aa r X i v : . [ qu a n t - ph ] J un Hardy’s paradox according to non-classical semantics
Arkady Bolotin ∗ Ben-Gurion University of the Negev, Beersheba (Israel)
October 16, 2018
Abstract
In the paper, using the language of spin-half particles, Hardy’s paradox is examined withindifferent semantics: a partial one, a many-valued one, and one defined as a set of weak valuesof projection operators. As it is shown in this paper, any of such non-classical semantics canresolve Hardy’s paradox.
Keywords:
Quantum mechanics; Hardy’s paradox; Truth values; Partial semantics; Many-valued semantics; Weak values of projection operators.
In essence, Hardy’s paradox can be reduced to a case of contradiction in classical logic.Indeed, let the letters A and B denote respectively a positron and an electron entering their corre-sponding superimposed Mach-Zehnder interferometers depicted in Figure 1 (drawn in conformitywith [1, 2, 3]). These interferometers have non-overlapping arms (denoted as N A and N B ) in addi-tion to the overlapping arms (which are denoted as O A and O B ). Besides, each interferometer isequipped with two detectors (represented by the symbols D A1 , D A2 as well as D B1 , D B2 ) capable ofdetecting the exit of the particle from the interferometer.Consider at first each interferometer separately. By way of the adjustment of the arm lengthsbetween the beam-splitters (BS A1 and BS A2 or BS B1 and BS B2 ) it is possible to make the particle Aemerge only at D A2 or the particle B only at D B2 . Then again, the detector D A1 or D B1 can be alsotriggered if an obstruction is present in the arm O A or O B .According to the design of Hardy’s thought experiment, if each interferometer is considered sep-arately, the particle A can be detected only at D A2 and the particle B only at D B2 . However, dueto the presence of the overlapping arms in the setup of the experiment (and, for that reason, thepossibility of the obstruction), the detectors D A1 and D B1 may be triggered.Thus, from the clicking of D B1 one can infer that the particle A has gone through the overlappingarm O A obstructing the particle B (and because of this the particle B was not able to get to D B2 ). ∗ Email : arkadyv @ bgu.ac.il S BS BS BS BA N B O B N B O B O A N A O A N A D D D A D Figure 1: Diagram representing the design of Hardy’s thought experiment.Similarly, the click of D A1 would mean that the particle B went through the overlapping arm O B obstructing the particle A (which, as a result, could not reach D A2 ).Obviously, if both the particle A and the particle B went through the overlapping arms O A andO B , they would annihilate. So, classically speaking, the simultaneous clicking of the detectors D A1 and D B1 is impossible.Let v be a valuation , that is, a mapping from a set of propositions {⋄} (where the symbol ⋄ standsfor any proposition, compound or simple) to a set of truth-values V N = { v } having the cardinality N and the range with the upper bound 1 (representing the truth ) and the lower bound 0 (repre-senting the falsehood ). As it is customary [4], the double-bracket notation [[ ⋄ ]] v is used to denotethe valuation v .Based on the setup of the interferometers, one can derive the next equalities:[[ D B1 ]] v = [[ O A ]] v , (1)[[ D A1 ]] v = [[ O B ]] v . (2)They mean, for example, that the proposition asserting the passage of the particle A through theoverlapping arm O A has the same value as the proposition asserting the click of the detector D B1 ;2n other words, these propositions are both true or both false.From these equalities it immediately follows[[ D ]] v = [[ O ]] v , (3)where the symbol D stands for the proposition of the simultaneous clicking of the detectors D B1 and D A1 , while the symbol O denotes the proposition that both particles travelled through theoverlapping arms O B and O A in their respective interferometers.On the other hand, the situation where both particles were in the overlapping arms O B and O A cannot occur because of the annihilation process. Accordingly, in the initial state of the experi-ment, i.e., before the verification of the proposition D , the proposition O is false, namely, [[ O ]] v = 0.Let P [[[ ⋄ ]] v = 1] ∈ [0 ,
1] be the value that represents the probability that the proposition ⋄ is true .Assume that the following conditions are satisfied:[[ ⋄ ]] v = 1 ⇐⇒ P [[[ ⋄ ]] v = 1] = 1 , (4)[[ ⋄ ]] v = 0 ⇐⇒ P [[[ ⋄ ]] v = 1] = 0 . (5)Then, in terms of a probabilistic logic [5, 6], the equality (3) would imply that the probability ofthe simultaneous clicking of the detectors D B1 and D A1 must be zero:[[ O ]] v = 0 = ⇒ P [[[ D ]] v = 1] = 0 . (6)However, in accordance with quantum mechanics, the probability P [[[ D ]] v = 1] is different fromzero, that is, sometimes the particles do emerge simultaneously at D B1 and D A1 . This constitutesHardy’s paradox, namely, [[ O ]] v = 0[[ D ]] v = [[ O ]] v (cid:27) = ⇒ P [[[ D ]] v = 1] = 0 . (7)As one can see, the paradox arises because of the implicit assumption supposing that any propo-sition relating to a quantum system is either true or false . Mathematically, it is equivalent to thestatement that a logic lying behind quantum phenomena has a non-partial bivalent semantics thatis defined as a set of bivaluations [[ ⋄ ]] v ∈ V = { , } .This suggests that the said paradox might not appear within semantics that are partial or many-valued (or both).Really, consider a “gappy” semantics where the valuation v is the function from propositions {⋄} into the set V = { , } such that v is not total . In this case, even if the proposition O is false,3ome propositions, e.g., D , might have no truth-values at all, which can explain the deviation ofthe probability P [[[ D ]] v = 1] from zero.Let us demonstrate a resolution of the paradox (7) based on the aforesaid gappy semantics as wellas gapless many-valued semantics. Let ˆ P ⋄ be the projection operator on the Hilbert space H associated with some quantum systemsuch that ˆ P ⋄ corresponds to the proposition ⋄ related to this system. Assume that the valuationalaxiom hold v ( ˆ P ⋄ ) = [[ ⋄ ]] v , (8)where v is the truth-value assignment function.To find out how this function works, let us take a quantum system prepared in a pure state | Ψ α i lying in the column space (range) of the projection operator ˆ P α . Since being in ran( ˆ P α ) meansˆ P α | Ψ α i = 1 · | Ψ α i , one can assume that in the state | Ψ α i ∈ ran( ˆ P α ), the truth-value assignmentfunction v assigns the truth value 1 to the projection operator ˆ P α and, in this way, the proposition α , specifically, v ( ˆ P α ) = [[ α ]] v = 1. Contrariwise, if v ( ˆ P α ) = [[ α ]] v = 1, then one can assume that thesystem is prepared in the state | Ψ α i ∈ ran( ˆ P α ). These two assumptions can be recorded togetheras the following logical biconditional: | Ψ α i ∈ ran( ˆ P α ) ⇐⇒ v ( ˆ P α ) = [[ α ]] v = 1 . (9)On the other hand, the vector | Ψ α i is in the null space of any projection operator ˆ P β orthogonalto ˆ P α . Since being in ker( ˆ P β ) means ˆ P β | Ψ α i = 0 · | Ψ α i , one can assume then | Ψ α i ∈ ker( ˆ P β ) = ran(ˆ1 − ˆ P β ) ⇐⇒ v ( ˆ P β ) = [[ β ]] v = 0 . (10)Bringing the last two assumptions into a union, one can write down the following bivaluations | Ψ i ∈ (cid:26) ran( ˆ P ⋄ )ran( ¬ ˆ P ⋄ ) ⇐⇒ v ( ˆ P ⋄ ) = [[ ⋄ ]] v ∈ V , (11)where the operation ¬ ˆ P ⋄ = ˆ1 − ˆ P ⋄ is understood as negation of ˆ P ⋄ .Suppose by contrast that the system is prepared in the state | Ψ i that does not lie in the column ornull space of the projection operator ˆ P ⋄ , i.e., | Ψ i / ∈ ran( ˆ P ⋄ ) and at the same time | Ψ i / ∈ ran( ¬ ˆ P ⋄ ).Then, under the bivaluations (11), the truth-value function v must assign neither 1 nor 0 to ˆ P ⋄ ,that is, v ( ˆ P ⋄ ) = 1 and v ( ˆ P ⋄ ) = 0. This means that the proposition ⋄ associated with ˆ P ⋄ cannot be4ivalent under the function v , namely, [[ ⋄ ]] v / ∈ V .Using a gappy yet two-valued semantics , the failure of bivalence can be described as the truth-valuegaps, explicitly, | Ψ i / ∈ (cid:26) ran( ˆ P ⋄ )ran( ¬ ˆ P ⋄ ) ⇐⇒ { v ( ˆ P ⋄ ) } = { [[ ⋄ ]] v } = ∅ . (12)Observe that under the bivaluations (11), the projection operators ˆ1 and ˆ0 are true and false,respectively, in any arbitrary state | Ψ i ∈ H , namely, | Ψ i ∈ (cid:26) ran(ˆ1) = H ran( ¬ ˆ0) = H ⇐⇒ (cid:26) v (ˆ1) = 1 v (ˆ0) = 0 . (13)Therefore, under those bivaluations, the operator ˆ1 can be equated with “ the super-truth ” since itcan be assigned the value of the truth in all admissible states of the quantum system. Similarly,the operator ˆ0 can be equated with “ the super-falsity ” because it can be assigned the value of thefalsity in all admissible states of the quantum system. Accordingly, one can call gappy semanticsdefined as the set of the bivaluations (11) and the truth-value gaps (12) quantum supervaluationism (for other details of such semantics see, for example, [7, 8] and also [9, 10]). In the language of spin-half particles, particle A’s states | O A i and | N A i in the overlapping andnon-overlapping arms of the Mach-Zehnder interferometer can be represented by the normalizedeigenvectors of Pauli matrices corresponding to the eigenvalues +1 and −
1, namely, | O A i def = (cid:20) (cid:21) , | N A i def = (cid:20) (cid:21) . (14)Obviously, kets | O B i and | N B i can be represented in the same way.Using such a representation, the projection operators corresponding to the propositions O A andN A are defined by ˆ P O A = | O A ih O A | = (cid:20) (cid:21) , (15)ˆ P N A = | N A ih N A | = (cid:20) (cid:21) . (16)Consequently, the projection operator ˆ P O relating to the proposition O can be expressed as5 P O = ˆ P O A ⊗ ˆ P O B = | O A ih O A | ⊗ | O B ih O B | = (cid:20) (cid:21) ⊗ (cid:20) (cid:21) = . (17)The column and null spaces of this operator areran( ˆ P O ) = a : a ∈ R , (18)ran( ¬ ˆ P O ) = bcd : b, c, d ∈ R . (19)According to the bivaluations (11), for the proposition O to have the value of the falsity in thestate | Ψ ¬ O i , the last-named must lie in the null space of the projection operator ˆ P O : | Ψ ¬ O i ∈ ran( ¬ ˆ P O ) ⇐⇒ v ( ˆ P O ) = [[ O ]] v = 0 . (20)Provided b = c = d , this state | Ψ ¬ O i can be written down as | Ψ ¬ O i = 1 √ = 1 √ (cid:18) | O A i ⊗ | N B i + | N A i ⊗ | O B i + | N A i ⊗ | N B i (cid:19) (21)and taken as the initial state of the system (i.e., the state prior to the verification).The detectors in particle A’s interferometer verify the values of the operators ˆ P D A2 and ˆ P D A1 , whoseprojections on the states | O A i and | N A i can be defined by the following superpositions | D A2 i = 1 √ (cid:18) | O A i + | N A i (cid:19) = 1 √ (cid:18)(cid:20) (cid:21) + (cid:20) (cid:21)(cid:19) = 1 √ (cid:18)(cid:20) (cid:21)(cid:19) , (22) | D A1 i = 1 √ (cid:18) | O A i − | N A i (cid:19) = 1 √ (cid:18)(cid:20) (cid:21) − (cid:20) (cid:21)(cid:19) = 1 √ (cid:18)(cid:20) − (cid:21)(cid:19) . (23)These superpositions (and equivalent ones for particle B’s states | O B i and | N B i ) allow one toidentify the projection operators ˆ P D A2 and ˆ P D A1 (along with ˆ P D B2 and ˆ P D B1 )ˆ P D A2 = | D A2 ih D A2 | = 12 (cid:20) (cid:21) , (24)6 P D A1 = | D A1 ih D A1 | = 12 (cid:20) − − (cid:21) , (25)and, then, the projection operators ˆ P D and ˆ P D ˆ P D = ˆ P D A2 ⊗ ˆ P D B2 = 14 , (26)ˆ P D = ˆ P D A1 ⊗ ˆ P D B1 = 14 − − − − − − − − , (27)whose column and null spaces areran( ˆ P D ) = aaaa : a ∈ R , ran( ¬ ˆ P D ) = − b − c − dbcd : b, c, d ∈ R , (28)ran( ˆ P D ) = a − a − aa : a ∈ R , ran( ¬ ˆ P D ) = b + c − dbcd : b, c, d ∈ R . (29)As follows, the null space of the projection operator ˆ P O cannot be a subset or superset of thecolumn space or the null space of the projection operator ˆ P D , namely, bcd : b, c, d ∈ R a − a − aa : a ∈ R b + c − dbcd : b, c, d ∈ R . (30)Under the truth-value gaps (12), this necessitates absolutely no truth value for the proposition D in the initial state | Ψ ¬ O i : | Ψ ¬ O i = 1 √ / ∈ (cid:26) ran( ˆ P D )ran( ¬ ˆ P D ) ⇐⇒ { v ( ˆ P D ) } = { [[ D ]] v } = ∅ . (31)7his in turn implies that, in accordance with the probability postulation (5), the a priori probabilityof the simultaneous clicking of the detectors D B1 and D A1 must not be equal to zero : | Ψ ¬ O i ∈ ran( ¬ ˆ P O ) ⇐⇒ [[ O ]] v = 0 { [[ D ]] v } = ∅ (cid:27) = ⇒ P [[[ D ]] v = 1] = 0 . (32)Contrariwise, when the detectors D B1 and D A1 click simultaneously and thus the system is found inthe state | Ψ D i , namely, | Ψ D i = | D A1 i ⊗ | D B1 i = 12 − − ∈ ran( ˆ P D ) = ⇒ v ( ˆ P D ) = [[ D ]] v = 1 , (33)the proposition asserting that both particles have passed the overlapping arms in their respectiveinterferometers would possess no value at all: | Ψ D i / ∈ (cid:26) ran( ˆ P O )ran( ¬ ˆ P O ) ⇐⇒ { v ( ˆ P O ) } = { [[ O ]] v } = ∅ . (34)Accordingly, in supervaluationist (i.e., gappy and yet bivalent) semantics, the question “Which waydid the particle take?” has no sense. As it is mentioned in [11], for gappy semantics, one can construct a gapless (non-classical) semanticsin which different degrees of truth would fill out the truth-value gaps.
To that end, instead of the truth-value gaps (12) consider the gapless valuations, namely, | Ψ i / ∈ (cid:26) ran( ˆ P ⋄ )ran( ¬ ˆ P ⋄ ) ⇐⇒ v P ( ˆ P ⋄ ) = h Ψ | ˆ P ⋄ | Ψ i , (35)where the function v P assigning the truth value to the projection operator ˆ P ⋄ in the state | Ψ i is deter-mined by the probability P [[[ ⋄ ]] v = 1] = h Ψ | ˆ P ⋄ | Ψ i . According to [12, 13], the value v P represents thedegree to which the proposition ⋄ is true before its verification. As h Ψ | ˆ P ⋄ | Ψ i ∈ { x ∈ R : 0 < x < } ,a semantics defined by the set of the bivaluations (11) and the valuations (35) is infinite-valued .Therewithal, this semantics respects functionality of the evaluation relation; what is more, in thissemantics the evaluation relation is a total relation.8ithin the said semantics, the truth value of the proposition D in the initial state | Ψ ¬ O i and thetruth value of the proposition O in the final state | Ψ D i are given by | Ψ ¬ O i = 1 √ / ∈ (cid:26) ran( ˆ P D )ran( ¬ ˆ P D ) = ⇒ v P ( ˆ P D ) = h Ψ ¬ O | ˆ P D | Ψ ¬ O i = 112 , (36) | Ψ D i = 12 − − / ∈ (cid:26) ran( ˆ P O )ran( ¬ ˆ P O ) = ⇒ v P ( ˆ P O ) = h Ψ D | ˆ P O | Ψ D i = 14 . (37)Both of these truth values differ from the classical truth values “false” and “true”, i.e., 0 and 1.Therefore, in the infinite-valued (i.e., gapless but non-bivalent) semantics the proposition of thesimultaneous clicking of the detectors D B1 and D A1 in the initial state | Ψ ¬ O i is neither true nor false .Along the same lines, the proposition affirming that the particles have taken the overlapping waysO B and O A is neither true nor false in the final state | Ψ D i . Another possibility to construct a gapless semantics presents weak values of the projection opera-tors. To be sure, consider the “weak” valuations | Ψ i / ∈ (cid:26) ran( ˆ P ⋄ )ran( ¬ ˆ P ⋄ ) ⇐⇒ v w ( ˆ P ⋄ ) = h Φ | ˆ P ⋄ | Ψ ih Φ | Ψ i , (38)where the bra h Φ | is some vector of the dual space H ∗ associated with the given quantum system.Had the quantum system been prepared in the state | Ψ i lying in the column or null space of theprojection operator ˆ P ⋄ , such valuations would have coincided with the bivaluations (11). What ismore, had the bra h Φ | been the Hermitian conjugate of the ket | Ψ i , i.e., h Φ | = h Ψ | , the value v w ( ˆ P ⋄ )would have lain in the range { x ∈ R : 0 < x < } .Given that neither of these statements is fulfilled, one can deduce that a semantics defined as theset of the bivaluations (11) and the weak valuations (38) is not bivalent and, unlike many-valuedsemantics, may include “truth degrees” lying beyond the range [0 , v w ( ˆ P ⋄ ) can be interpreted as the divergence (conformity) of the proposition ⋄ from (to) the false , i.e., a trivial proposition whose truth value being always false.9ake, for example, the proposition D in the initial state | Ψ ¬ O i and the proposition O in the finalstate | Ψ D i : Under the weak valuations (38), they can be evaluated as | Ψ ¬ O i / ∈ (cid:26) ran( ˆ P D )ran( ¬ ˆ P D ) = ⇒ v w ( ˆ P D ) = h Ψ D | ˆ P D | Ψ ¬ O ih Ψ D | Ψ ¬ O i = 1 , (39) | Ψ D i / ∈ (cid:26) ran( ˆ P O )ran( ¬ ˆ P O ) = ⇒ v w ( ˆ P O ) = h Ψ ¬ O | ˆ P O | Ψ D ih Ψ ¬ O | Ψ D i = 0 . (40)In line with the said interpretation, these weak truth degrees mean that despite the falsity of theproposition O in the initial state | Ψ ¬ O i , the proposition of the simultaneous clicking D cannotbe regarded as false in this state. By the same token, the fact that in the final state | Ψ D i theproposition D diverges from the false does not imply that in the same state the proposition Odiffers from the false as well.That is to say, in the weak-valued (i.e., gapless and non-bivalent) semantics, it holds that[[ O ]] v = 0 ; [[ D ]] v w = 0 , (41)[[ D ]] v = 0 ; [[ O ]] v w = 0 . (42)As it follows, any of the mentioned in this paper non-classical semantics resolve Hardy’s paradox. Acknowledgment
The author would like to express acknowledgment to the anonymous referee for the constructiveremarks contributing to the improvement of this paper.
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