� � 8.04 Quantum Physics Lecture XXIVDirac notation of photon polarization states Ahorizontally polarized photon is denoted by|h�, a vertically polarized photon by |ν�.A Figure I: Definition of the photon polarization bases |h�, |v� and |+45◦�, |−45◦�. photon polarized along ±45◦ is in the state 1 �� |±45◦� = √2 |h�± |ν� . (24-1) Right and left circularly polarized photons are described by the states 1 �� |R� = √2 |h� + i|ν� (24-2) 1 �� |L� = √2 |h�− i|ν� (24-3) Aphoton polarized at an angleθ is denoted by |θ� = cos θ|h� + sin θ|ν�. (24-4) The probability amplitude of a verticallypolarized photon to pass through a polarizer at angle θ is given by the projection �θ|ν� = cos θ�h| + sin θ�ν| ·|ν� = sin θ�ν|ν� = sin θ. (24-5) The probability is given by sin2θ etc. Different polarization states can be converted into each other by using material where the index of refraction depends on the polarization direction. λ πplate: object for which accumulated phase along one axis (the “slow” axis) is • 42 less than for other(fast) axis. If the slow andfast axis are oriented at ±45◦, then |h�→ |R�, |ν�→ |L� ��� � 45◦� + e= (24-6)|h� = √12 |45◦� + |−45◦� →√12 |i π 2|−45◦� |R� |ν� = √12 � |45◦� − |−45◦� � →√12 � |45◦� + e−i π 2|−45◦� � = |L� (24-7) Massachusetts Institute of Technology XXIV-1 8.04 Quantum Physics Lecture XXIVFigure II:Fast and slow axis ona λ/4-plate. λ plate: object for which accumulatedphase along slow andfast axesdiffersby π.A • 2 λ plate at angle θ converts h into linearly polarized light at angle2θ.2 The concept of uncertainty in hidden-variable theories and in the standard interpretation of QM In the standard interpretation of QM, certain combinations of measurable quantities (those that do not commute with each other) are “not simultaneously predictable”, e.g., we cannot predict with certainty both the outcome of a position and of a momentum measurement. This is an intrinsic feature of the standard interpretation, and not due to some limitation in the resolution of the measurement apparatus etc. It is a feature of the state of the system, and the structure of the vectorspace of states. In contrast, a “hidden-veriables” theory would postulate that randomness arises becaaus we do not know the complete state of the system. (There is some information about the system that is not contained in the wavefunction, but that in principle exists.) This corresponds to a randomness somewhat similar to that encountered in classicalstatistical mechanics. According to hidden-variables theories, the quantum mechanical state descriptiio is thus an incomplete representation of the physical reality. Randomness then arises not as an inherent featureof thephysical reality,butof the QM description of the reality. This is similar to classical statistical mechanics, where, not having access to or not being able to track the variables of 1023 particles, we resort to average values. Then fluctuatiion around these average values naturally result from our incomplete knowledge of the microscopic dynamics. For many decades, it was believed that hidden-variables theories cannot be distinguiishe in their predictions, let alone experimentally, from the standard interpretation of QM.TheJohnBellcameupwithasimple situationwherethe predictionsdiffer,and where experiments can decide between the two views. Massachusetts Institute of Technology XXIV-2 8.04 Quantum Physics Lecture XXIVExample: angular momentum and uncertainty Let us consider a system with l = 1ina standard state(m = 1) The standard interpretation Figure III: ??? of QM asserts that, in this state, the components Lx and Ly (and, hence, the direction of L) are uncertain in the sense that they are “unknowable” or “unpredictable”, and that the QM specification of the quantum numbers of the operators that commute(L, Lz)tells us all that can be possibly known about the system: Two systems with the same quantum numbersl, m areidentical,evenifa measureemen of, say Lx on the two systems yields different outcomes. The uncertaiint is inherent in that it is not dependent on the “exactness’ of preparation of the system. An uncertainty of a different kind appears in statistical mechanics or classically chaotic systems.Asmall uncertainty ε in the initial preparation leads to an exponentially growing uncertainty in time, so that after a very short time the state of the system is unpredictable: Two systems, that yield different measurement outcomes were not prepared identically, had we known the exact preparation parameters, we could have predicted the measurement outcome, and two sufficiently identically prepared systems would yield the same results when subjected to some measurement. Some physicists, among them Einstein, believed that the uncertainty in QM should be of the same kind, mainly based on philosophical grounds: Massachusetts Institute of Technology XXIV-3 8.04 Quantum Physics Lecture XXIVIt should be possible to assign “physical reality” to the component Lx of angulla momentum that will be measure before the measurement is made. Accordiin to this view of “local realism”, if two systems prepared in the same state |l, m� yield different outcomes where Lx is measured, it is because they were actually not prepared identically. Local realistic hidden theories assume that thereisa“hiddenvariable” not contained within QM that specifies the Lx component of angular momentum. If we knew the value of this hidden variable, we could predict Lx. According to this view, QM is incomplete in that it does not specify the state of the systemcompletely,it is moresome sort of “average theory” just like statistical mechanics. The problem is sharpened further if we consider: Entangled states of two particles For two particles, QM allows us to prepare entangled states where the measurement on each particleyieldsan uncertain outcome,butthe outcomesofthetwo measuremewntsare correlated. As a simple example, consider the following polarization state of two photons: 1 �� |Bell� = √2 |h�A|ν�B −|ν�A|h�B (24-8) We call this state a Bell state in honor of the Irish physicist John Bell who first showed the non-equivalence of hidden-variable theories and the standard interpretation of QM. Here, the subscripts A, B labelthetwophotons.Astateoftwoparticlesis definedtobe entangled (theexpressionwasinventedbySchr¨odinger, German:’Verschr¨ankung’)ifno basisexists where the state can be written as product state |... �A|... �B. The states of the two photons are correlated if person A (Alice) measures her photon to be horizontally polarized, then person B (Bob) will measure his photon to be vertically polarized, and vice versa. Based on her measurement, the outcome of which is completely uncertain, Alice can predict the outcome of Bob’s measurement. However, the state is mote correlated than in a classical system where, say,a blue anda red ball are distributed between Alice and Bob,but wedo notknowwhogot which.Toseethis,letuswritetheBell stateinthe |±45◦� basis: Massachusetts Institute of Technology XXIV-4 � � 8.04 Quantum Physics Lecture XXIVSince, 1 �� 1 �� |h� = √2 |+45◦� + |−45◦� = √2 |�� + |�� (24-9) 1 �� 1 �� |ν� = √2 |+45◦� − |−45◦� = √2 |�� − |�� (24-10) |Bell� = |h�A|ν�B −|ν�A|h�B (24-11) 1 � � �� � � = 2√2 |��A + |��A |��B + |��B − |��A − |��A |��B + |��B (24-12) 1 �� = 2√22|��A|��B − 2|��A|��B (24-13) 1 �� = √2 |−45◦�A|+45◦�B −|+45◦�|−45◦� (24-14) The states of the two particles are also orthogonally polarized in the ±45◦ basis! Infact, one can show that they are orthogonal in anybasis. This means that if Alice chooses to measure her photon in the ±45◦ basis, she can predict the outcome of Bob’s measurement in that basis. Note. Right after Alice’smeasurementofa |h�-polarized photon, Bob’sphoton’sstate is |ν�, while immediately after Alice’s measurement in the |45◦� basis, say with outcome |+45◦�, Bob’s photon’s state will be |−45�, even if Bob is light years away. Although Alice cannot use this fact to transmit information faster than the speed of light (if Bob does not communicate with Alice, his probabilities are 12 for measurements in any basis, and he gains no information), we are reaching treacherous ground: Does Alice’smeasurement constitute immediate actionatadistance, i.e.,is QM non-local? After all,Bob canmeasure immediately afterAlice(theycouldhave synchronized their clocks initially), both can measure randomly in either the |h�, |ν� or the |±45◦�, basis and when they compare notes later, they will find that whenever they happened to measure inthe same basis, Bob hadthe opposite polarization from that of Alice. Local theories are very deartophysicists,andwedonotliketogiveupthe notionof locality easily. Example: Local vs. global conservation laws Global charge conservation law ”The total charge in the universe is conserved.” Such conservation laws are useless for all practical purposes (cannotbefalsified). Massachusetts Institute of Technology XXIV-5 � � 8.04 Quantum Physics Lecture XXIVFigureIV:Ifalocal conservationlawisvalid,achargeleavingavolumemustpassthrough its bounding surface. Global conservation laws, where a charge disappears from a volume without passing through the surface, and appears elsewhere in the universe, are useless. Local charge conservation law If charge disappears from a volume, it has to flow through the surface into the neighboring volume ∂ρ ∂t + �· j = 0 → differential form (24-15) ∂ ρdV + j dA = 0 integral form (24-16)∂t ·→ of continuity equation describing local conservation of charge. Figure V: Charge Q in a volume bounded by a surface A. Bell’s argument and inequality If a local hidden parameter exists that completely determines the results of the measuremeent of Alice and Bob, then there must exist a function A(λ, eˆ)with valuesA(λ, eˆA)= ±1 for Alice’s measurements and B(λ, eˆB)= ±1for Bob’s measurements. HereˆeA(ˆeB)defines the direction along which Alice (Bob) chooses to measure (sets up her (his) polarizer), and we call the value of the function +1if the photon passes through the polarizer, and−1ifit does not. The main point is that the local hidden-variables theory assumes that Bob’s outcome function B(λ, eˆB)does not depend on the direction in which Alice set up her polarizer, or what outcome she measured. We define a correlation function E(ˆeA, eˆB)as the product of Massachusetts Institute of Technology XXIV-6 � � � � � 8.04 Quantum Physics Lecture XXIVAlice’s and Bob’s outcome functions. For a hidden-variable theory with some unknown distribution function P(λ)for the variableλ, with P(λ)≥ 0, dλP(λ)= 1, we can write E(ˆeA, eˆB)= P(λ)A(λ, eˆA)B(λ, eˆB) (24-17) Note. P(λ)does not depend on the analyzer angles ˆeA,ˆeB chosen by Alice and Bob: the photon pair can be prepared before Alice and Bob decide how to set their polarizers. Bell’s theorem For a local hidden-variable theory, the quantity S = E(ˆeA, eˆB)+ E(ˆe�A, eˆ�B)+ E(ˆeA, eˆ�B)− E(ˆe�A, eˆ�B) (24-18) for anychoiceof measurement anglesˆeA,ˆe�A,ˆeB,ˆe�B satisfies |S |≤ 2. (24-19) This inequality canbe violatedby the predictionsof QM. Proof. For hidden-variables theories we define the quantity S (λ)= A(λ, eˆA)B(λ, eˆB)+ A(λ, eˆ�A)B(λ, eˆB)+ A(λ, eˆA)B(λ, eˆ�B)− A(λ, eˆ�A)B(λ, eˆB) (24-20) with S = dλP(λ)S (λ). S (λ)can also be written as S (λ)= A(λ, eˆA)[B(λ, eˆB)+ B(λ, eˆ�B)] + A(λ, eˆ�A)[B(λ, eˆ�B)− B(λ, eˆ�B)] (24-21) Since Bob’s outcome function is always B =+1or B = −1, either the first or the second term always vanishes. Consequently, S = dλP(λ)S (λ)= dλP(λ)2A(λ, eˆ(A �)). (24-22) Since A takes on only the values ±1, −2 ≤ s ≤ 2. Thus for hidden-variables theories, |s|≤ 2. � ForQM, consider the Bell state 1 �� |Bell� = √2 |h�A|ν�B −|ν�A|h�B (24-23) and the polarizer angles. What is the QM prediction for the correlation function E(ˆeA, eˆB)? Consider Alice’s measurement alongˆea: Massachusetts Institute of Technology XXIV-7 8.04 Quantum Physics Lecture XXIV(a) Polarizer angle for Alice. (b) Polarizer angle for Bob. Figure VI: Polarizer angles. If she measures her polarization along ˆeA (with probability 12), corresponding to her photon being ν polarized, and her outcome functionvalue being A =+1, then she projects the Bell state into 1 |Bell�→− √2|ν�A|h�B (24-24) Consequently, Bob has a has a horizontally polarized photon and if he chooses the measurement axis ˆeB, he will find a photon along that axis(B = 1) with a probability cos238 π = sin2 π 8. If he choose ˆeB instead, he will find B = 1 with probability cos2 π 8 etc. Generally, we can write for an angle α of Alice’s polarizer relative to the same axis, with the notation α⊥ = α + π 2, β⊥ = β + 2π : |h�A = cos α|eˆα�A + sin α|eˆα⊥�A (24-25) |ν�A = − sin α|eˆα�A + cos α|eˆα⊥� (24-26) |h�B = cos β|eˆβ�B + sin β|eˆβ⊥�B (24-27) |ν�B = − sin β|eˆβ�B + cos β|eˆβ⊥�B (24-28) In terms of these polarization states, the Bell state can be rewritten as (substitute this into Massachusetts Institute of Technology XXIV-8 � � � � � � � � � � � � � � � 8.04 Quantum Physics Lecture XXIVdefinition of Bell state) |Bell� = √12 |h�A|ν�B −|ν�A|h�B : 1 �� � |Bell� = √2 − cos α sin β + sin α cos β |eˆα�A|eˆβ�B + �− sin α sin β − cos α cos � β |eˆα⊥�A|eˆβ�B + cos α cos β + sin α sin β |eˆα�A|eˆβ⊥�B � + sin α cos β − cos α sin β |eˆα⊥�A|eˆβ⊥�B 1 � = √2 sin(α − β)|eˆα�A|eˆβ�B − cos(α − β)|eˆα⊥�A|eˆβ�B + cos(α − β)|eˆα�A|eˆβ⊥�B − sin(α − β)|eˆα⊥�A|eˆβ⊥�B Sinceˆeα (ˆeβ)is associated with outcome functionA = 1(B = 1) andˆeα(ˆeα⊥⊥with A = −1(B = −1). The Bell state yields for the correlation function 1E(ˆeA, eˆB)= sin2(α − β)(+1)(+1)2· 1 + cos2(α − β)(−1)(+1)2· 1 + cos2(α − β)(+1)(−1)2· 1 + sin2(α − β)(−1)(−1)2· 1�� = sin2(α − β)− cos2(α − β)2 = − cos(2α − 2β) Consequently, the Bell parameter is S = E(ˆeA, eˆB)+ E(ˆe�A, eˆ�B)+ E(ˆe�A, eˆB)− E(ˆeA, eˆ�B) = − cos(2α − 2β)− cos(2α�− 2β�)− cos(2α�− 2β)+ cos(2α − 2β�) For the suggested choice of angles, (p. XXIV-8). 3π ππ π 3ππS = − cos π − 4 − cos2 − 4 − cos4 − 4 + cos π − 4 �π��π�� π��3π� = − cos4 − cos4 − cos −4 + cos 4 = −321√2− 21√2= −2√2≤−2 (24-29) (24-30) )is associated (24-31) (24-32) (24-33) (24-34) (24-35) (24-36) (24-37) (24-38) Massachusetts Institute of Technology XXIV-9 8.04 Quantum Physics Lecture XXIV(a) α = π 2,α� = 4 π (b) β = 34 π,β� = π 8 Figure VII: Polarizer angles. So QM indeed violates Bell’s inequality. The different predictions by hidden-variables theories(|S |≤ 2always) andQM (|S | > 2possible, one can however show that|S |≤ 2√2 for QM)can be tested experimentally if one has a source of entangled photons. Such sourcesexist,e.g.,adoublyexcitedatomthatcandecaytotheground stateviatwodifferent pathways emits entangled photon pairs: In suchacascade the photons A and B alwayshave FigureVIII:Adecayfroma secondexcited stateinanatomviatwo possibledecaypaths can produce entangled photons. tohaveopposite polarizations,buteachofthemcouldbeleftorright circularly polarized. This corresponds to a Bell state such as the one we have considered. Experiments violate the hidden-variables prediction |S |≤ 2, withupto20 standarddeviations,but arein perfect agreement with QM. Massachusetts Institute of Technology XXIV-10 8.04 Quantum Physics Lecture XXIVWhat does this mean? It means that for entangled states Alice’s chosen measurement direction and outcome ’influeence Bob’s measurement outcome beyond what could be possible if you assumed that once Bob has its photon, the properties of that photon are fixed. This is true even if Bob’s measurement lies outside the light cone of Alice’s measurement. Entanglement cannot be used for communicationfaster than the speed of light,but it can be used for quantum cryptography, i.e., cryptographythat is protected by the laws of QM. Massachusetts Institute of Technology XXIV-11