When the words Quantum Theory are spoken it immediately conjures images of Schro ̈dinger’s cat or of bearded men writing furiously on a black board whilst students scratch their heads bewildered by what is going on. Whilst these images are quite accurate, it does not fully demonstrate the power or the elegance of the theory. In modern day physics there are arguably two great theories, Quantum and General Relativity. However, each theory deals with quite opposite things. The former deals with phenomena of small pieces of matter, called particles. Whilst general relativity deals with phenomena from large pieces of matter on the scale of galaxies. I have pondered on why these theories give rise to such different predictions, as I am sure many of you have done. The answer is one of very deep meaning. If this pondering can be answered then one would be able to solve all the issues surrounding the development of quantum technologies. Here we have raised our second very important question, what is a quantum technology? Well, if one considers it as a technology that works because of quantum theory, one could similarly consider a classical technology, as a technology that works without the need for quantum theory. Such as a car, a lift mechanism, an archimedian screw.
As can be seen classical technology is mechanical in nature, it is somewhat primitive in today’s society, as a classical world would not have the use of computers. A very early quantum technology is a computer, or more ac- curately, a transistor. If it was not for quantum theory we would not have a working understanding of conducting electrons and thus semi-conductors and so the transis- tor would not work as eciently. By research standards though, transistors are a very primitive technology. With the development of science researchers have managed to invent a new wave of quantum technology. Focussing on a less esoteric invention, the quantum computer, one can understand it by drawing parallels with its so called ’clas- sical’ cousin. In a ’classical’ computer, the way in which the computations are carried out is classical. The infor- mation in the computer is coded, via the spin of electrons, to the hard drive by a series of 00s or 10s, a bit. This is how the computer can understand what the human user is asking it to do. Whereas, in a quantum computer this information is not held upon a single bit, but instead the bits are in superposition states, being both 00s and 10s simultaneously, coined a qubit. A qubit is maintained by the entanglement capabilities of particles, this is where the real diculty of the technology comes in. The dif- ference between a classical and quantum state is this su- perposition state, so to create a very ecient quantum technology this superposition must be maintained. Nev- ertheless, the world in which we inhabit is classical, we are classical, everything on the macroscopic is inherently classical. This is part of the reason we find quantum the- ory so hard to understand because it is non-sensical. If we place a quantum object into a classical environment it is obvious that the environment will e↵ect the quantum object in some way. The way in which it a↵ects the sys- tem is the key to creating an ecient technology. These interactions can be modelled using quantum theory, a very successful way of understanding this is called ’Mas- ter Equations’. These equations show how the system evolves in time when in contact with an environment, usually called a bath in this modelling technique. But as mentioned earlier how can we mathematically model the classical interactions with a quantum system? The answer is something called ’Decoherence’.
Decoherence is the bane of many physicists, it destroys any well constructed entanglement and ultimately cre- ates a classical system (so when we say size is the only thing that matters, this is misleading, decoherence is the only thing that matters). The truth of decoherence is un- known, we know what it does and how to model it in our master equations but we do not know what it actually is or where it originates from. It was brought into equa- tions of dynamics to eliminate the need for an intelligent observer to destroy the superposition of a quantum state. This superposition is sometimes called the wavefunction in many texts. What is clear however is that there is a certain limit to which quantum states can exist, this limit is marked by the decoherence destroying the superposi- tion of the system, the so called ’Correspondence Limit’. Although not explicitly known, this limit is usually asso- ciated with a system becoming a certain size, thus size in a certain aspect does matter. To understand the con- cept of decoherence imagine a person in a swimming pool pushing the water with their hand. It forms a clear wave front, this can be thought of as a coherent state, a quan- tum object. Now, if we add in two more people either side of the first person and they too push the water, they destroy the clear and coherent primary wave so we are left with a superposition of the three water packets. We can think of the environment around a quantum object as adding this superposition and as such decoherence.
In conclusion, to understand the effects of an environment on the quantum system it houses we must further understand the nature of decoherence. Decoherence destroys the quantum state, we can model it but we can only model it in terms of the interactions of the environment and the quantum state. If we want to further develop our technology then we must move to deepen our knowledge on the collapse of the superposition present in the quantum system and the nature of decoherence.
PACS numbers: 03.65.-w, 03.67.-a
Since the emergence of Quantum Theory there has been considerable misunderstandings on the ideas and theorems that evolve from the theory; from the incorrect inference of the uncertainty principle to the paradoxi- cal behaviour of the EPR experiment . Nevertheless, a small community of dedicated physicists have enabled the development of new technologies, new theorems and a new way of thinking by deepening their understanding of the quantum world. From the first major development in technology due to quantum theory, the transistor, to the development of apparent ’sci-fi’ technologies, the tele- porter or even the quantum computer. This paper’s aims to present the issues surrounding quantum technologies and how these problems stop this technology being used outside of the laboratory environment.
When considered, there are not many technologies to- day that do not have a root, direct or indirect, in quan- tum theory. The computer run by transistors would not be possible without a proper understanding of atomic processes. Fire, even though it occurs because of chemi- cal reactions, would not be understood without a knowl- edge of the way in which molecules behave on an atomic level. For this we can thank quantum theory. Neverthe- less, these technologies work in any environment because the external environment they inhabit has little or no ef- fect on the system. Thus the models that are used in the lecture theatre, represent the system to a very high degree of accuracy. However, when one moves from these ’simple’ examples to a more esoteric technology, for in- stance superconductivity. The SQUID (Superconduct- ing QUantum Interference Device) has many real world applications, in medical science and further afield. The diculty is to sustain the superconductivity in an every day environment. In a laboratory it is easy to cool the superconductor to 9K  but in the environment this is a risk to the users safety and so presents many problems. Likewise, the quantum phenomena of entanglement has been known about since the first presentation of Quan- tum Theory, in a Copenhagen interpretation (then it was called Spooky Action at a Distance, as coined by Einstein ), however, apart from in a laboratory there has been no ecient real life application of this theory, because of
the diculty in blocking the environmental e↵ects on the system.
Entanglement is key to the development of quantum computing, and this technology shall be the focus of this paper. However, the diculty in creating an entangled particle can be thought of as trying to balance a pencil on the tip in a hurricane. It is very dicult and many di↵erent external environmental e↵ects need to be taken into account, the thermal e↵ects, the decoherence of the environment exchanging to the coherence of our system and the noise to name a few. This is where modelling techniques are so important, if one correctly models the way in which a system and environment interact (cou- ple) then one can build technologies that can work in the outside world without the need for a laboratory like con- dition. In many research circumstances master equations are used to model these behaviours. In these equations, one uses baths to model the coupling of the system to the external environments, where the bath replaces the environment [4, 5] . These baths will have certain proper- ties depending on the environment the system inhabits. These have proven to be very powerful tools in under- standing the coupling of system to environment and will be the sole modelling technique discussed here.
II. MASTER EQUATIONS A. Theory
Before any theory of master equations is discussed it is important to first outline the three main pictures of quantum dynamics. The first is the Schr ̈odinger Picture, named after Erwin Schr ̈odinger who is credited with the equation for the dynamical evolution of the state vector in time. Thus in this picture the evolution of the state vector is described by time independent hermitian oper- ators but with a time dependent state vector and fixed basis vectors. So the evolution in time can be described by the Schr ̈odinger equation (Equation 1).
H| ni=i ̄h@| ni (1) @t
Environmental E↵ects on Quantum Systems
Luke De Clerk
Loughborough University, Department of Physics, Leicestershire, LE11 3TU
(Dated: April 16, 2018)
In this paper we discuss the environmental e↵ects on quantum systems and how these e↵ects alter the precision and accuracy of quantum technologies, mainly focussing on a static probability distribution. We pay particular attention to the Master Equation modelling technique with refer- ence to the example of Quantum Computing. We critically evaluate how this technique is useful in modelling environmental e↵ects on quantum technologies and summarise how to develop our understanding of these e↵ects for more precise and accurate quantum technologies.
The second picture is that of Heisenberg. Here instead of a time dependent state vector one uses time depen- dent hermitian operators as the basis vectors and uses a fixed state vector. The state vector is still a solution of Sch ̈odinger’s equation but here the Hamiltonian di↵ers by the picture we choose to describe it in. To get from one picture to another mathematically one uses a uni- tary operator, pre multiplying the state vector by this operator. Nevertheless, the two pictures are physically identical.
The third and final picture is that of the interaction picture. Here it is used when the interactions of systems and environments or sub systems is needed to be taken into account. The main di↵erence here is to do with the hamiltonian; H = Hos + Vs. Here the time dependency may explicitly be held in Vs, this is to do with the interac- tion energy of the operator of the system. Whereas, Hos is time independent and can easily be used to solve equation 1 for its eigenvectors. This picture is solely used when one wants to consider the e↵ects of interactions with the time evolution of the system1.
It is now appropriate for the introduction of the den- sity operator. In quantum mechanics there are two types of states. A pure state, in which the state vector is all that is needed to describe the time evolution of the sys- tem and a statistical mixture state, where the probability of the states have to be considered. The probabilities of each state are related to the state vector of each state. Therefore, it is not rigorous enough for us to simply de- fine the state vectors, we must use a density operator to describe the dynamics of the latter systems. Never- theless, we can also describe a pure state by the same operator. Equation 2 shows the mathematical definition of the operator  . It is worth noting that each quan- tum mechanical picture has its own density operator and one can transform each into the next, with the use of the unitary operator as mentioned above. Here we shall focus on a general operator.
⇢(t)=| (t)ih (t)| (2)
The density operator is a way of expressing a quan- tum system in a canonical ensemble, the same way that a partition function can be used to describe a classical system in the same ensemble. As the density operator now describes the system at any one time, we need to know how it evolves with time, thus deriving the master equation for an isolated system. First we di↵erentiate with respect to time as seen in equation 3.
d⇢(t) =(d| (t)i)h (t)|+| (t)i(dh (t)|) (3) dtdt dt
1 All definitions and working were taken from [6–13] and were ad- justed for the purpose of this paper.  was a very elegant source, if the reader would like a more in-depth understanding we sug- gest looking at this text in particular.
Then using Schr ̈odinger’s equation we can substitute into our equation. For the second term we take the com- plex conjugate of equation 1 and again substitute., gain- ing equation 4.
d⇢(t) = 1 H| (t)ih (t)| 1 | (t)ih (t)|H (4) dt i ̄h i ̄h
We notice that equation 4 is the commutator of the density operator and the hamiltonian, meaning the equa- tion can be rearranged into equation 5 [6–13].
d⇢(t) = i [H (t), ⇢(t)] (5) dt ̄h
As mentioned this master equation is for an isolated system in seemingly no environment. One would need to add in the coupling terms and interaction terms of the system to the environment. As well as the two di↵erent types of states there are two main properties of statistics that govern the dynamics of a quantum system; Marko- vian and Non-Markovian. The former is applied for a random process whose future probability can be deter- mined by its present values, a memoryless system, , the latter occurs when the systems future is determined by the past values of the system. These systems are very tricky to deal with because there are a lot of variables to be taken care of in the maths . The properties are applied to stochastic systems. That is one that fol- lows stochastic statistics whereby, for a random process its probability distribution can be analysed statistically but a prediction cannot be made precisely . From these definitions one can immediately see the issues sur- rounding quantum systems, they can be predicted but not precisely known. This property is not unique to only quantum systems because stochastic analysis is used in many di↵erent areas including economics. Thus one can- not argue a measurement problem approach.
Equation 5 is called the Von-Neumann Master equa- tion, for a master equations to be applicable to an ob- servable system it must be in Lindblad form. There are three main reasons as to why this transformation is un- dertaken. The first is so that ⇢ still has linear dynamics in the system, the sum of the Lindblad operator’s eigen- value are always one, meaning it is trace-free. And the final reason is it leaves ⇢ hermitian, which means it is left with real eigenvalues thus being an observable [18, 19]. Once we have the master equation for our system we must now solve it to enable us to properly know how to treat our system and environment to gain the dynamics of our quantum system.
Systems that have a statistical nature are in general very dicult to solve exactly, what is needed is a ’small deformation’ on the system so it can be solved. This is
called perturbation theory and is central to the solutions of master equations. In the most simplistic of cases a hamiltonian will have an unperturbed term and a per- turbed potential. To find the energy eigenvalues and the eigenfunctions of the system, the perturbed term is ex- panded under a Taylor series and then placed into the Scho ̈dinger equation , where the unperturbed term is still a solution. It is worth noting that the Schr ̈odinger equation is a special case of the more general Lindblad equation, thus perturbation theory is a solution of a unique case . Moreover, perturbation theory does have many other limitations, if a system has a large per- turbation term, for example, if in the potential if it is appropriate to include coupling constants for our system, then these terms can become too large for the perturba- tion method to be accurate enough to use. Moreover, the theory fails when we try and describe a system or state that has been created via non-adiabatic processes. An example being the production of Cooper Pairs in su- perconductivity . In the context of condensed phase molecules it has been shown by Fitzgerald in  that Density Functional Theory can be more appropriate and more accurate than perturbation theory. Fitzgerald as- serts that it is more popular and versatile in many body systems, especially when dealing with electronic configu- rations. In many superconducting applications, the elec- trons are the system, therefore a method that deals with specifically electronic configurations would be more suit- able . Furthermore, if we just use it to deal with quantum technologies as many body systems then it is worth reflecting upon the, Hartree-Fock method. Here we find the eigenvectors by treating the systems particles in their respective statistics, fermions in Fermi-Dirac statis- tics and bosons in Bose-Einstein statistics. As shown in  by Abdulsattar it provides a link between the more abstract mathematics, Schr ̈odinger’s equation, and a real-world application.
In  it is asserted by Fleming et al. that 100% ac- curate master equations for stochastic dynamics of open systems are out of reach, that is they will never be cre- ated. Their paper focusses on entanglement and how the perturbative master equations, namely of the second or- der are in adequate for explaining the death of entangled systems at low temperatures. As well as critically assess- ing these perturbative equations, Fleming et al. show how the interaction of the system and environment is modelled by tracing out the environments degrees of free- dom in the master equation. This mathematical trick is essential for their chosen system, it allows them to de- velop a sound and detailed description for entanglement. However, because of this trick the solution has lost ac- curacy and as they mention in the paper, exact master equations for open systems are out of reach. This shows the diculty in developing an entangled system in an environment. Until one can fully model an environment accurately, then the system will not work as intended. Thus we see our first major issue of environmental ef- fects on the quantum system, setting up and solving our
mathematical construct is extremely dicult.
This however has not stopped attempts to be made in solving the master equation in its most general form. As can be seen from , Zhao et al. describe how if a system and its environment are no longer weakly cou- pled the master equation is no longer Markovian, an ef- ficient way of describing open systems, and so become non-Markovian. In this case the systems and environ- ments operators commute, this means the quantities are conserved. Moreover, Zhao et al. go on to detail how the non-Markovian quantum state di↵usion approach al- lows measurements of the system to be made with much higher precision. This does mark a very big breakthrough in the solution of open systems. Nevertheless, an open system is one in which the system and environment can exchange particles and in most cases in quantum tech- nology the system is not open. Therefore, one cannot use this state di↵usion approach to deduce solutions of these master equations. This presents further issues from the modelling of the systems to inferring real-world im- plications, proving the diculty of having a functional quantum system in a non-laboratory environment.
Many other examples of e↵orts to solve the stochas- tic master equation exist. In  it is maintained by Santos et al. that most master equations are unsolved, this is not too hard to believe due to the diculty in solving a di↵erential equation with many operators. The paper depicts a method for solving the master equation using what is coined the ”phenomenological approach”. Here Santos et al. use corrections from their observations to create a master equation, they do not derive it from first principles. This approach is a sort of cyclical rea- soning, we observe it therefore it must be so. Now this is a good approach for purely ’blue-skies’ research but for a solid understanding and a precise working quan- tum technology it is not ideal as we cannot make predic- tions o↵ the theory, which is a key part of the scientific process, . Whilst in  Dickman creates a numeri- cal analysis method for solving transitional systems. His motivation for doing such a feat is because the master equation, in his view, is a central tool in developing un- derstanding of systems in ”discrete state space, in contin- uous time”. There is no denying the power of the master equation, but there is a limit to the extent which it is contributing to the development of the quantum tech- nologies. For instance, Dickman’s approach can only be used on Markovian Stochastic master equations, if the system has Non-Markovian property it cannot be solved via this new method. Moreover, Dickman also advocates that the state must have a ”stationary probability distri- bution”, if this is not the case then it cannot be solved by the iterative method described by him. These stationary distributions only describe Markovian property statistics, which have been discussed as much simpler dynamic of the quantum world. If a more complex system is needed then we must pursue a new method for solution, thus it can be argued that if we must invent a new method for solving such equations every time we require a new mas-
ter equation. It becomes too inecient at modelling a situation and highlights how understanding environmen- tal e↵ects on a quantum object is so dicult.
III. QUANTUM TECHNOLOGIES
Quantum computing for some is deemed to be the most promising technology to come from quantum theory. As many will know D-Wave  have reportedly created the world’s first quantum computer, but to the extent it can be called quantum is in doubt. Nevertheless, it does rep- resent a major advancement in the applications of quan- tum technologies. The main issue of using entangled par- ticles to store data is keeping them in this superposition state. That is, without the system gaining decoherence from the environment. Moreover, a quantum computer needs a macroscopic superposition state [31–35]. Conse- quently, to do this there needs to be a macroscopic co- herence in the system as a whole, for such delicate states an in-depth knowledge of the interactions of the environ- ment and the system is needed. Thus, a very detailed master equation is needed.
to our engineering of the quantum computer. A more robust modelling technique that would allow the appli- cation to be applied the technologies would be the use of say Density Functional Theory as discussed above. The modelling using the master equations and quantum the- ory in general are so abstract that when we try to apply them to real life this does not always yield the solution we were seeking. Decoherence is the main problem in many quantum technologies not just quantum comput- ing, this is due to the abstract nature of the concept. When you ask any physicist what decoherence is they will tell you what it does to a system but not what it truly is. This highlights the problem of quantum sys- tems, we know what the environments do on the systems but not the true nature of these phenomena.
IV. DISCUSSION AND DEVELOPMENT
There are two things we must now discuss, decoher- ence and measurement. If we look at  Cresser ex- presses a link between master equations and measure- ment. Cresser writes ”….quantum trajectories realisa- tions correspond to the collapse into a state.” Here the collapse into state that is discussed corresponds to the collapse of the wave vector into one of its eigenvectors. We have already run into a problem, in all the master equations we have discussed we have used decoherence to model the collapse. What must be stated is that the decoherence does not generate the wave function itself, rather it provides an explanation for the apparent col- lapse into an eigenstate. This eloquent definition comes from , where Bacon highlights how the quantum com- puter needs isolated evolution of the systems coherence in order to perform its function. However, if we instead use a di↵erent interpretation of quantum mechanics (many worlds, decoherence or Copenhagen) then we also change the way the wavefunction collapses and we change the meaning of said action.
The problem with these di↵erent interpretations of quantum theory is that they are all physically equiva- lent up to the point of wavefunction collapse, then after- wards they diverge. This divergence cannot be proven by experiment, thus we cannot prove one over the other. Hence we cannot improve our modelling techniques be- cause we do not fully understand the underlying reality. Our everyday experience is tainted by the classical world, we do not see any quantum phenomena so we assume the world is classical, it is only when we delve into sin- gle atoms or very limited particle systems that we see the manifestation of quantum e↵ects. This can be un- derstood by decoherence, in  Gerry et al. describe how a quantum system is translated into a classical one by decoherence. In classical optics, the phase relation of light is understood by how coherent two sources are. The more in phase these sources are the more coherent they are. A similar thing is understood, a quantum system can be thought of as a superposition of many eigenvec-
@⇢ =L[⇢]⌘1i[H,⇢]+LD[⇢] @ t ̄h
In equation 6 a master equation for the so called ’semi- group’ approach of quantum computing as seen in , it bears many similarities as equation 5, however, as can be seen an additional term is present. This term is the Lindblad operator accounting for the dynamics of deco- herence in our system, this operator can be seen in figure 1.
FIG. 1. The Lindblad Operators for the Semi-Group Ap- proach of Quantum Computing
However, in  Lidar introduces the semi-group ap- proach, where the decoherence arises from the Hilbert space these qubits exist in. However, Lidar presents the qubits as taken into a subspace and due to the symme- try of the system, the decoherence seems to disappear in the qubits themselves, creating a ”decoherence free sub- space”. Therefore, it can be easily seen that the biggest problem surrounding the advancement of quantum com- puting are the properties of decoherence. The problem is whilst the mathematics from the master equations is elegant and solves the problem of decoherence, it cannot be transferred to condensed matter problems as advo- cated in  by Du↵us et al., that is it cannot be applied
tors, these eigenvectors, imagined as waves, are all in phase and so coherent. When the system is exposed to an environment decoherence seeps into the system, the superposition is destroyed and the system becomes inher- ently classical. So in everyday life we see the decoherence of the atoms that form reality. We can further develop this point and relate it directly to measurement by try- ing to understand Schlosshauer’s work in , here it is asserted that decoherence is so successful because it of- fers an alternative mechanism for wavefunction collapse without altering Schr ̈odinger’s equation. It carries on to explain the weak measurement issue too, whereby the amount of decoherence introduced into the system by measuring it is determined by the strength of coupling between the measuring device and the system, a similar view held by Clarke in . Nevertheless, these sources do not provide a way of dealing with decoherence, it does not tell of how decoherence travels or what the properties are, it just states what it is. A more rigorous theory of decoherence would allow the master equation modelling discussed to have real world impacts. Thus allowing us to have quantum technologies that were more ecient and e↵ective in real world situations.
Similarities can be drawn from Maxwell’s equations of electromagnetic (EM) radiation. Before these very little was known of the EM spectrum, we only knew that it existed. After these equations it was possible to produce the phenomena of light and many other radiation on a commercial scale. For example, the transformer would not be possible without an understanding of Faraday’s law of induction. This, the author realises, is a very loose comparison because quantum technologies are infinitely more complex but the principle is still the same. If one wants to produce a result consistently, then she needs to understand what makes the result work and most impor- tantly why it does not work.
In  Du↵us et al. discuss how a master equation is only relevant for the environment it is modelled to be in, it is argued that there is no general environment in the real world in which we can extend all of our mod- els to. It could therefore be inferred as the environment holds an important place in the model as a whole. The correspondence limit is not well defined and if modelled by decoherence then it is just the point at which the su- perposed state ceases to exist. It has been argued many times, does the collapse of a wavefunction need an intel- ligent observer to collapse the wavefunction? This was the main need for decoherence, to remove the observer from the equation. Nevertheless, there is a phenomena in quantum theory that does solely rely on observation, the Zeno e↵ect [43, 44]. The author is not insinuating that the observer  holds an important role in the cor- respondence principle, but instead of the observer, is it that each atom carries a certain quantity of decoherence, is decoherence a fundamental concept like charge or spin? Is it something that each particle has, then when atoms are added together in matter does the decoherence sum and destroy the coherence?
To conclude, the modelling techniques that have been discussed at length in this paper are a very useful tools in understanding how Markovian stochastic quantum sys- tems interact with the environments in which they re- side. The solutions to these models are extremely com- plex and are often not solved to the degree of exactness that we are used to. This raises a lot of issues, namely that quantum technologies can never be built to perfectly suit the environment they are being designed for because we have no complete modelling of the coupling present. Moreover, many believe the exact solutions and complete master equation modelling techniques to be out of reach of modern science. If we wish to understand all quan- tum systems not just Markovian systems, then we must expand our knowledge to Non-Markovian systems. This has been shown to be extremely dicult, the property of the statistics used exacerbate the diculty due to the nature of the probability distribution. The models and solutions discussed at length focus on a static or at least quasi-static probability distribution. To introduce a dif- ferent type of probability would need a development of this paper and more in-depth research.
With the problem not solely being the modelling tech- nique being used but rather the understanding of the world in which the systems exist. If one does not fully understand the way in which the environment interacts with the system at a very basic level then one cannot expect to have a fully ecient system. This is one of the main problems surrounding quantum technologies today. However, if one were to try and determine the nature of decoherence and that of nature itself, we raise the issue as to which interpretation we use. If we use many worlds we are met by parallel universes and a multitude of dif- ferent ones for every time the wavefunction of any system were collapsed. In  Hall explains how you would have to travel ”a distance of 1 followed by a hundred thousand trillion trillion zeros” to meet yourself in another parallel universe. If a theory can deal with these kind of numbers and if it can be used to calculate such a esoteric quantity can it not be used to understand the interactions of a quantum system and its environment? Nevertheless, it is our conclusion that if one wants to gain a fully func- tional quantum technology one must first understand the behaviour of the environment, namely the behaviour of the collapse of a quantum system, be that decoherence or a wave function collapse. This is the major barrier stop- ping quantum systems being more ecient and thus we must deduce that this is the diculty in understanding environmental e↵ects on quantum technologies.
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