Entangle me! A game to demonstrate the principles of Quantum Mechanics
aa r X i v : . [ phy s i c s . e d - ph ] J a n Entangle me!A game to demonstrate the principles of Quantum Mechanics
Andrea L´opez-Incera ∗ and Wolfgang D¨ur † Institut f¨ur Theoretische Physik und Institut f¨ur Fachdidaktik, Bereich DINGIM,Universit¨at Innsbruck, Technikerstrasse 21a, A-6020 Innsbruck, Austria
We introduce a game to illustrate the principles of quantum mechanics using a qubit (or spin-first)approach, where students can experience and discover its puzzling features first-hand. Students takethe role of particles and scientists. Scientists unravel underlying rules and properties by collectingand analysing data that is generated by observing particles that act according to given rules. Weshow how this allows one to illustrate quantum states, their stochastic behavior under measurementsas well as quantum entanglement. In addition, we use this approach to illustrate and discussdecoherence, and a modern application of quantum features, namely quantum cryptography. Wehave tested the game in class and report on the results we obtained.
I. INTRODUCTION
Quantum mechanics is a subject that is notoriouslydifficult to teach in class. In addition to the concep-tual difficulties that are intrinsically connected to itscounter-intuitive features, an in-depth understanding re-quires advanced mathematics. In contrast to other areasof physics, it is hard to find contexts from everyday expe-rience, making it difficult to provide motivating or illus-trating examples. Furthermore, hands on experiments,or even demonstration experiments, are unavailable orseriously limited. Perhaps for these reasons, concepts toteach quantum mechanics at high-school level are far lessnumerous and less developed as in other areas of physics(see however [1–17]).Here we suggest a game to illustrate the principles ofquantum mechanics (see also [18] for an alternative ap-proach), where students can experience first-hand howthe rules of quantum mechanics work, and what theyimply for the properties of systems. Students will playthe role of both scientists and quantum particles, simu-lating a real laboratory. This allows them not only toplay and behave like scientists, but helps them to inter-nalize the non-classical features and strange properties ofa quantum world. The role-play in science teaching [19]has attracted interest over the past years, and there existseveral examples [20–22] in the literature of different ap-plications of kinesthetic activities used to teach conceptsof physics. Kinesthetic activities provide direct illustra-tions of the physical concepts, which makes it easier forthe students to create image schemas that help under-standing. At the same time, these type of activities en-hance motivation and enjoyment. In addition, the gamewe propose gives the student the opportunity of obtain-ing the same experimental results that they would obtainin a real laboratory.The game we introduce has three parts, where (i) quan-tum superposition states (together with Heisenberg’s un-certainty relation); (ii) quantum entanglement and (iii)decoherence are illustrated. This covers more than isoften taught in introductory courses on quantum me-chanics at undergraduate level. Nevertheless, we believethat with our approach these three fundamental princi- ∗ [email protected] † [email protected] ples which are at the core of quantum mechanics can beillustrated and experienced. With the same rules, alsoquantum cryptography [12, 23, 24] can be explored in agame-like fashion.We stress that no previous mathematical backgroundis needed. In fact, it is even desirable that the studentsdo not have any previous knowledge of quantum mechan-ics, so that they can come up with fresh and imaginativetheories after analyzing their own measurement results. A. Theoretical background
We make use of the qubit (or spin first) approach toquantum mechanics, [12, 13, 16, 25] where the simplestquantum mechanical system, a two-level system or qubit,is used to demonstrate the basic features of quantum the-ory. In contrast to a classical two-level system, a qubitcan be in a superposition of two states, | ψ i = α | i + β | i , (1)where α and β are complex numbers. Thus, there ex-ist infinitely many superposition states. In this work, weconcentrate only on four of these states to reduce thecomplexity of the game and make it understandable andtractable for the students. In particular, in addition tothe (classical) basis states {| i , | i} , we work with super-position states of the form,[12, 13] |±i = 1 √ | i ± | i ) . (2)If one associates states e.g. with two different positions,this means that such a superposition state describes asituation where the system is essentially at both placessimultaneously. For the polarization degree of freedomof a single photon, the states | i , | i correspond to hori-zontal and vertical polarization, while the superpositionstates |±i are ± ◦ polarized. The Bloch sphere is nicelysuited to illustrate a qubit: [12, 13] standard basis statesare represented by unit vectors pointing in ± z direction,while superposition states point in ± x direction. Noticethat orthogonal vectors are anti-parallel in this picture.Measurements may be illustrated by a slit oriented in acertain direction (e.g. z for z -property). [12, 13] Withinthis picture, measurement eigenstates [26] are easy to vi-sualize, since they are unit vectors pointing in the mea-surement direction, i.e. they can pass unaltered throughthe slit, giving the corresponding outcome deterministi-cally. For instance, if one performs a measurement in the z direction, the state | i passes unaltered through theslit, giving the outcome +1, while a superposition state | + i pointing in x -direction has to flip up or down to pass.Thus, the outcome one gets when the state | + i is mea-sured is random and cannot be predicted. In addition,the state of the qubit is changed -the vector points in+ z or − z direction afterwards. Measurements of differ-ent properties are possible, e.g. the x − property, wherethe slit is oriented in x -direction. Crucially, a quantumsystem cannot possess a deterministic z and x property si-multaneously -the properties are complementary–, whichis the basis of Heisenberg’s uncertainty relation.[12]Apart from the states | i , | i , | + i , |−i (and all the pos-sible superpositions given by eq. (1)) described above –called pure states–, there can also exist probabilistic mix-tures of pure states, called mixed states. As an example,let us consider an ensemble of N qubits, that are eitherin the | i or the | i state with equal probability. Thismixed state is represented in the Bloch as the null vec-tor. If one performs a series of measurements on the N qubits, the outcomes will be random in whatever direc-tion one measures. However, a pure state of N qubits,e.g. all in the state | i , will always give the outcome +1when measured in the z -direction.For systems of two qubits, the superposition princi-ple leads to the possibility of entangled states, e.g. of theform | φ + i = ( | i| i + | i| i ) / √ z -property, butalso for the x -property, as can be seen by noting that | φ + i = ( | + i| + i + |−i|−i ) / √
2. The latter is what makesthis a unique quantum feature that does not exist in aclassically system.Finally, entanglement (with environmental particles)can also be seen as source of decoherence. [28] Decoher-ence [29, 30] is a mechanism that allows one to explainthe absence of quantum effects in large or poorly isolatedsystems, and is thus an essential building block to under-stand the power and limitations of quantum mechanics.Entanglement that is built up due to interaction of a sys-tem with its environment leads to random behavior ofthe system due to lack of control on the environment.[28] This is a different kind of randomness as for purestates that are in a superposition state, or for entangledsystems. [12, 28]
B. Learning objectives
The game is divided in three parts, each one designedto teach a specific set of concepts about Quantum Me-chanics in a hierarchical way, i.e. the complexity of theconcepts increases from part 1 to part 2, and from part2 to part 3. This structure gives the teacher the flexibil-ity to choose either to first play all parts and then workwith all the concepts, or to play and work with each partseparately.The learning objectives of each part are the following:1. Single particles (Sec. II A). • Quantum superposition states. • Preparation and measurement processes (in-cluding measurement in different bases) of sin-gle quantum particles.2. Entanglement (Sec. II B). • Preparation and measurement processes of en-tangled pairs of particles. • Difference between classical and quantum cor-relations.3. Decoherence (Sec. II C). • Effect of decoherence on single-particle states(difference between pure and mixed states). • Effect of decoherence on entangled states.Furthermore, the game is designed so that the studentscan also learn to work and think as scientists, which is offundamental importance not only for their future careersbut also for their personal development as citizens. Thegame can be used as an active learning approach (seeSec. II) to teach Quantum Mechanics and enhance criticalthinking. Among other skills, we highlight that studentscan learn to: • Perform measurements. • Analyze measurement results from a critical per-spective, identifying the differences between whatthey expected from their previous knowledge andwhat they obtain. • Come up with explanations for the obtained results. • Predict results based on these explanations and testthem afterwards.In addition, we explain how the game can be ex-tended to simulate Quantum Cryptography in Sec. III.In Sec. IV, we report the experience and conclusions oftesting the game in class with 16-year high-school stu-dents.
II. IMPLEMENTATION OF THE GAME
In this section, we explain how the game is played, giv-ing more details for the specific parts in sections II A, II Band II C. The students are initially arranged in twogroups: one playing the role of scientists and the otherone playing the role of quantum particles. In part 3, athird group (the environment) is introduced to deal withdecoherence.The goal of the particles is to avoid being measured bythe scientists. The measurement process is illustrated bythe scientists trying to hit the particles with a ball, anal-ogously to e.g. sending photons. The particles can onlymove along two lines painted on the floor, following therules specified in each part (see Sec. II A, II B and II C).The role of the scientists is to figure out rules (i.e.physical laws) that properly describe the behavior ofparticles by observation, i.e. by preparing particles inspecific states and by performing measurements. Theyhave a source available, where they can push one out offour buttons in part 1. Depending on the chosen button(there is a one-to-one correspondence to the four states {| i , | i , | + i , |−i} listed above which however only theparticles know), the particle starts in a certain state andbegins to run along the lines to avoid the balls thrown bythe scientists. Once the scientist hits a particle with theball, he/she has to choose which property of the particlehe/she wants to measure: either the leg (associated to z -measurements) or the arm property (associated to x -measurements). Then, the particle tells the measurementoutcome to the scientist. The scientists should performas many measurements as possible and should take notesof their experiment results, in order to come up later witha description of what is happening. That is, the scientistsshould figure out the rules and make predictions for newexperiments, which they then test. In part 2, the scien-tists discover a hidden button on the machine. When thisbutton is pressed, two particles emerge together in an en-tangled state. Now, the scientists work in pairs, whereeach scientist measures one particle.In the following sections, the detailed rules for the par-ticles in each part are explained. These rules are ex-plained to the particles separately (the scientists cannotknow them) at the beginning of the game. In this way,the scientists can experience how it is to do research whilethey learn (active learning) and can develop a criticalthinking. [31] A. Part 1: Single Particles
In order to play the role of quantum particles (qubits inour case), students should be able to stay not only in oneof the two levels of the qubit, but also in a superpositionstate of both. To do so, we can paint two lines on thefloor: one on the left and one on the right, and standingwith both legs on one line represents the basis states | i and | i . However, the students can be at the two lines atthe same time by just putting a foot on each line, whichrepresents the superposition state. In addition, parti-cles act according to the following simple rules (that onlythey know), and that correspond to the actual behav-ior of quantum particles. They can be prepared in fourdifferent states, {| i , | i , | + i , |−i} where they stand withboth legs and outstretched arms on the left line (state | i ) or right line (state | i ), or with legs on different lineswhere arms point forward (state | + i ) or backward (state |−i ) (see Fig. 1). [32] Thus, the general rule for the parti-cles in this part is that if they stand with separated legs,the arms should be together, and vice versa. Note thatseparated legs and outstretched arms represent that thecorresponding property is unspecified.The difference between pure and mixed states can alsobe addressed. Pure states have one property completelyspecified, i.e. either legs or arms together, leading to a de-terministic outcome (always the same) if the correspond-ing property is measured. However, mixed states leadto random outcomes for whatever property one measures(see Sec. I A). Since separated limbs represent an unspeci-fied property and lead to random measurement outcomes,a mixed state can be represented by the student standingwith outstretched arms and separated legs. FIG. 1. Representation of the different states. For the state | i ( | i ), the student stands on the left (right) line with bothfeet on it and outstretched arms. For the state | + i ( |−i ), thestudent stands with one foot on each line and arms pointingforward (backward).
1. Measurement process
Once a particle is measured by a scientist (hit withthe ball), the scientist has to choose between two dif-ferent kinds of measurements: legs (corresponding to z -measurement, with outcomes left and right), or arms (corresponding to x -measurement, with outcomes frontand back). If the leg property is measured, and the par-ticle stands on one line, the student announces the result(left line or right line). If the particle is standing on bothlines, the student can choose to jump with the two feet tothe left or right line randomly (outstretched arms now),and announces the result (Fig. 2(a)). Similarly, if the arm property is measured, the student tells if her/his arms arepointing forward (front) or backward (back). If the armsare outstretched, the student chooses randomly if her/hisarms will point forward or backward, changing positionso that the legs are on different lines (see Fig. 2(b)), andannounces the resulting state. (a) | + i = √ ( | i + | i ) → | i (b) | i = √ ( | + i + |−i ) → | + i FIG. 2. Measurement of the (a) leg or (b) arm property onsuperposition states. When the scientist measures a particlethat is in a superposition state on the chosen property (basis),the particle has to randomly jump to either one state or theother of the superposition.
B. Part 2: Entanglement
In this part, the scientists prepare pairs of entangledparticles. In the entangled state, the particles are facingeach other, holding arms and standing with their legs ondifferent lines (see Fig. 3(a)). Note that now, the particlesreact randomly to a measurement of both leg and armproperties, since they stand with both separated legs andarms. That is, if the first particle is measured, the particlechanges the state and announces a result accordingly (seerules for part 1). But now there is a new rule for theparticles: the second particle is entangled to the first, soit also changes its state in the same way as its other mate-particle did, without being measured (see Fig. 3(b)). Dueto this first measurement, the entanglement between thetwo particles is broken –they no longer hold hands–. Ifthe second particle is subsequently measured, it behavesaccording to the rules specified in part 1.The game proceeds in the same way as before: the twoparticles emerge in an entangled state and move along thelines. Once one particle is hit by a scientist, the scientistchooses to measure either the leg or the arm property.Then also the other entangled particle can be measuredby another scientist. Again, the role of the scientists is to (a) | φ + i = √ ( | i| i + | i| i ) = √ ( | + i| + i + |−i|−i )(b) | φ + i → | i| i FIG. 3. (a) State of two particles that are entangled ( | φ + i =( | i| i + | i| i ) / √ | + i| + i + |−i|−i ) / √ leg property ( z -measurement) of an entangledstate. Both particles have to do the same –in this example,jump either to the left or to the right line–. figure out rules of the behavior, make predictions and, inparticular, announce if they find out something specialabout the behavior of the two particles. This behavioris even more counter intuitive if one considers entangle-ment between particles that are far away. This can berepresented by two long strings or ropes both particleshold. If one of the particles is measured, the other onestill behaves in the same way as the measured one. Thesituation is the same as the one explained in this sec-tion, but now the entangled particles are far apart. Stu-dents should encounter that there are strong, non-localcorrelations between measurement outcomes that are notpossible in a classical system. C. Part 3: Decoherence
We now move on to a more realistic situation whereparticles are not perfectly isolated, but interact withother particles from the environment. Usually scientistswork hard in their laboratory to prevent this, becausequantum features disappear due to such decoherence ef-fects.In this part, a third group of students is needed to playthe role of environment particles. (a) (b)(c) √ ( | i| i| i + | i| i| i ) → | i| i| i (d) √ ( | + i ⊗ | φ + i ,E + |−i ⊗ | φ − i ,E ) → | + i ⊗ | φ + i ,E . FIG. 4. (a) A single particle in a state | + i gets entangled with a particle from the environment (represented by a stick figurewith a cap in the picture). (b) One of the particles of the entangled pair interacts with the environment. (c) Measurement ofthe leg property of a particle from the entangled state that has interacted with the environment. (d) Measurement of the arm property of a particle from the entangled state that has interacted with the environment. From figures (c) and (d), one can seethat the measurement outcomes of the initially entangled pair are no longer correlated in both the arm and the leg properties,i.e. the interaction with the environment has broken the entanglement. See also the model of the interaction.[33] We first consider a single particle that is prepared inthe state | + i ( arms front ). When it moves around, iteventually collides with another particle from the envi-ronment (represented with a cap in the pictures) and thetwo get entangled, i.e. they are in an entangled state asconsidered in part 2 (see Fig. 4(a)). This process canbe played as a catch-the-particle game, where the parti-cles have to avoid being caught by the environment. Theenvironment-particles are not under control of the scien-tists and cannot be measured. The particle follows thesame rules as before when measured. Note however, thatonce the environment gets entangled to it, the particlehas both legs and arms separated, i.e. scientists obtainrandom outcomes for all measurements. Thus, the par-ticle that was in a pure state ( | + i ), is in a mixed statewhen it interacts with the environment (see Sec. I A).In a similar way, one can investigate what happens toinitially entangled states when one of the two particlescollides with an environment-particle. Then, the entan-glement between the initial particles is broken and allthree particles together are now in an entangled state(see Fig. 4(b) and a more detailed explanation of themodel[33] we use for describing the interaction with theenvironment).Due to the interaction with the environment, the scien-tists obtain very different results from the previous entan- gled case. If the leg property is measured on one particle,the particle randomly chooses to jump either to the leftor the right line, and the other particle has to jump to thesame state, as it is shown in Fig. 4(c). From this point for-ward, the environment particle is also disentangled andcan move away. If the arm property is measured, theparticle randomly chooses to change either to the frontor back state, while the other particle and the environ-ment particle remain entangled (Fig. 4(d)). Hence, ifthe second particle is subsequently measured, one finds arandom outcome (no longer correlated to the result of thefirst particle). Therefore, the initially entangled pair ofparticles, that showed correlated results when either theleg or the arm property was measured, now shows non-correlated results when the arm property is measured.The entanglement has been broken due to the interac-tion with the environment. III. QUANTUM CRYPTOGRAPHY
The game presented in the previous section can be fur-ther used to work with more applications of quantumtechnology, e.g. quantum cryptography. [23, 24] Proto-cols like the BB84 for quantum cryptography rely on theprinciples of Quantum Mechanics described in Part 1 ofthe game, that is, quantum superposition states and mea-surements in different bases. The goal of such protocolsis to establish a secret key, i.e. a random sequence of bits,only known to the sender, Alice, and the receiver, Bob.With help of this random key, an arbitrary message canthen be reliably encrypted.In the BB84 protocol, Alice randomly sends one ofthe four states {| i , | i} (eigenstates of z -basis), {| + i , |−i} (eigenstates of x -basis) to Bob, who measures the statein a randomly chosen basis (either z or x ). This is re-peated N times. The preparation (Alice) and measure-ment (Bob) bases are then announced publicly (not themeasured values!), and only the set of bit values obtainedwhen the two bases coincided are used for creating thekey.Since only two bases ( x and z -bases) are needed, thisprotocol can be played with only part 1 of the game. Thestates Alice sends are simply { legs left, legs right, armsfront, arms back } , and the corresponding bases are the leg and the arm properties. With this correspondence andthe measurement rules specified in Sec. II A, the BB84protocol can be played directly. One student can playthe role of Alice and another one the role of Bob. Therest of the students play the role of quantum particles.Further extensions can be made to include the role of aneavesdropper, Eve, that wants to intercept the secret keyand get access to the message. IV. ENTANGLE ME! IN CLASS
In this section, we present the results obtained fromthe experience we had testing the game in class. First,we give a brief description of how the game was played,and then we present the results of a brief survey given tothe students after the session.
A. The game
The game was tested in May 2018 with the three sci-ence classes of 1 Bachillerato (a total of 73 high schooljuniors –16-year students–) at Colegio JOYFE in Madrid.The students had no previous background on advancedalgebra and quantum mechanics. The session was dividedinto two slots of one hour each: the first one for playingthe game and the second one for the discussion of theresults and a brief presentation in which we explainedthe concepts of Quantum Mechanics and their relation tothe game. The game was played outside, in the court-yard of the school, within a space of approx. 20 − m (19-30 students). The lines on the floor were made of col-ored adhesive tapes. The class was divided in two groups–particles and scientists–, with more scientists than par-ticles (approx. 2:1). The instructions were given to eachgroup separately, so that the scientists do not know theparticles’ rules. The scientists were also given a sheetof paper for writing down the results, with a table ofthe form: “Preparation state”, “Limbs you measured”,“Measured state”. Due to the time resources we had,only part 1 –Single Particles– (Sec. II A) and part 2 –Entanglement– (Sec. II B) of the game were played. Thespecific instructions for the entangled particles in part 2 were given just after part 1 was finished, to make it easierfor the particle-students to remember the rules.The game was played as described in Sec. II. The sci-entists were arranged in two rows, from where they threwthe balls at the particles. Every time they hit a particle(measurement), they wrote down the result and went tothe end of the row to wait for the next turn.Regarding the particles, in part 1, a group of thementered the lines in the state prepared by the scientists.Every time one particle was measured, it had to leave thelines. Once all the particles of the first round were out,the next round of particles entered. In part 2, only onepair of entangled particles entered the lines. When thispair was measured, the following pair entered.In part 2, we proposed the following exercise to thescientists: “Collaborate in pairs, each of you measuresone of the two particles of the entangled pair and writesdown the results. Make as many combinations of mea-surements as possible so that you can compare resultswith your colleague afterwards”.After the game, we discussed the results in groups, sothat the students could organize and interpret all the re-sults together and share opinions. First, the studentswere asked if they had found any unusual behavior in themeasurement results. If needed, more specific questionscan be made to guide the discussion, such as: “How werethe results when you prepared legs and measure the legproperty? And if you measured the arm property?”, “Inpart 2, compare your results with the ones obtained byyour colleague; is the behavior you observe the same ifyou measured arms and legs or not?”. Finally, the stu-dents were asked to propose possible explanations for thephysics behind the results, as if they were in a scientificcongress.From this experience, we highly recommend to planthe session for small groups of students supervised by ateacher (around 15 students per supervisor), so that thegame is more dynamic and the students can participatemore in the discussion. In addition, we have observedthat the discussion encourages the students to be criti-cal when analyzing the results and to be creative whenimagining possible explanations for the results.We concluded the discussion with a brief presentationthat highlighted the key concepts of quantum mechanicsthat can be learnt with a spin first approach, [12, 13, 16]to help the students understand the relation of their find-ings to these central concepts. We emphasized that quan-tum superpositions, stochastic behavior and state changeunder measurement, as well as Heisenberg’s uncertaintyrelation, were illustrated in part 1 of the game, while thecentral concept of entanglement was illustrated in part 2. B. Feedback from students
In this section, we summarize the feedback of the stu-dents we collected after the session. We carried out anopinion survey to test the students’ perception of thegame, in order to evaluate the main difficulties they hadand their suggestions for improvements.The students were presented with a set of sentencesthat they could evaluate from 1 to 4, where 1 correspondsto full disagreement and 4 to full agreement.
The instructions were complicated 60% 32% 4% 4%The game was easy to perform 1% 6% 41% 52%I have enjoyed the game 0% 6% 30% 64%I find the game useful to understand the new concepts 0% 15% 32% 53%TABLE I. Results of the opinion survey given to the students after the game. A total of n = 73 students were surveyed. Theevaluation of each sentence goes from 1 (I fully disagree) to 4 (I fully agree). The survey results presented in Table I show a goodacceptance by the students. In addition, when asked fordifficulties they had during the game, some of the scien-tists reported that they were too far from the particles sothey could not hit them enough times to perform all themeasurements they wanted to. The students were alsoasked for suggestions to improve the game. The mostfrequent suggestions were to have more time to get moreresults, and to make it more dynamic so that the scien-tists do not have to wait long for their turn. We considerthat these suggestions could be fulfilled by having lessstudents per game (we worked with groups of 28, 26 and19 students in a limited space), so that each student canplay more often.With these results, we can conclude that the game canbe done successfully, and that students do not find itcomplicated. In addition, it follows from the received re-sponses that the students have fun with the game, whichwas also a crucial goal for us.
V. SUMMARY
To summarize, we have introduced a game with simplerules that allows one to illustrate the basic principles ofquantum mechanics, and to directly experience them. Inaddition, students can act and work like a real scientist,develop theories and test them. This should not onlyallow students to better remember rules and features ofquantum systems, but also to grasp their significance anddifferences to classical systems. We have also shown thatadvanced quantum features such as entanglement and de-coherence can in principle be illustrated in the same way,and even modern applications such as quantum cryptog-raphy can be treated.
ACKNOWLEDGMENTS
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Phys. , 715–775 (2003).[31] If the students already have some background knowledgeabout Quantum Mechanics or if the teachers find it moresuitable for their purposes, it is also possible to just tell the rules of both scientists and particles to all the studentsat the beginning of the game, so that they can experienceboth roles.[32] The body positions of the states may resemble the Blochvectors of the states. The z axis in the Bloch sphere can bethought of as the left-right axis and the x axis as the front-back axis. So far, the representation introduced by thebody positions does not account for the phase differencebetween | + i + |−i and | + i−|−i explicitly in the arms rep-resentation, only by the leg position ( | i = √ ( | + i + |−i )(legs left) and | i = √ ( | + i − |−i ) (legs right)). If moretechnical considerations want to be taught, one can alsodistinguish these states by facing different directions (onedirection will correspond to a plus sign, and the other to aminus sign). However, we consider that this increases thecomplexity of the game and leads to difficulties in part2 and part 3 of the game (e.g. it leads to the entangledstate | φ − i instead of | φ + i ).[33] The system particles are initially in a | φ + i state, andthe environment particle is in | i state. One can use asimple model for the system-environment interaction andassume that the second system particle and the envi-ronment particle interact via a CNOT operation of theform: | i| i E → | i| i E ; | i| i E → | i| i E . This yieldsa GHZ state ( | i + | i ) / √
2, which can equivalentlybe written as √ ( | + i ⊗ | φ + i ,E + |−i ⊗ | φ − i ,E ) =( | + i ⊗ ( | + i | + i E + |−i |−i E ) + |−i ⊗ ( | + i |−i E + |−i | + i E )) / √
2. Measurement results of the system par-ticles in the z -basis (legs) are random and perfectly corre-lated, while xx