 𝐣 , or 𝐅 → (𝐹 𝑥 , 𝐹 𝑦 ) . Similarly, a quantum representation of a ket can be given by a column vector as, |𝜓⟩ 𝑆 𝑧 basis → (⟨+𝐳|𝜓⟩⟨−𝐳|𝜓⟩) = (𝑐 + 𝑐 − ) (17) Now in this vector form, the state |+𝐳⟩ for instance, can be given by, |+𝐳⟩ 𝑆 𝑧 basis → (⟨+𝐳|+𝐳⟩⟨−𝐳|+𝐳⟩) = (10) (18) and the state |−𝐳⟩ is represented by, |−𝐳⟩ 𝑆 𝑧 basis → (⟨+𝐳|−𝐳⟩⟨−𝐳|−𝐳⟩) = (01) (19) If a particle in the state |+𝐱⟩ is sent through a magnetic field in the 𝑧 direction, and measurements of 𝑆 𝑧 gives probabilities of 50% for both ℏ 2⁄ and − ℏ 2⁄ , we have, |⟨+𝐳|+𝐱⟩| = (20) |⟨−𝐳|+𝐱⟩| = (21) which allows to express Equation (5) in the vector form as, |+𝐱⟩ 𝑆 𝑧 basis → (⟨+𝐳|+𝐱⟩⟨−𝐳|+𝐱⟩) = 1√2 (11) (22) In order to write Equation (15) in a vector/matrix form, and accommodate the bra vector for ⟨𝜓| , the relation can be written as, ⟨𝜓|𝜓⟩ = (⟨𝜓|+𝐳⟩, ⟨𝜓|−𝐳⟩) (⟨+𝐳|𝜓⟩⟨−𝐳|𝜓⟩) (23) It can be seen that the appropriate vector form for a bra is a row vector to satisfy the matrix multiplication of the ket and bra as in (23), and thus, ⟨𝜓| 𝑆 𝑧 basis → (⟨𝜓|+𝐳⟩, ⟨𝜓|−𝐳⟩) (24) A rotation operator can be used to represent the transformation of a ket from one state to another. For instance, if the magnetic moment of a spin- particle in the state |+𝐳⟩ placed in a magnetic field is in the 𝑥 direction, the spin will rotate in the 𝑥 - 𝑧 plane and as time progresses, the particle will be in the state |+𝐱⟩ . This rotation of the state |+𝐳⟩ in the magnetic field pointed in the 𝑥 direction can be denoted by 𝑅̂ . The rotation can be given by, |+𝐱⟩ = 𝑅̂ ( 𝜋2 𝐣) |+𝐳⟩ (25) where 𝐣 indicates the unit vector along the axis of rotation y. Therefore, operator 𝑅̂ rotates |+𝐳⟩ about the unit vector 𝐣 with an angle of 𝜋2 . Such an operator rotates an arbitrary state of a spin- particle |𝜓⟩ , when placed in a magnetic field in the 𝑥 direction, as given by, 𝑅̂ (𝜋2 𝐣) |𝜓⟩ = 𝑅̂ (𝜋2 𝐣) (𝑐 + |+𝐳⟩ + 𝑐 − |−𝐳⟩)= 𝑐 + 𝑅̂ (𝜋2 𝐣) |+𝐳⟩+ 𝑐 − 𝑅̂ (𝜋2 𝐣) |−𝐳⟩ = 𝑐 + |+𝐱⟩ + 𝑐 − |−𝐱⟩ (26) In Equation (26), it is assumed that the state |𝜓⟩ is initially in superposition of the states |+𝐳⟩ and |−𝐳⟩ as in Equation (6). Note that the amplitudes 𝑐 + and 𝑐 − are numbers, which can be placed before or after the operator. An infinitesimal rotation, for instance, by an angle 𝑑𝜙 about the 𝑧 axis can be expressed by an operator 𝑅̂(𝑑𝜙𝒌) given by,
𝑅̂(𝑑𝜙𝒌) = 1 − 𝑖ℏ 𝐽̂ 𝑧 𝑑𝜙 (27) 𝐽̂ 𝑧 is an operator that generates rotations about the 𝑧 axis. As seen from (25), when 𝑑𝜙 → 0 , 𝑅̂(𝑑𝜙𝒌) → 1 . 𝐽̂ 𝑧 have the same dimensions of as ℏ , or angular momentum. The Schrödinger Equation
Time evolution in quantum mechanics is introduced by the Hamiltonian operator, which gives the time translations in quantum systems. The time-evolution operator
𝑈̂(𝑡) translates a ket vector forward in time as given by,
𝑈̂(𝑡)|𝜓(0)⟩ = |𝜓(𝑡)⟩ (28) where |𝜓(𝑡)⟩ is the state of the system at time 𝑡 , with the initial state |𝜓(0)⟩ at time 𝑡 =0. By adopting the rotation operator in (27), a time translation, as an operator, and knowing that |𝜓(𝑡)⟩ is a unit vector, 𝑈̂(𝑡) must be a unitary transformation and therefore, the first-order Taylor expansion of
𝑈̂(𝑡) can be given by,
𝑈̂(𝑡) ≈ 1 − 𝑖ℏ 𝐻̂𝑑𝑡 (29) where the operator 𝐻̂ is the generator of time translations. The operator 𝐻̂ translates an initial ket to a ket at a different time. By substituting (29) into (28) we get, |𝜓(𝑡)⟩ = (1 − 𝑖ℏ 𝐻̂𝑑𝑡)|𝜓(0)⟩ (30) Equation (30) can be rewritten as, |𝜓(𝑡)⟩ − |𝜓(0)⟩ = (− 𝑖ℏ 𝐻̂𝑑𝑡) |𝜓(𝑡)⟩ (31) Or 𝑖ℏ 𝑑𝑑𝑡 |𝜓(𝑡)⟩ = 𝐻̂|𝜓(𝑡)⟩ (32) Equation (32) is known as the Schrödinger Equation. This equation describes the state of the system at its position at a corresponding time. This is analogous to Newton’s second law of motion, which describes the state of a physical system at each time instant by solving for the position and momentum of the system as a function of the applied forces. Controls in the Context of Quantum States
Dynamics of a robotic system, and of a robotic control task, as an example of a control of multi-agent systems in a network of an autonomous system, has been presented in Section 3. The dynamics of a robotic platform can be given by Equations (1) and (2). For a multi-agent networked system, the equations of motion of the agents can be given by the system of equations, which in a matrix form can be represented by {𝐅} = [𝐌]{𝐚} . The present paper aims to integrate the classical dynamics problems with quantum capabilities such as quantum entanglement and cryptography, to enhance the performance of dynamic systems, through an interdisciplinary approach. The integrated quantum and classical control block diagram of a dynamic system is presented in Fig. 3.
Fig. 3. The control block diagram for an autonomous dynamic system with an integrated quantum control system.
The Laplace transform of the integrated classical and quantum feedback control system in Fig. 3 is of interest. The Laplace transform approach for classical dynamic systems is readily available. The Laplace transform approach for quantum states is presented below, and further, it is proposed for the integrated control system. The Schrödinger Equation in (32) is repeated by, 𝑖ℏ 𝑑𝑑𝑡 |𝜓(𝑡)⟩ = 𝐻̂|𝜓(𝑡)⟩ - + 𝑒 𝑐(|𝜓(𝑡)⟩, 𝑒) H [ 𝐫 𝚯 ] 𝑜 [ 𝒓 𝜣 ] 𝑖 𝑭 𝐵,𝑑𝑟𝑎𝑔 , 𝑴
𝐵,𝑑𝑟𝑎𝑔 + - 𝑭 𝐵,𝑝𝑟𝑜𝑝𝑢𝑙 𝑴 𝐵,𝑝𝑟𝑜𝑝𝑢𝑙 𝒗̇ 𝑩 (𝑡) 𝝎̇ 𝐵 (𝑡) Dynamics Actuation
Classical and/or quantum sensing
Controller
Mechanical/Quantum disturbances
The Hamiltonian in the Schrödinger Equation can be expressed by
𝐻̂ = 𝐻̂ + 𝐻̂′ , where 𝐻̂′ is the perturbation. By applying the Laplace transformation to the Schrödinger Equation [53], we get,
ℒ[𝐻̂|𝜓(𝑡)⟩] = ℒ [𝑖ℏ 𝑑𝑑𝑡 |𝜓(𝑡)⟩] (33) This can be written as [53], 𝐻̂ ℒ[|𝜓(𝑡)⟩] + ℒ[𝐻̂′|𝜓(𝑡)⟩]= −𝑖ℏ|𝜓(0)⟩+ 𝑖ℏℒ [ 𝑑𝑑𝑡 |𝜓(𝑡)⟩] (34) Therefore, 𝐻̂ |Ψ(𝑠)⟩ + ℒ[𝐻̂′|𝜓(𝑡)⟩]= −𝑖ℏ|𝜓(0)⟩ + 𝑖ℏ𝑠|Ψ(𝑠)⟩ (35) where ℒ[|𝜓(𝑡)⟩] is denoted by |Ψ(𝑠)⟩ . The time-independent and time-dependent cases of
𝐻̂′ are presented now. In the first case, where
𝐻̂′ is time-independent, the solution to the expression in (35) can be given by, (𝐻̂ + 𝐻̂ ′ − 𝑖ℏ𝑠)|Ψ(𝑠)⟩ = −𝑖ℏ|𝜓(0)⟩ (36) Equation (36) represents linear variation of a system with the generalized inertia matrix (Laplace transform of the system matrix), 𝑀 𝑙 = ⟨Ψ(𝑠)|𝐻̂ − 𝑖ℏ𝑠|Ψ(𝑠)⟩ + 𝑖ℏ⟨Ψ(𝑠)|𝜓(0)⟩+ 𝑖ℏ⟨𝜓(0)|Ψ(𝑠)⟩ (37) Equation (34) is obtained from the extremals of |𝑀 𝑙 | . Inverse Laplace transform of the time-independent case can be obtained by rewriting Equation (36) as, (𝐻̂ + 𝜆𝐻̂ ′ − 𝑖ℏ𝑠)|Ψ(𝑠)⟩ = −𝑖ℏ|𝜓(0)⟩ (38) where 𝜆 is a perturbation parameter, and by expanding |Ψ(𝑠)⟩ in 𝜆 𝑛 , as, |Ψ(𝑠)⟩ = ∑ 𝜆 𝑛𝑛=0 |Ψ 𝑛 (𝑠)⟩ (39) By substituting (39) into (46) and separating the terms of each order in 𝜆 we have, (𝐻̂ − 𝑖ℏ𝑠)|Ψ (0) (𝑠)⟩ = −𝑖ℏ|𝜓(0)⟩ (𝐻̂ − 𝑖ℏ𝑠)|Ψ (1) (𝑠)⟩ + 𝐻̂ ′ |Ψ (0) (𝑠)⟩ = 0 (𝐻̂ − 𝑖ℏ𝑠)|Ψ (2) (𝑠)⟩ + 𝐻̂ ′ |Ψ (1) (𝑠)⟩ = 0 (40) For initial condition |𝜓(0)⟩ = 𝜑 , the coupled equations in (40) are solved by expanding |Ψ 𝑛 (𝑠)⟩ in terms of eigenfunctions of 𝐻̂ as [55], |Ψ 𝑛 (𝑠)⟩ = ∑ 𝐶 𝑚(𝑛) 𝑠 𝑚 𝜑 (41) By substituting (41) into (40), we get, 𝐶 = 𝑖ℏ𝑖ℏ𝑠 − 𝐸 𝐶 = 0 (42) and also, 𝐶 𝑚(1) = 𝑖ℏ⟨𝜑 𝑚0 |𝐻̂ ′ |𝜑 ⟩(𝑖ℏ𝑠 − 𝐸 𝑚0 )(𝑖ℏ𝑠 − 𝐸 ) (43) which gives the first-order approximation as, |Ψ(𝑠)⟩= 𝑖ℏ𝜑 𝑖ℏ𝑠 − 𝐸 + 𝜆 ∑ 𝑖ℏ⟨𝜑 𝑚0 |𝐻̂ ′ |𝜑 ⟩𝜑 𝑚0 (𝑖ℏ𝑠 − 𝐸 𝑚0 )(𝑖ℏ𝑠 − 𝐸 ) 𝑚 + ⋯ (44) Finally, the inverse Laplace transform is determined from (44), as, |𝜓(𝑡)⟩= [1 − 𝑖𝑡𝜆ℏ⟨𝜑 |𝐻̂ ′ |𝜑 ⟩] 𝜑 exp (− 𝑖𝐸 𝑡ℏ )+ 𝜆 ∑ 𝑖ℏ⟨𝜑 𝑚0 |𝐻̂ ′ |𝜑 ⟩𝜑 𝑚0 (𝐸 𝑚0 − 𝐸 ) 𝑚 {exp (− 𝑖𝐸 𝑡ℏ )− exp (− 𝑖𝐸 𝑡ℏ )} 𝜑 𝑚0 (45) For the second case, 𝐻̂′ is time-dependent and given by [54], 𝐻̂ ′ = ∑ 𝑉 𝑙 (𝒓)𝑒𝑥𝑝(−𝑖𝑙𝜖𝑡/ℏ) 𝑙 (46) where 𝑙 ranges from −∞ to ∞ . By applying the Laplace transformation to the (32), we get, 𝐻̂ |Ψ(𝑠)⟩ + ∑ 𝑉 𝑙 (𝒓)ℒ[𝑒𝑥𝑝(−𝑖𝑙𝜖𝑡/ℏ)|𝜓(𝑡)⟩]= −𝑖ℏ|𝜓(0)⟩ + 𝑖ℏ𝑠|Ψ(𝑠)⟩ (47) By Taylor's expansion of the second term in (47) is expressed as, 𝐻̂ |Ψ(𝑠)⟩ + ∑ 𝑉 𝑙 (𝒓)𝑒𝑥𝑝(−𝑖𝑙𝜖𝐷 𝑠 /ℏ) 𝑙 |Ψ(𝑠)⟩= −𝑖ℏ|𝜓(0)⟩ + 𝑖ℏ𝑠|Ψ(𝑠)⟩ (48) where 𝐷 𝑠 denotes the deferential operator 𝜕𝜕𝑠 . Equation (48) can be rewritten as, [𝐻̂ + ∑ 𝑉 𝑙 (𝒓)𝑒𝑥𝑝(𝑖𝑙𝜖𝐷 𝑠 /ℏ) 𝑙 − 𝑖ℏ𝑠] |Ψ(𝑠)⟩= −𝑖ℏ|𝜓(0)⟩ (49) which is the linear variation of 𝑁 𝑙 = ⟨Ψ(𝑠)|𝐻̂ + + ∑ 𝑉 𝑙 (𝒓)𝑒𝑥𝑝(𝑖𝑙𝜖𝐷 𝑠 /ℏ) 𝑙 − 𝑖ℏ𝑠|Ψ(𝑠)⟩ + 𝑖ℏ⟨Ψ(𝑠)|𝜓(0)⟩+ 𝑖ℏ⟨𝜓(0)|Ψ(𝑠)⟩ (50) The extremals of |𝑁 𝑙 | obtain Equation (49). The transfer function of the feedback control system in Fig. 3 may be expressed as, 𝑇𝐹= 𝑐(|Ψ(𝑠)⟩, 𝑒)𝐴𝑐𝑡({𝐅(s), |Ψ(𝑠)⟩})Dyn({𝐅(s)})/1+ 𝑐(|Ψ(𝑠)⟩, 𝑒)𝐴𝑐𝑡({𝐅(s)}, |Ψ(𝑠)⟩)Dyn({𝐅(s)})𝐇 (51) The transfer function in (51) is a generalized feedback control system, which integrates the classical dynamics of the system, actuation, and sensing, denoted by
Dyn({𝐅(s)} , 𝐴𝑐𝑡({𝐅(s)} , and 𝐇 , respectively, with the quantum states |Ψ(𝑠)⟩ . The response in the time domain is obtained by applying a desired input to the transfer function and eventually through the inverse Laplace of the function. Quantum Cryptography for Automation
An automation system that includes robotic manipulators, and an instrumented conveyor with integrated sensors and actuators, and autonomous mobile platforms, is shown in Fig. 4. In this section, it is desired to implement automated control tasks in such automation scenarios by using quantum cryptography techniques. A generalized feedback control system has been introduced in Section 7. Quantum cryptography is applied to guarantee security against cyber-physical attacks while performing efficient control tasks for autonomy. A schematic diagram of a quantum cryptography process is given in Fig. 5. Fig. 6 and Fig. 7 show the components of the robotic systems in more detail. Fig. 8 introduces an alternative mobile platform as an aerial system rather than a ground platform. Experimental quantum cryptography is carried out by polarizing photons, passing them through a polarizing beamsplitter cube, and detecting the photons’ polarization.
Fig. 4 An integrated automation and robotic system with quantum cryptography instrumentation.
Fig. 5 Quantum cryptography experimental setup.
Laser
Polarizing Rotator
PBS Bea 0 Polarizing Rotator
Laser
Polarizing
Rotator
PBS Bea 0 Polarizing
Rotator “Alice Robot”
Eve “Bob Robot”
An automated facility Robotic manipulator Bob Robot Conveyor belt Cryptography equipment Gimbal Alice Robot Manipulator Polarizer Alice Drone (An alternative to the ground Robot depending on the application) Laser source
Fig. 6 The Bob Robot.
Fig. 7 The Alice Robot.
End effector Robotic manipulator installed on Bob robot Sensors Polarizer PBS Servo motors Gimbal Automated Manufacturing workspace Laser source Polarizer Bob Robot Alice Robot Robotic manipulator Manipulator’s servo motors Conveyor belt Manufacturing Objects
Fig. 8 The Alice drone.
The primary operations in a quantum cryptography experiment include: Generating and sending single photons by a laser diode. Spontaneous Parametric Down-Conversion (SPDC) process. Polarizing photons: performed by sending photons through 𝜆2 (half-wave) plate. The process of detecting the polarizations: the polarized photons are sent to a polarizing beamsplitter (PBS) cube, where the PBS passes the horizontal polarizations and reflects vertical polarizations, and two sensors, each dedicated to vertical or horizontal polarization sensing, respond to the receiving of the photons. For the laboratory demonstration of experimental quantum cryptography, we assume that the laser diode is producing single photons, by applying the SPDC process, and the sensors detect single photons. For demonstration purposes, instead of single photons, laser pulses are used, which is sufficient for the experimental demonstration of the concept of experimental quantum cryptography here. The process of generating and counting (detecting) single photons is presented in the entanglement experiment in Section 9. The mobile robots in conjunction with quantum cryptography tools are used for cooperative tasks and control of the mobile robot applications. The “Alice robot” contains a laser diode and a 𝜆2 polarizing rotator plate. The “Bob robot” contains a 𝜆2 polarizing rotator plate, a polarizing beamsplitter cube, and two sensors. Eve, eavesdrop attacker, tries to: a) intercept and detect the information sent from Alice, and b) duplicate the information and send it to Bob. In the quantum cryptography, Eve is exposed to Alice and Bob after a few exchanges of photons, because a single photon cannot be partially detected (as once a single photon is detected by a sensor, it is trapped). Also, the polarization of a single photon (e.g., vertical and horizontal) can be produced by a variety of polarization combinations. When a photon passes through two separate polarizers, one on Alice and the other on Bob, the result of two polarizers are combined. So, by having two polarizers (one on Alice, and the other on Bob), there are various ways of combining polarizations to achieve one result. This is discussed next. The “ket” symbol | ⟩ is used to denote a quantum state. The combination of polarization associated with the half-wave plates at Alice and Bob are given in [49]-[50], where “+” basis corresponds to |0 o ⟩ , and |90 o ⟩ , polarizations (or horizontal and vertical polarizations) and “ × ” basis corresponds to |−45 o ⟩ and |45 o ⟩ polarizations produced by the half-wave plate polarizers. The polarization states of the photons (i.e., |−45 o ⟩ , |0 o ⟩ , |45 o ⟩ , and |90 o ⟩ ) from Alice to Bob are not known publicly. However, Alice and Bob publicly share the information about the bases (“+” or “×”) they used. More details on the quantum cryptography procedure are found in [50]. The polarization of the photons reaching the detectors on the Bob Robot is converted to zeros and ones, where zeros and ones are associated with the horizontal and vertical polarizations, respectively. The zeros and ones are Alice Drone (An alternative to the ground robot)/ or a leader robot (depending on the application) Polarizer Laser sent to a microcontroller, as digital commands, for control and autonomy applications. Quantum Entanglement for Automation
Quantum entanglement can be achieved by the Spontaneous Parametric Down-Conversion (SPDC) process, where two particles are entangled, and predicts non-local behavior. Photons can be the states of vertical or horizontal polarization, and the entangled photons can be specified by the superposition of two photons that are in orthonormal linear polarizations (vertical and horizontal). Quantum mechanics only predicts that the photons are in the vertical and horizontal polarization states, simultaneously, but the state of the polarization of the photons cannot be individually labeled for each photon. When a measurement is made, we can find each of the two photons in one of two states with a corresponding probability. By knowing the polarization of one photon, the polarization of the other photon can be predicted as the polarizations of the photons are orthogonal in the SPDC process. However, the two photons are only in the entangled state until the measurement is made. Once the measurement is made, the photons are no longer correlated, and therefore they are no longer entangled, due to the non-local property of quantum mechanics (Violation of Bell’s inequalities [63]). Non-local properties have led to many remarkable applications such as quantum teleportation (e.g., [64]-[68]), and the rise of the new field of quantum information (e.g., [69]-[71]). A quantum entanglement experiment is shown in the setup of Fig. 9 (in this figure, the notations are: FM: Flipper mirror; M: Mirror; MA: Mirror A; MB: Mirror B; HWP: Half waveplate; AP: Autonomous (Mobile) Platform). The physical experimental setup is presented in Fig. 10. The SPDC process converts one photon of higher energy into a correlated pair of photons with lower energy. By sending a laser beam (e.g., a 100 mW laser, with 405 nm wavelength) through a nonlinear crystal (i.e., BBO: beta Barium Borate), one photon of higher energy is converted into a correlated photon pair (with 810nm wavelengths), producing entangled states, with orthogonal horizontal and vertical polarizations. The two single-photon counter module modules (SPCM50A/M [59]), in Fig. 10, installed on two robots, count photons (by detecting the incident photon), which is used for evaluating entanglement. The SPCM50A/M module specifications include: Wavelength Range of 350 - 900 nm, Typical Max Responsivity of 35% at 500 nm, and Active Detector Size 50 µm ([59]). The 810 nm narrow band-pass filters, with the Bandwidth of 30 nm, shown in Fig. 10, will allow only 810 nm photon pairs produced by the SPDC process to reach the Single Photon Counter Modules (SPCMs). The process of identifying the entangled correlated photon pairs entails the detection of the photon pairs that reach the two SPCM detectors at the same time [74]. The signals from the two SPCM detectors can be sent to an electronic coincidence unit. Each SPCM detector assigns a coincidence to any pair of detected pulses that arrive within a specified time (basically an AND gate). Another possibility could be to have a circuit that assigns a time to the arrival of the photons so that through a classical channel, Alice and Bob can compare the arrival times, and those that arrive within a specified time can be considered in as coincidences. The SPCM detectors on Alice and Bob should synchronize their clocks to within tens of nanoseconds or perhaps pick the time from a wireless signal, and then record the time of arrival of the detector photons. Alice and Bob need to share the basis information through the classical channel (e.g., BB84 encoding), and also the photon arrival times [74]. If the HWP and the polarizer in Fig. 9 and Fig. 10 are used to manipulate the polarization state of one entangled photon (e.g., Alice), along the way when the photon travels from the BBO to the polarizing beamsplitter, the polarization of the other corresponding entangled photon (Bob) is affected simultaneously due to the entanglement phenomenon; this retains the photons in an entangled state (until before the measurement of a photon is made). This is where the non-classical phenomenon of the quantum entanglement appears when Alice and Bob are at any arbitrary distance apart. When the single-photon counters detect entangled photon pairs, a digital signal from the counters can be sent to the robot onboard microcontroller, which then can be translated to a digital task for control of the autonomous system (such as a system of cooperating robots). It should be noted that although the wave functions of the entangled photons collapse after the detections, by converting the detected entangled photons to digital signals, the signals are now available for digital control purposes of the autonomous system. Once the Alice and Bob robots are in an entangled state, any desired information can be exchanged (e.g., by the quantum cryptography process) between a leader robot and the entangled robots for the two robots (Fig. 9 and Fig. 10) to perform the desired robotic or autonomous tasks. The quantum cooperative autonomous platforms presented in the cryptography and entanglement section of the present paper, and a combination of scenario of entanglement and cryptography for autonomy, (where entanglement triggers the process of Cryptography for multiple robotic systems) perhaps, can be the most sophisticated technique in cooperative robotics and unmanned systems technology. This is due to the ultimate speed of photon propagation for robotic control applications, true guaranteed security and immunity against cyberattacks, and the possibility of having access to the entanglement capabilities (that does not exist in the classical domain). An alternative way of quantum communication between autonomous systems is Quantum Teleportation (e.g., [64]-[68]). A brief description of this technique is given now. Assume a two-robot problem where Alice robot is to teleport a quantum state, for example |𝜓⟩ , to Bob. Alice and Bob share a quantum entangled state via SPDC process (Fig. 9 and Fig. 10). Alice, with the state |𝜓⟩ to teleport, carries out a measurement on the entangled photon that is received from the SPDC process. Alice then communicates the outcome of the measurement classically to Bob. This is done by converting Bell States (e.g., horizontal and vertical) to corresponding digital codes (e.g., zero and one) and transferring the zero and one through a classical channel. Finally, Bob robot performs a single-qubit operation based on this information and retrieves the initial unknown state |𝜓⟩ [75]. It should be noted that in this Quantum Teleportation technique, the state of |𝜓⟩ is teleported from Alice to Bob, where the intermediate entanglement process that sends a pair of entangled photons (e.g., with vertical and horizontal polarizations) to Alice and Bob allows this quantum state teleportation process. As supplemental material, some videos of the quantum robotic experiments, presented in this paper, are available in References [72] and [73].
Fig. 9 Quantum entanglement experimental setup diagram.
Laser (Blue)
HeNe Alignment Laser o o SPCM
SPCM
HWP
Polarizer
Nonlinear Crystal (BBO) M A M B M FM Autonomous Platform (AP)
AP (e.g. Robot)
Leader AP (e.g. Robot)
810 nm Filters
Coincidence count device
810 nm Filter
SPCM
SPCM
810 nm Filter
PBS
PBS
Fig. 10 Quantum Entangled robot experimental setup.
Leader Robot
Entangled Robots
BBO Crystal
Laser diode
Motion tracking camera
SPCMs
Photon Counter Software (for both robots)
810 nm Narrow Band-pass Filters
810 nm correlated photon pairs
810 nm correlate photon pairs
405 nm incident photons
Photon Polarizer (for the photons on the right side)
The photon pairs (on the left side) are altered and remain orthogonal with the correlated photons on the right side due to entanglement of the photons, when the polarization of the photons on the right side change by the polarizer.
810 nm correlated photon pair with Horizontal Polarization (as an example) 810 nm correlated photon pair with Vertical Polarization (as an example)
PBS cubes
Beamspliter cubes pass the photons with horizontal polarization and reflect the photons vertical polarizations
Single Photon Counter Module (SPCM)
SPCM Conclusions
An integrated quantum and classical control system for autonomy was introduced in this paper. The dynamics and control system for a generalized automation scenario was presented as an application example. An introduction to analyzing the quantum states was given. A transfer function for the integrated quantum and classical feedback control system of the automated scheme was proposed. Quantum cryptography and entanglement were explored with applications to autonomous systems, and the corresponding experimental setups and procedures were described. The present paper proposed and discussed the integrated quantum and classical feedback control transfer function for the autonomy of mechanical systems, at non-atomistic scales, for the first time. The future work of this research will include presenting and investigating of the use of quantum teleportation for control of autonomous and dynamical systems.
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Farbod Khoshnoud , PhD,
CEng, PGCE, HEA Fellow, is a faculty member in the college of engineering at California State Polytechnic University, Pomona, and a visiting associate in the Center for Autonomous Systems and Technologies, in the department of Aerospace Engineering at California Institute of Technology. His current research areas include Self-powered and Bio-inspired Dynamic Systems; Quantum Multibody Dynamics, Robotics, Controls and Autonomy, by experimental Quantum Entanglement, and Quantum Cryptography; and theoretical Quantum Control techniques. He was a research affiliate at NASA’s Jet Propulsion Laboratory, Caltech in 2019; an Associate Professor of Mechanical Engineering at California State University, 2016-18; a visiting Associate Professor in the Department of Mechanical Engineering at the University of British Columbia (UBC) in 2017; a Lecturer in the Department of Mechanical Engineering at Brunel University London, 2014-16; a senior lecturer at the University of Hertfordshire, 2011-14; a visiting scientist and postdoctoral researcher in the Department of Mechanical Engineering at UBC, 2007-11; a visiting researcher at California Institute of Technology, 2009-11; a Postdoctoral Research Fellow in the Department of Civil Engineering at UBC, 2005-2007. He received his Ph.D. from Brunel University in 2005. He is an associate editor of the Journal of Mechatronic Systems and Control. He was the guest editor of the Quantum Engineering special issue of the Journal of Mechatronic Systems and Control. Emails: [email protected]; [email protected]
Ibrahim I. Esat has BSc and PhD from Queen Mary College of London University, after completing his PhD, he worked on mechanism synthesis at Newcastle university for several years, after this he moved to a newly formed University in Cyprus spending one and half year before returning back to UK joining to the University College of London University where he worked on designing “snake arm” robot. After this he moved to the Dunlop Technology division as a principal engineer working on developing CAD packages in particular surface modellers, following this he returned back to academia as a lecturer at Queen Mary College and later moving to Brunel University. He continued working closely with industry. He continued developing bespoke multi body dynamics software with user base in the UK, Europe and USA. Currently he is a full professor at Brunel University. Email: [email protected]
Shayan Javaherian received his B.S. in mechanical engineering from University of California, Berkeley. His passion is in the areas of soft robotics, controls, quantum robotics, and AI. He worked on underwater communication, ROVs, and laser tractions as his undergrad research at UC Berkeley and also as his graduate research at California State Polytechnic University, Pomona, which he is the project lead for the Intelligence & Autonomous Vehicles laboratory. Email: [email protected]
Dr. Behnam Bahr received Ph.D. in Mechanical Engineering from the University of Wisconsin-Madison in. His teaching and research are in the area of Biologically Inspired Robotics, Automation, and Autonomous Systems, Computer Aided Engineering, and Controls. He has authored and coauthored more than seventy journals and conference papers, and has been the advisor for nine Ph.D., and more than fifty master’s degree students. He is currently the PI of the $2.6 million grant, Mentoring, Educating, Networking, and Thematic Opportunities for Research in Engineering and Science from the Department of Education. He served as the Associate Dean for Research and Graduate Studies from 2012-2016 in the College of Engineering at California State Polytechnique University-Pomona (Cal Poly Pomona). In that capacity he developed seminar series for faculty in the areas of Teaching, Research and Safety, and was instrumental in developing three new Master’s emphasis in “System Engineering”, “Environmental and Water Resource Engineering,” and “Materials Engineering.” He was also the Co-PI of the Cal Poly Pomona on thereceived Ph.D. in Mechanical Engineering from the University of Wisconsin-Madison in. His teaching and research are in the area of Biologically Inspired Robotics, Automation, and Autonomous Systems, Computer Aided Engineering, and Controls. He has authored and coauthored more than seventy journals and conference papers, and has been the advisor for nine Ph.D., and more than fifty master’s degree students. He is currently the PI of the $2.6 million grant, Mentoring, Educating, Networking, and Thematic Opportunities for Research in Engineering and Science from the Department of Education. He served as the Associate Dean for Research and Graduate Studies from 2012-2016 in the College of Engineering at California State Polytechnique University-Pomona (Cal Poly Pomona). In that capacity he developed seminar series for faculty in the areas of Teaching, Research and Safety, and was instrumental in developing three new Master’s emphasis in “System Engineering”, “Environmental and Water Resource Engineering,” and “Materials Engineering.” He was also the Co-PI of the Cal Poly Pomona on the