Probability Theory with Superposition Events: A Classical Generalization in the Direction of Quantum Mechanics David Ellerman University of Ljubljana, Slovenia Abstract In nite probability theory, events are subsets S  U of the outcome set. Subsets can be represented by 1-dimensional column vectors. By extending the representation of events to two dimensional matrices, we can introduce "superposition events." Probabilities are introduced for classical events, superposition events, and their mixtures by using density matrices. Then probabilities for experiments or measurementsof all these events can be determined in a manner exactly like in quantum mechanics (QM) using density matrices. Moreover the transformation of the density matrices induced by the experiments or measurements is the Lüders mixture operation as in QM. And nally by moving the machinery into the n-dimensional vector space over Z2, di¤erent basis sets become di¤erent outcome sets. That non-commutativeextension of nite probability theory yields the pedagogical model of quantum mechanics over Z2 that can model many characteristic non-classical results of QM. Contents 1 Introduction: Probability Theory with Superposition Events 1 2 The Density Matrix Representations 2 3 Computing Measurementor Trial Probabilities with Density Matrices 4 4 How MeasurementTransforms Density Matrices 5 5 Measurement and Logical Entropy 7 6 The Pedagogical Model of Quantum Mechanics over Z2 8 7 Concluding Remarks 10 1 Introduction: Probability Theory with Superposition Events The purpose of this paper is to introduce new concepts such as "superposition events" into nite probability theory. Let U = fu1; :::; ung be the outcome set or sample space of outcomes with the respective point probabilities of p = (p1; :::; pn). Classical events are represented by subsets S  U with probabilities Pr (S) = P ui2S pi where the conditional probability of the event T given the event S is Pr (T jS) = Pr(S\T )Pr(S) . 1 2 The Density Matrix Representations To generalize classical events to superposition events, we need a richer mathematical representation than just the notion of a subset. The mathematical information in a classical event S (for convenience, always non-empty) could be represented in a (normalized) column vector jSi with ith entry being q pi Pr(S)S (ui) (where S : U ! f0; 1g is the characteristic or indicator function for S, S (ui) = 1 if ui 2 S and 0 otherwise). The same information could be represented in two dimensions by the diagonal n n matrix  (S) with the diagonal entries piPr(S)S (ui), i.e.,  (S)i = pi Pr(S)S (ui). But the richer two-dimensional matrices allows us to dene the superposition event S associated with S as being represented by the n  n matrix  (S) (writing the transpose jSit = hSj) by multiplying the n 1 column vector jSi times the 1 n transpose jSit = hSj:  (S) = jSi hSj with the entries  (S)ik = q pi Pr(S) pk Pr(S)S (ui)S (uk). Note that singleton events S = fuig have no distinct elements to superpose and accordingly  ( fuig) =  ( fuig). Both  (S) and  (S) are examples of real density matrices which can be dened abstractly as symmetric matrices  = t over the reals with trace (sum of diagonal elements) tr [] = 1; and with non-negative eigenvalues. But for practical purposes, density matrices (over the reals unless otherwise stated) may be taken to be any probabilistic mixtures of matrices of the form  (S). That is, for any probability distribution q = (q1; :::; qm) and classical events Sj  U for j = 1; :::;m, the convex combination Pm j=1 qj (Sj) is also a density matrix. A density matrix  is said to be pure if 2 = , and otherwise mixed. For instance,  (S) is pure while  (S) is a mixture unless S is a singleton event fuig since  ( fuig)2 =  ( fuig)2 =  ( fuig) =  ( fuig) trivially. A partition  = fB1; :::; Bmg on U is a set of non-empty mutually disjoint subsets fBjgmj=1 whose union is U . [2] The partition  is represented by the density matrix:  () = Pm j=1 Pr (Bj)  (Bj) Density matrix associated with a partition  on U that is mixed unless  is the indiscrete partition 0U = fUg since  (0U ) =  (U). With a suitable interchange of rows and columns, any density matrix  () dened by a partition would be blockdiagonal according to the partition blocks Bj 2 . For the discrete partition 1U = ffu1g ; :::; fungg on U ,  (1U ) =  (U). Thus the two extreme partitions at the top (discrete partition 1U ) and bottom (indiscrete partition 0U ) in the lattice of partitions (ordered by renement) on U correspond to the two extreme density matrices  (U) and  (U), and all the intermediate partitions  have density matrices that are mixtures of the pure density matrices  (Bj) for their blocks. For the discrete partition on a subset S, 1S = ffuiggui2S and the indiscrete partition 0S = fSg on a subset S,  (1S) =  (S) and  (0S) =  (S). The discrete partition 1S on a set S  U distinguishes all the elements of S from each other in singleton blocks, and thus the density matrix  (1S) associated with that partition is the statistical mixture of the singleton events for elements of S:  (1S) =  (S) = P ui2S pi Pr(S) ( fuig). In contrast, the superposition event S associated with S represented by  (S) blurs, blobs, or coheres together, i.e., superposes, the elements of S. For equal probabilities 1jSj , the elements of S are equally superposed. Otherwise, we may say ui; uk 2 S are superposed with an amplitude of  (S)ik = q pi Pr(S) pk Pr(S) . The entries in the density matrices associated with S, namely  (S) and  (S), have the same diagonal elements and di¤er only in 2 the o¤-diagonal elements. When an o¤-diagonal entry  (S)ik is non-zero, then it indicates that the corresponding elements ui; uk 2 S are cohered together with that non-zero amplitude. All the o¤diagonal elements in  (S) are zero indicating that the elements of S are completely distinguished or decohered from each other. For a suggestive visual example, consider the outcome set U as a pair of isosceles triangles that are distinct by the labels on the equal sides and the opposing angles. Figure 1: Set of distinct isosceles triangles The superposition event U is denite on the properties that are common to the elements of U , i.e., the angle a and the opposing side A, but is indenite where the two triangles are distinct, i.e., the two equal sides and their opposing angles. Figure 2: The superposition event U . Consider the partition  = fB1; B2g = ff};~g ; f|;gg on the outcome set U = f|;};~;g with equiprobable outcomes like drawing cards from a randomized deck. For instance, the superposition event associated with B1 = f};~g, is pure since (rows and columns labelled in the order f|;};~;g):  (B1) = 1 Pr(f};~g) 2664 0 0 0 0 0 Pr(f}g) p Pr(f}g) Pr(f~g) 0 0 p Pr(f~g) Pr(f}g) Pr(f~g) 0 0 0 0 0 3775 = 2664 0 0 0 0 0 12 1 2 0 0 12 1 2 0 0 0 0 0 3775 equals its square, but density matrix for the half-half mixture of the two suit-color pure events: 1 2 (B1) + 1 2 (B2) = 12 2664 0 0 0 0 0 12 1 2 0 0 12 1 2 0 0 0 0 0 3775+ 12 2664 1 2 0 0 1 2 0 0 0 0 0 0 0 0 1 2 0 0 1 2 3775 = 2664 1 4 0 0 1 4 0 14 1 4 0 0 14 1 4 0 1 4 0 0 1 4 3775 is a mixture since it does not equal its square. Intuitively, the interpretation of the superposition event represented by  (B1) =  ( f};~g) is that it is denite on the properties common to its elements, e.g., in this case, being a red suite, but indenite on where the elements di¤er. The indeniteness is indicated by the non-zero o¤-diagonal elements that indicate that the diamond suite } is blurred, cohered, or superposed with the hearts suite ~ in the superposition state  f};~g. 3 3 Computing Measurementor Trial Probabilities with Density Matrices A (real-valued) random variable on the outcome space U is a function f : U ! R with values of f1; :::; mg. The inverse image of f is a partition  = fBjgmj=1 where Bj = f 1 (j). In ordinary classical probability theory, the conditional probability of getting the value j given the event S in a trial is Pr (j jS) = Pr(Bj\S)Pr(S) . But now we have two versions of S, the classical event and the superposition event. Since they have di¤erent density matrices, we can take the given conditioning event as a density matrix . Let PT for T  U be the diagonal projection matrix with the diagonal entries (PT )ii = T (ui). Projection matrices are idempotent, i.e., PTPT = PT and equal their transpose PT = P tT . The usual conditional probability of the classical event T given the classical event S can be computed as: Pr (T jS) := Pr(S\T )Pr(S) = tr [PT  (S)]. In general, the probability of getting the value j conditioned by the density matrix  is dened as: Pr (j j) := tr  PBj  . In particular, starting with the conditioning event being the superposition event corresponding to S, that probability is: Pr (j j (S)) = tr  PBj (S)  = Pr(Bj\S) Pr(S) . This yields the perhaps surprising result that the probabilities for the values of a random variable (or any given event T ) are the same if the conditioning event is the classical event S represented by the mixed  (S) or the superposition event S represented by the pure  (S): Pr (j j (S)) = tr  PBj (S)  = Pr(Bj\S) Pr(S) = tr  PBj (S)  = Pr (j j (S)). But the interpretation is quite di¤erent. The classical trial starting with the subset S represented by  (S) picks out the subset Bj \ S represented by  ( (Bj \ S)) with probability Pr (j jS) = tr  PBj (S)  . However, the measurementof the superposition event S represented by  (S) sharpens or projects that indenite event to the more denite superposition event  (Bj \ S) represented by  ( (Bj \ S)) with probability Pr (j jS) = tr  PBj (S)  . In either case, the followup trial or measurement returns the same value j with probability 1, i.e., Pr (j jBj \ S) = tr  PBj ( (Bj \ S))  = tr  PBj ( (Bj \ S))  = 1. In the classical case, all the elements of Bj \ S have the value j so the conditioning classical event Bj \ S occurs with probability 1. In the superposition case, the property of having the value j is denite on the superposition event  (Bj \ S) represented by  ( (Bj \ S)), so no sharpeningoccurs and projection PBj restricted to Bj \ S is the identity so the measurement returns the same event  (Bj \ S) with probability 1. Let us illustrate this result with the case of ipping a fair coin. The classical set of outcomes U = fH;Tg is represented by the density matrix:  (U) =  1 2 0 0 12  . Figure 3: Classical event: trial picks out heads or tails 4 The superposition event U , that blends or superposes heads and tails, is represented by the density matrix:  (U) =  1 2 1 2 1 2 1 2  . Figure 4: Superposition event: Measurement sharpens to heads or tails. The probability of getting heads in each case is: Pr (Hj (U)) = tr  PfHg (U)  = tr  1 0 0 0   1 2 0 0 12  = tr  1 2 0 0 0  = 12 Pr (Hj (U)) = tr  PfHg (U)  = tr  1 0 0 0   1 2 1 2 1 2 1 2  = tr  1 2 1 2 0 0  = 12 and similarly for tails. Thus the two conditioning events U and U cannot be distinguished by performing an experiment or measurement that distinguishes heads and tails. But this actually should not be too surprising since the same thing occurs in quantum mechanics. For instance, a spin measurement along, say, the z-axis of an electron cannot distinguish between the superposition state 1p 2 (j"i+ j#i) with a density matrix like  (U) and a statistical mixture of half electrons with spin up and half with spin down with a density matrix like  (U) [1, p. 176]. The states can only be distinguished by measuring in a di¤erent basis, and we will show in a later section how probability theory with superposition events can be further enriched to demonstrate that possibility. It might be further noticed that the average value of a random variable can also be computed in that same manner as in QM. If Of is the n  n diagonal matrix with diagonal entries f (ui) which represents f : U ! R, then the average value of the random variable restricted to a subset S,P ui2S Pr (ijS) f (ui), is: hfiS = tr [Of (S)] = tr [Of (S)]. Average value of random variable f on S. The probability Pr (T jS) = tr [PT  (S)] = tr [PT  (S)] is just the average value of the characteristic function T : U ! f0; 1g on S considered as a random variable on U , i.e., OT = PT . In particular, Pr (S) = tr [PS (U)] = tr[PS (U)] is the average value of S on U . 4 How MeasurementTransforms Density Matrices Since events, classical or superposition and any probability mixture thereof, are now dealt with using density matrices, we need to dene the resulting change in the density matrix when a trial, an experiment, or a measurement of a random variable occurs. Since the density matrix  (S) is constructed as jSi times its transpose hSj, the corresponding transformation by the projection matrix PT is: 5 PT  (S)P t T = PT jSi hSjPT = Pr(T\S) Pr(S)  ( (T \ S)) since the preand post-multiplying by PT zeros all the entries in jSi hSj except the ones q pi Pr(S) pk Pr(S) = 1 Pr(S) p pipk for ui; uk 2 T \ S, and  ( (T \ S)) has the entries 1Pr(T\S) p pipk for the same ui; uk 2 T \ S, so Pr(T\S)Pr(S) 1 Pr(T\S) p pipk = 1 Pr(S) p pipk giving the result. When T = Bj = f 1 (j), PBj (S)PBj = Pr(Bj\S)) Pr(S)  ( (Bj \ S)). When the outcome of the experiment is j with probability Pr (j jS) = Pr(Bj\S)Pr(S) , then the superposition event S represented by the density matrix  (S) is transformed into the superposition event  (Bj \ S) represented by the density matrix  ( (Bj \ S)). The partition induced on S by  = fBjgmj=1 =  f 1 (j) m j=1 is   S, the partition of all the non-empty blocks Bj \ S for j = 1; :::;m. The density matrix associated with all the probabilistic results is the mixed sum of the density matrices  ( (Bj \ S)) weighted by their probabilities Pr (j jS) = Pr(Bj\S)Pr(S) which is denoted by  (  S). Thus we have:  (  S) := Pm j=1 Pr (j jS)  ( (Bj \ S)) = Pm j=1 Pr(Bj\S) Pr(S)  ( (Bj \ S)) = Pm j=1 PBj (S)PBj The Lüders mixture operation:  (S)  (  S). The operation of experimenting with or measuringthe random variable f : U ! R starting with the superposition event S represented by the pure density matrix  (S) transforms it into the mixture  (  S) = Pm j=1 PBj (S)PBj , and that transformation is called the Lüders mixture operation [1, p, 279] in quantum mechanics. As an example, let us take S = f|;};g  U = f|;};~;g and take f : U ! f0; 1g  R as a random variable that distinguished the color of the suits so  = fB1; B2g =  f 1 (0) ; f 1 (1) = ff};~g ; f|;gg. Then we have:  (S) = 2664 1 3 1 3 0 1 3 1 3 1 3 0 1 3 0 0 0 0 1 3 1 3 0 1 3 3775. And the probability in an experiment of getting a black suite where B2 = f 1 (1) = f|;g is: Pr (1jS) = tr [B2 (S)] = tr 2664 2664 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3775 2664 1 3 1 3 0 1 3 1 3 1 3 0 1 3 0 0 0 0 1 3 1 3 0 1 3 3775 3775 = tr 2664 1 3 1 3 0 1 3 0 0 0 0 0 0 0 0 1 3 1 3 0 1 3 3775 = 23 . The experiment of measuring the suite-colors starting with S transforms the density matrix  (S) according to the Lüders mixture operation:  (  S) = P2 j=1 PBj (S)PBj = 2664 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 3775 2664 1 3 1 3 0 1 3 1 3 1 3 0 1 3 0 0 0 0 1 3 1 3 0 1 3 3775 2664 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 3775 + 2664 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3775 2664 1 3 1 3 0 1 3 1 3 1 3 0 1 3 0 0 0 0 1 3 1 3 0 1 3 3775 2664 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 775 6 = 2664 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 3775+ 2664 1 3 0 0 1 3 0 0 0 0 0 0 0 0 1 3 0 0 1 3 3775 = 2664 1 3 0 0 1 3 0 13 0 0 0 0 0 0 1 3 0 0 1 3 3775. 5 Measurement and Logical Entropy The logical entropy of a partition [4]  = fB1; :::; Bmg on U is: h () := Pm j=1 Pr (Bj) (1  Pr (Bj)) = 1  Pm j=1 Pr (Bj) 2 = P j 6=j0 Pr (Bj) Pr (Bj0) and the logical entropy of any probability distribution q = fq1; :::; qmg is similarly: h (q) = 1  Pm j=1 q 2 j = P j 6=j0 qjqj0 = 2 P j<j0 qjqj0 . The interpretation of the logical entropy of  is the probability in an ordered pair of independent draws or trials to get elements distinguished by  (i.e., elements from di¤erent blocks of ) or di¤erent qjs. The logical entropy of any density matrix  is: h () = tr [ (1  )] = 1  tr  2  . The trace of any density matrix squared is the sum of all the squared entries (or the absolute squares in complex density matrices of QM): tr  2  = Pn i;k=1 jikj 2 [6, p. 77]. When the partition  is represented by the density matrix  () = P j Pr (Bj)  (Bj), then a simple calculation shows that: h ( ()) = 1  tr h  () 2 i = 1  Pm j=1 Pr (Bj) 2 = h (). Since the trace of any density matrix is 1 and for any pure density matrix, 2 = , tr  2  = tr [] = 1 so the logical entropy of any pure density matrix is 0. Logical entropy measures distinctions, and in a pure superposition event S, there are no distinctions between the superposed or cohered outcomes. When an o¤-diagonal element of a density matrix is non-zero, that means the corresponding diagonal elements cohere together or are superposed in a superposition. But when the experiment or measurement operationdistinguishes (or decoheres) those elements, the corresponding o¤-diagonal elements are zeroed. Since the logical entropy measures distinctions, the logical entropy created by the measurement operation can be computed as the squares of the o¤-diagonal elements zeroed in the Lüders mixture operation on the density matrices. Theorem 1 The logical entropy created in the measurement of  (S) by , i.e. h ( (  S))   h ( (S)) [which equals h ( (  S)) since  (S) is pure], is the sum of the squares of the o¤diagonal elements in  (S) that are zeroed in the Lüders mixture operation  (S)  (  S). Proof: All elements in the density matrix  (S) either have the same value (e.g., all diagonal elements and some o¤-diagonal elements) or are zeroed (e.g., some o¤-diagonal elements) by the projections in the Lüders mixture operation. Hence the sum of squares of the o¤-diagonal elements that are zeroed is:Pn i;k=1  (S) 2 ik   Pn i;k=1  (  S) 2 ik = tr h  (S) 2 i   tr h  (  S)2 i =  1  tr h  (  S)2 i    1  tr h  (S) 2 i = h (  S)  h ( (S)).  This theorem holds,mutatis mutandis, for quantum logical entropy and the Lüders mixture operation in quantum information theory where the squares are absolute squares [5]. To illustrate the theorem, consider the previous suite-color measurement where S = f|;};g, The logical entropy of the pure  (S) is 0, and: 7  (  S)2 = 2664 1 3 0 0 1 3 0 13 0 0 0 0 0 0 1 3 0 0 1 3 3775 2664 1 3 0 0 1 3 0 13 0 0 0 0 0 0 1 3 0 0 1 3 3775 = 2664 2 9 0 0 2 9 0 19 0 0 0 0 0 0 2 9 0 0 2 9 3775 so h ( (  S)) = 1  tr h  (  S)2 i = 1  59 = 4 9 . Comparing the before and after matrices,  (S) = 2664 1 3 1 3 0 1 3 1 3 1 3 0 1 3 0 0 0 0 1 3 1 3 0 1 3 3775 2664 1 3 0 0 1 3 0 13 0 0 0 0 0 0 1 3 0 0 1 3 3775 =  (  S), we see that four entries of 13 are zeroed (since the di¤erent colors were distinguished by the color measurement) and the sum of their squares is also 49 as per the theorem. For illustrative purposes, we might represent the matrix associated with the superposition event S for S = f|;};g represented by  (S) as: 2664 f|;|g f|;}g 0 f|;g f};|g f};}g 0 f};g 0 0 0 0 f;|g f;}g 0 f;g 3775 so it is clear that the four o¤-diagonal elements zeroed by the measurement (that distinguished color) are the four that cohered di¤erent colored suites together in the superposition. The suit-color partition  = ff};~g ; f|;gg restricted to S = f|;};g is   S = ff}g ; f|;gg. In two independent ordered draws from S, the probability of getting elements from di¤erent blocks of   S is 13 2 3+ 2 3 1 3 = 4 9 = h ( (  S)) ;and that is the general interpretation of h (), the probability in two ordered draws of getting elements in distinct blocks of . 6 The Pedagogical Model of Quantum Mechanics over Z2 The previous results including the fundamental theorem connecting measurement and logical entropy holdmutatis mutandisin quantum mechanics (QM) when superposition states are being measured using a given (orthonormal) basis U = fu1; :::; ung of an observable.[5] But many results in QM require consideration of di¤erent bases. The above results about probabilities using superposition events can be extended in the pedagogical model of quantum mechanics over Z2 (QM/Sets) [3] where the state space is Zn2 and where the n-ary zero-one vectors are considered as subsets of the basis set with equiprobable outcomes. Then U is just one basis which could be taken as the computational basis, but there are many other bases. By Gausss formula [7, p. 71], the number of ordered bases for Zn2 are: (2n   1)   2n   21  :::   2n   2n 1  and the number of unordered bases is obtained by dividing by n!. For n = 2; there are   22   1    22   21  1 2! = 3 (unordered) bases of Z 2 2. In the coin-ipping example where U = fH;Tg was taken as the outcome set, there is another basis U 0 = fH 0; T 0g where fH 0g = fH;Tg and fT 0g = fTg which is a basis since fH 0g + fT 0g = fH;Tg + fTg = fHg (mod 2 addition) and fT 0g = fTg. The third basis is for U 00 = fH 00; T 00g where fH 00g = fHg and fT 00g = fH;Tg. Since we have di¤erent bases for Z22, we can consider a ket as an abstract vector that can be represented in di¤erent bases, e.g., fHg, fH 0; T 0g, and fH 00g all represent the same abstract vector in di¤erent bases. Then we can form a ket-table where each row represents a ket. In Z22, there are 22   1 = 3 non-zero abstract vectors, each corresponding to a row in the ket-table. 8 U -basis U 0-basis U 00-basis fH;Tg fH 0g fT 00g fHg fH 0; T 0g fH 00g fTg fT 0g fH 00; T 00g Figure 5: ket-table for Z22. Each ket or abstract vector is a superposition in one basis and a singleton event in the other two bases. We saw previously that we could not distinguish the classical mixture event U associated with  (U) from the superposition event U associated with  (U) when measured in the U -basis. For instance, the probability of getting heads in the two cases is: Pr (Hj (U)) = tr  PfHg (U)  = tr  1 0 0 0   1 2 0 0 12  = tr  1 2 0 0 0  = 12 Pr (Hj (U)) = tr  PfHg (U)  = tr  1 0 0 0   1 2 1 2 1 2 1 2  = tr  1 2 1 2 0 0  = 12 . But the two events can be distinguished when measured in a di¤erent basis such as the U 0-basis. The vector fHg is expressed in the U -basis by the column vector  1 0  U (the subscript indicating the basis) and in the U 0-basis by the column vector  1 1  U 0 . The basis conversion matrix is CU!U 0 =  1 0 1 1  so  1 0 1 1   1 0  U =  1 1  U 0 . Hence converting the superposition  1 1  U or fH;Tg to the U 0-basis gives: CU!U 0  1 1  U =  1 0 1 1   1 1  U =  1 0  U 0 or fH 0g so its density matrix (computing in the reals) is 1 0  U 0  1 0  U 0 =  1 0 0 0  U 0 . The classical mixed event U is the half-half mixture of fHg and fTg. The basis conversion for fHg gives CU!U 0  1 0  U =  1 0 1 1   1 0  U =  1 1  U 0 so the associated real density matrix is: " 1p 2 1p 2 # U 0 h 1p 2 1p 2 i U 0 =  1 2 1 2 1 2 1 2  U 0 and for fTg, CU!U 0  0 1  U =  1 0 1 1   0 1  U =  0 1  U 0 so its real density matrix is:  0 1  U 0  0 1  U 0 =  0 0 0 1  U 0 . Their half-half mixture has the density matrix in the U 0-basis: 1 2  1 2 1 2 1 2 1 2  U 0 + 12  0 0 0 1  U 0 =  1 4 1 4 1 4 3 4  U 0 . We then measure by the partition  = ffH 0g ; fT 0gg with half-half probabilities so the probability of H 0 for the superposition event fH;Tg or fH 0g in the U 0-basis is: tr  PfH0g  1 0 0 0  U 0  = tr  1 0 0 0  U 0  1 0 0 0  U 0  = tr  1 0 0 0  U 0 = 1 9 and for the classical mixture of half fHg and half fTgwhich in the U 0-basis is the mixture of half fH 0; T 0gand half fT 0g, is: tr  PfH0g  1 4 1 4 1 4 3 4  U 0  = tr  1 0 0 0  U 0  1 4 1 4 1 4 3 4  U 0  = tr  1 4 1 4 0 0  U 0 = 14 . The rst calculation makes intuitive sense since the superposition fH;Tg in the U -basis is the singleton event fH 0g in the U 0-basis, so measuring in the U 0-basis for the event fH 0g will give fH 0g with probability 1. The second calculation makes intuitive sense since it is half-half in the mixture whether we get the fT 0g event or the fH 0; T 0g event and then the probability of getting H 0 is zero for the fT 0g event and 12 for the fH 0; T 0g event so the overall probability of fH 0g is 14 . Thus the two events, the classical mixture of half fHg and half fTg, and the superposition fH;Tg, which cannot be distinguished by measurements in the U -basis, can be distinguished by measurement in the U 0-basis. 7 Concluding Remarks Ordinary nite probability theory can be extended to include superposition events by using the two-dimensional representations of:   (S) for the classical event S  U , where the outcomes in S are kept discrete and completely decohered, and   (S) for the superposition event S that superposes or coheres together the outcomes in S. The calculation of probabilities for classical events in ordinary nite probability theory can be computed using the density matrices in the form  (S) for classical events S. Thus the extension to include superposition events just extends to using density matrices of the form  (S), and the density matrix formalism also represents classical mixtures of superposition events. Ordinary nite probability theory sticks with one outcome or sample space U . But the whole machinery can be developed in Zn2 where U is just one among many basis sets and then it is part of the pedagogical model of quantum mechanics over Z2 or QM/Sets. That pedagogical model of QM over Z2 could also be viewed as just the non-commutative extension of nite probability theory with superposition events (since the bases do not in general commute in QM/Sets). Many characteristic QM results can be modeled in this non-commutative probability theory such as the double-slit experiment, the indeterminacy principle, quantum statistics for identical particles, and even Bells Theorem.[3] Our purpose has been to illustrate, in a rather classical setting, the notion of a superposition event, where all the outcomes in the event cohere together (with various amplitudes), so the event is objectively indenite between those outcomes. The notions of objective-indeniteness and superposition are the essentials in what Abner Shimony called the "Literal" or objectively-indenite interpretation of QM, an interpretation that is routinely neglected in the literature that focuses on fantasies about many worlds or hidden variables. From these two basic ideas alone  indeniteness and the superposition principle  it should be clear already that quantum mechanics conicts sharply with common sense. If the quantum state of a system is a complete description of the system, then a quantity that has an indenite value in that quantum state is objectively indenite; its value is not merely unknown by the scientist who seeks to describe the system. [8, p. 47] But the mathematical formalism ... suggests a philosophical interpretation of quantum mechanics which I shall call "the Literal Interpretation." ...This is the interpretation 10 resulting from taking the formalism of quantum mechanics literally, as giving a representation of physical properties themselves, rather than of human knowledge of them, and by taking this representation to be complete. [9, pp. 6-7] To understand or interpret QM, one needs to better understand the notions of objective indeniteness and superposition as well as the related notion of a (distinguishing) measurement that sharpens an indenite superposition event to a mixture of more denite ones. We have shown that the concepts of superposition, objective-indeniteness, and measurement can be illustrated in a very small extension of classical nite probability theorywhich should help to intuitively understand those notions in quantum mechanics. References [1] Auletta, Gennaro, Mauro Fortunato, and Giorgio Parisi. 2009. Quantum Mechanics. Cambridge UK: Cambridge University Press. [2] Ellerman, David. 2014. An Introduction to Partition Logic.Logic Journal of the IGPL 22 (1): 94125. https://doi.org/10.1093/jigpal/jzt036. [3] Ellerman, David. 2017. Quantum Mechanics over Sets: A Pedagogical Model with NonCommutative nite Probability Theory as Its Quantum Probability Calculus. Synthese 194 (12): 486396. [4] Ellerman, David. 2017. Logical Information Theory: New Foundations for Information Theory. Logic Journal of the IGPL 25 (5 Oct.): 80635. [5] Ellerman, David. 2018. Logical Entropy: Introduction to Classical and Quantum Logical Information Theory.Entropy 20 (9): Article ID 679. https://doi.org/10.3390/e20090679. [6] Fano, Ugo. 1957. Description of States in Quantum Mechanics by Density Matrix and Operator Techniques.Reviews of Modern Physics 29 (1): 7493. [7] Lidl, Rudolf, and Harald Niederreiter. 1986. Introduction to Finite Fields and Their Applications. Cambridge UK: Cambridge University Press. [8] Shimony, Abner 1988. The reality of the quantum world. Scientic American. 258 (1): 46-53. [9] Shimony, Abner. 1999. Philosophical and Experimental Perspectives on Quantum Physics. In Philosophical and Experimental Perspectives on Quantum Physics: Vienna Circle Institute Yearbook 7. Dordrecht: Springer Science+Business Media: 1-18.