Manuscript Draft. Title: A Many-valued Approach to Quantum Computational Logics. Keywords: Quantum logics; quantum tomography; logical gates.

Transcription

1 an Systems Elsevier Eitorial System(tm) for Fuzzy Sets Manuscript Draft Manuscript Number: FSS-D--00R Title: A Many-value Approach to Quantum Computational Logics Article Type: SI: LINZ 0 Keywors: Quantum logics; quantum tomography; logical gates. Corresponing Author: Dr. Roberto Leporini, PhD Corresponing Author's Institution: Università egli Stui i Bergamo First Author: Maria Luisa Dalla Chiara, Prof. Orer of Authors: Maria Luisa Dalla Chiara, Prof.; Roberto Giuntini, Prof.; Roberto Leporini, PhD; Giuseppe Sergioli, PhD Abstract: Quantum computational logics are special examples of quantum logic where formulas are suppose to enote pieces of quantum information (qubit-systems or mixtures of qubit-systems), while logical connectives are interprete as reversible quantum logical gates. Hence, any formula of the quantum computational language represents a synthetic logical escription of a quantum circuit. We investigate a many-value approach to quantum information, where the basic notion of qubit has been replace by the more general notion of quit. The quit-semantics allows us to represent as reversible gates some basic logical operations of Lukasiewicz many-value logics. In the final part of the article we iscuss some problems that concern possible implementations of gates by means of optical evices.

2 Abstract Quantum computational logics are special examples of quantum logic where formulas are suppose to enote pieces of quantum information (qubitsystems or mixtures of qubit-systems), while logical connectives are interprete as reversible quantum logical gates. Hence, any formula of the quantum computational language represents a synthetic logical escription of a quantum circuit. We investigate a many-value approach to quantum information, where the basic notion of qubit has been replace by the more general notion of quit. The quit-semantics allows us to represent as reversible gates some basic logical operations of Lukasiewicz many-value logics. In the final part of the article we iscuss some problems that concern possible implementations of gates by means of optical evices.

3 *Manuscript A Many-value Approach to Quantum Computational Logics M.L. Dalla Chiara a, R. Giuntini, G. Sergioli b, R. Leporini c, a Dipartimento i Lettere e Filosofia, Università i Firenze, via Bolognese, I-0 Firenze, Italy. b Dipartimento i Peagogia, Psicologia, Filosofia, Università i Cagliari, via Is Mirrionis, I-0 Cagliari, Italy. c Dipartimento i Ingegneria Gestionale, ell Informazione e ella Prouzione, Università i Bergamo, viale Marconi, I-0 Dalmine (BG), Italy. Abstract Quantum computational logics are special examples of quantum logic where formulas are suppose to enote pieces of quantum information(qubit-systems or mixtures of qubit-systems), while logical connectives are interprete as reversible quantum logical gates. Hence, any formula of the quantum computational language represents a synthetic logical escription of a quantum circuit. We investigate a many-value approach to quantum information, where the basic notion of qubit has been replace by the more general notion of quit. The quit-semantics allows us to represent as reversible gates some basic logical operations of Lukasiewicz many-value logics. In the final part of the article we iscuss some problems that concern possible implementations of gates by means of optical evices. Keywors: Quantum logics, quantum tomography, logical gates.. Introuction The mathematical formalism of quantum theory has inspire the evelopment of ifferent forms of non-classical logics, calle quantum logics. In Corresponing author aresses: allachiara@unifi.it (M.L. Dalla Chiara), giuntini@unica.it, giuseppe.sergioli@gmail.com (R. Giuntini, G. Sergioli), roberto.leporini@unibg.it (R. Leporini) Preprint submitte to Fuzzy sets an systems December, 0

4 many cases the semantic characterizations of these logics are base on special classes of algebraic structures efine in a Hilbert-space environment. The prototypal example of quantum logic(create by Birkhoff an von Neumann) can be semantically characterize by referring to the class of all Hilbert-space lattices, whose support is the set P(H) of all projections of a Hilbert space H. The question whether the class of all Hilbert-space lattices can be axiomatize by a set of equations is still open. What is known is that the variety of all orthomoular lattices (which gives rise to a semantic characterization of a logic often terme abstract quantum logic ) oes not represent a faithful abstraction from the class of all Hilbert-space lattices. A characteristic example of an equation that hols in all Hilbert-space lattices, being possibly violate in orthomoular lattices is the orthoarguesian law [, ]. Interesting generalizations of Birkhoff an von Neumann s quantum logic are the so calle unsharp (or fuzzy) quantum logics that can be semantically characterize by referring to ifferent classes of algebraic structures whose support is the set of all effects of a Hilbert space. Accoring to the stanar interpretation of the quantum formalism, any projection P P(H) represents a sharp physical event to which any possible state of a physical system S (associate with the space H) assigns a probability-value. Such events are calle sharp because they satisfy the non-contraiction principle: P P = O (the infimum between P an its orthogonal projection P is the null projection O). Effects, instea, represent unsharp physical events that may violate the non-contraiction principle. The set E(H)of alleffects ofahilbert space H is efine asthe largest set of linear boune operators E for which a Born-probability can be efine. In other wors, for any ensity operator ρ of H (representing a possible state of a physical system S whose associate Hilbert space is H), we have: Tr(ρE) [0, ] (where Tr is the trace-functional). The number Tr(ρE) represents the probability that a quantum system S in state ρ verifies the physical event represente by the effect E. Of course, E(H) properly inclues P(H). Different kins of algebraic structures have beeninuceonthesete(h),givingrisetovariousformsofunsharp quantum logics [, ]. A ifferent approach to quantum logic has been evelope in the framework of quantum computational logics, inspire by the theory of quantum computation []. While sharp an unsharp quantum logics refer to possible

5 structures of physical events, the basic objects of quantum computational logics are pieces of quantum information: possible states of quantum systems that can store an transmit the information in question. Maximal pieces of information (which cannot be consistently extene to a richer knowlege) correspon to pure states, mathematically represente as unit vectors ψ of convenient Hilbert spaces. At the same time, pieces of information that o not necessarily express a maximal knowlege correspon to mixe states (or mixtures), mathematically represente as ensity operators ρ. Of course any pure state ψ correspons to a special case of a ensity operator: the projectionp ψ thatprojectsover the(one-imensional) closesubspace etermine by ψ. In this article we will investigate particular examples of quantum computational logics base on a many-value approach to quantum information, where the funamental notion of qubit has been replace by the more general concept of quit.. Qubits an quits As is well known, the basic concept of quantum information is the notion of qubit (or qubit-state): a possible pure state of a single quantum system, mathematically represente as a unit-vector ψ of the two-imensional Hilbert space C (base on the set of all orere pair of complex numbers). Accoringly, any qubit can be escribe as a superposition ψ = c 0 0 +c, where 0 = (,0) an = (0,) (the two elements of the canonical basis of C ) represent, in this framework, the two classical bits (0 an ) or, equivalently, the two classical truth-values (Falsity antruth). From an intuitive point of view, a qubit c 0 0 +c can be regare as an uncertain answer to a given question; an answer that might be false with probability c 0 an might be true with probability c. A natural many-value generalization of qubits is represente by quits: unit-vectors living in a space C, where. The elements of the canonical basis of C can be regare as ifferent truth-values, which can be conventionally inicate in the following way: 0 0 = = (,0,...,0) = (0,,0,...,0)

6 = (0,0,,0,...,0). = = (0,...,0,). While 0 an represent the truth-values Falsity an Truth, all other basis-elements correspon to intermeiate truth-values. A particularly interesting example of a quit-space is the qutrit-space C, where the truth-values are: 0 = 0 = (,0,0) = (0,,0) = = (0,0,). From a physical point of view, this space can be naturally use to represent the spin-values of bosons. The eigenvalues of the observable Spin z (the spin in the z-irection, corresponing to the z-component of the angular momentum) can be associate to the elements of the canonical basis; while the spin-observables in all other irections can be associate to ifferent bases of the space. Like classical bits, quits represent atomic pieces of information: answers (which, in the quantum case, are generally uncertain) to single questions. At the same time, complex pieces of quantum information can be naturally represente as possible states of composite quantum systems that can store the information in question. Accoringly, by using the quantum-theoretic formalism for the mathematical representation of composite systems (base on tensor-proucts), any piece of quantum information can be ientifie with a possible (pure or mixe) state of a quantum system: a ensity operator ρ living in a tensor-prouct space H (n) = C }... C {{}, where n. n times While represents the number of truth-values, n represents the number of the components of the quantum system that stores the information ρ. The canonical basis of H (n) (whose elements are calle registers) is the following set: { v,...,v n : v,..., v n are elements of the canonical basis of C }

7 (where v,...,v n is an abbreviation for the tensor prouct v... v n ). A quregister of H (n) is a pure state, represente by a unit-vector ψ (or, equivalently, by the corresponing ensity operator P ψ ). In any space H (n), each truth-value j etermines a corresponing, whose range is the close subspace spanne by truth-value projection P (n) j the set of all registers v,...,v n where v n = j of view, P (n) j. From an intuitive point represents the property having the truth-egree j. In particular, P (n) 0 an P (n) represent the Falsity-property an the Truth-property, respectively. On this basis, one can naturally apply the Born-rule an efine for any state ρ (of H (n) ) the probability that ρ satisfies the property P(n) j ( p j (ρ) := Tr ρp (n) j The probability tout court of ρ can be then efine as the weighte mean of all truth-egrees. Definition. The probability of a ensity operator ρ of H (n). We have: j= ). p () (ρ) := jp j (ρ). ( ) p () (ρ) = Tr ρ(i (n ) E), where I (n ) is the ientity operator (of H (n ) ) an E is the effect (of C ) represente by the following matrix: In theparticular case where ρ correspons to the qubit ψ = c 0 0 +c, we obtain that p () (ρ) = c. :

8 Due to the properties of the quantum-theoretic formalism, pieces of quantum information turn out to satisfy the characteristic holistic features of quantum states. Generally quantum information flows from a whole to its parts, an not the other way aroun, as happens either in classical information or in the semantics of many important logics (like classical logic, intuitionistic logic or stanar fuzzy logics). Consier a quregister ψ representing the pure state of a composite quantum system S consisting of n subsystems S,...,S n (say, an n-electron system), an let H (k +...+k n) be the Hilbert space associate to S. Accoring to the quantum formalism ψ etermines the reuce states Re () [k,...,k n] (P ψ ),...,Re (n) [k,...,k n] (P ψ ) of the subsystems S,...,S n. Generally, the state ψ of the global system S cannot be represente as the factorize state corresponing to the tensorprouct of the states of the subsystems S,...,S n. We may have: P ψ Re () [k,...,k n] (P ψ )... Re (n) [k,...,k n] (P ψ ). It may also happen that the states of all subsystems are one an the same proper mixture (while ψ is a pure state). Consequently, the states of the parts of S turn out to be inistinguishable. Furthermore, the information about the global system is more precise than the information about its parts. In such a case ψ is calle an entangle pure state. Entanglement-phenomena (which have for a long time been escribe as mysterious an potentially paraoxical) represent toay a powerful resource in quantum information; they are currently use, for instance, in quantum teleportation-experiments an in quantum cryptography.. Quantum logical gates Quantum information is processe by quantum logical gates (briefly, gates) that transform pure an mixe pieces of quantum information in a reversible way. When applie to quregisters of a quit-space H (n) a gate is a unitary operator G (n) : a reversible map that transforms all vectors of the space, preserving their length. At the same time, any unitary operator G (n) can be canonically extene to a unitary operation D G (n) that transforms all ensity operators ρ of the space in a reversible way. We have: D G (n) (ρ) := G (n) ρg (n),

9 where G (n) is the ajoint of G (n). In the particular case where ρ is a pure state P ψ, we obtain: D G (n) (P ψ ) = P G (n) ψ. For the sake of simplicity, we will call gate either a unitary operator G (n) or the corresponing unitary operation D G (n). Why is reversibility so important in quantum computation? The reason epens on the form of Schröinger s equation, where unitary operators play an essential role. An, from a physical point of view, any quantum computation can be escribe as the time-evolution of a particular quantum system that transforms a given information-input into an information-output. In the semantics of classical logic an of many important non-classical logics (incluing Birkhoff an von Neumann s quantum logic) the basic logical operations are generally ealt with as irreversible operations. In the (two-value) classical semantics negation only is efine as a reversible truthfunction: v := v, for any truth-value v {0,}. A the same time, both the conjunction an the isjunction are efine as irreversible truth-functions (for any pair of truth-values u, v): u v := min(u,v); u v := (u v ) = max(u,v). For our aims it is expeient to recall what happens in the semantics of many-value Lukasiewicz logics (which represent special examples of fuzzy logics). In this case the set TV of truth-values is ientifie either with the real interval [0, ] or with a finite subset thereof (conventionally inicate as a set { 0,,..., }, where ). The negation is efine like in the classical case: v := v, for any truth-value v TV. At the same time the conjunction is split into two ifferent irreversible operations, the min-conjunction (also calle lattice-conjunction) an the Lukasiewicz-conjunction : u v := min(u,v), u v := max(0,u+v ), for any u,v TV. While an arethesameoperationsinthetwo-valuesemantics, when > our two conjunctions turn out to satisfy ifferent semantic properties.

10 The min-conjunction gives rise to possible violations of the non-contraiction principle. We may have: v v 0. Hence, contraictions are not necessarily false, as happens in the case of most fuzzy logics whose basic aim is moeling ambiguous an unsharp semantic situations. At the same time, behaves as a lattice-operation in the truth-value partial orer (TV, ). The Lukasiewicz-conjunction, instea, is generally non-iempotent. We may have: v v v. Apparently, one is ealing with a kin of conjunction that can be usefully applie to moel semantic situations where repetita iuvant! ( repetitions are useful! ). Asexpecte, thetwoconjunctions an allowustoefinetwoifferent kins of isjunctions (via e Morgan-law): u v := (u v ) = max(u,v); u v := (u v ) = min(,u+v). All these logical operations (which are usually ealt with as irreversible) can be simulate in the many-value approach to quantum information by means of convenient (reversible) gates. Let us first consier pure pieces of quantum information: quregisters living in some quit-spaces. In such a case, gates correspon to particular examples of unitary operators. The logical negation has a natural gate-counterpart: the unitary operator NOT (n), which is efine in any quit-space H (n), where an n. Definition. The negation-gate of H (n). The negation-gate of H (n) is the linear operator NOT (n) that is efine for every element of the canonical basis of as follows: NOT (n) v,...,v n := v,...,v n v n. In the particular case of H () = C we obtain: NOT () v := v. Thus, NOT () behaves as the stanar fuzzy negation.

11 How to eal, in this framework, with the irreversible conjunctions an? A reversible counterpart for these operations can be obtaine by using two special versions of a gate that plays an important role in quantum computation: the Toffoli-gate. Let us first recall the efinition of the Toffoli-gate for qubit-spaces. Definition. The Toffoli-gate of a qubit-space. For any m,n,p, the Toffoli-gate T (m,n,p) of the qubit-space H (m+n+p) is the linear operator that is efine for every element of the canonical basis as follows: T (m,n,p) u,...,u m,v,...,v n,w,...w p := u,...,u m,v,...,v n,w,...,w p (u m v n ) + w p, where + is the aition moulo. Apparently the smallest qubit-space where the Toffoli-gate is efine is the space H (). In this case we have: T (,,) u,v,w = u,v,(u v) + w. The Toffoli-gate can be generalize to quit-spaces in ifferent ways. We will consier here two ifferent gates that will be calle the Toffoli-gate an the Toffoli- Lukasiewicz gate, respectively. Definition. The Toffoli-gate of a quit-space. For any m,n,p, the Toffoli-gate T (m,n,p) of the quit-space H (m+n+p) is the linear operator that is efine for every element of the canonical basis as follows: T (m,n,p) u,...,u m,v,...,v n,w,...w p := u,...,u m,v,...,v n,w,...,w p (u m v n ) + w p, where + is the aition moulo. Definition. The Toffoli- Lukasiewicz gate of a quit-space. For any m,n,p, the Toffoli- Lukasiewicz gate T L (m,n,p) of the quit-space H (m+n+p) is the linear operatorthat is efine for every element of the canonical basis as follows: T L (m,n,p) u,...,u m,v,...,v n,w,...w p := u,...,u m,v,...,v n,w,...,w p (u m v n ) + w p.

12 Clearly, T (m,n,p) an T L (m,n,p) are the same gate when =. By using the unitary operations D T (m,n,p) an D T L (m,n,p) (which correspon to the unitary operators T (m,n,p) an T L (m,n,p), respectively) one can now efine two ifferent kins of reversible conjunctions for any (pure or mixe) state of a quit-space. Definition. The Toffoli-conjunction in a quit-space. For any m,n an for any ensity operator ρ of a quit-space H (m+n) the Toffoli-conjunction AND (m,n) is efine as follows: AND (m,n) (ρ) := D T (m,n,) (ρ P 0 ). Definition. The Toffoli- Lukasiewicz conjunction in a quit-space. For any m,n an for any ensity operator ρ of a quit-space H (m+n) the Toffoli- Lukasiewicz conjunction LAND (m,n) is efine as follows: LAND (m,n) (ρ) := D T L (m,n,) (ρ P 0 ). In the efinition of both conjunctions the projection P 0 (which correspons to the bit 0) plays the role of an ancilla, which is transforme by the gates D T (m,n,) an D T L (m,n,) into the final truth-value of the conjunction. In the particular case of ensity operators corresponing to registers of H () we obtain: AND (,) (P u,v ) = P T u,v,0. (,,) Hence: { AND (,) P u,v,, if u = v = ; (P u,v ) = P u,v,0, otherwise. (In agreement with the classical truth-table of conjunction). In the general case, both the Toffoli an the Toffoli- Lukasiewicz conjunctions represent holistic forms of conjunction that reflect the characteristic holistic features of the quantum-theoretic formalism. Let ψ be a quregister of a space H (m+n). We may have: AND (m,n) (P ψ ) AND (m,n) (Re () [m,n] (P ψ ) Re () [m,n] (P ψ )); LAND (m,n) (P ψ ) LAND (m,n) (Re () [m,n] (P ψ ) Re () [m,n] (P ψ )). In other wors, the conjunction over a global piece of information (consisting of two parts) oes not generally coincie with the conjunction of the two 0

13 separate parts. Interesting counterexamples arise with entangle quregisters. Consier, for instance, the following (entangle) Bell-state: We have: Hence, ψ = ( 0,0 +, ). T (,,) ( ψ 0 ) = ( 0,0,0 +,, ). AND (,) (P ψ ) = P ( 0,0,0 +,, ), which is a pure state. At the same time, Re [,] (P ψ ) = Re [,] (P ψ ) = I(). Thus, AND (,) (Re () [,] (P ψ ) Re () [,] (P ψ )) = D T (,,) (Re () [,] (P ψ ) Re () [,] (P ψ ) P 0 ), which is a proper mixture. The gates Negation, Toffoli an Toffoli- Lukasiewicz are also calle semiclassical gates, because they are unable to create superpositions. Whenever the information-input is a register, also the information-output will be a register. Quantum computation, however, cannot help referring also to genuine quantum gates that can transform classical inputs (represente by registers) into genuine superpositions. An it is neeless to stress how superpositions play an essential role in quantum computation, being responsible for the characteristic parallel structures that etermine the spee an the efficiency of quantum computers. Let us first recall the efinition of two important genuine quantum gates of the qubit-space H () = C. Definition. The Haamar-gate of H (). The Haamar-gate(also calle square-root of ientity) is the linear operator I () thatisefineforeveryelementofthecanonicalbasisofh () as follows: I () v := (( ) v v + v ).

14 Definition. The square-root of negation of H (). The square-root of negation is the linear operator NOT () that is efine for every element of the canonical basis of H () as follows: where i is the imaginary unit. NOT () v := ((+i) v +( i) v ), Apparently, both I () an NOT () transform classical pieces of information (corresponing to the truth-values Falsity an Truth) into genuine superpositions. We have: I () 0 = ( 0 + ); I () = ( 0 ); () ( ) NOT 0 = (+i) 0 +( i). () ( ) NOT = ( i) 0 +(+i). In this way, all certain answers (Falsity or Truth) are transforme into maximally uncertain answers that assign probability-value either to the Falsity or to the Truth. Thegates I () an NOT () satisfythefollowingcharacteristicproperties that have suggeste their names ( square-root of ientity an square-root of negation ): I () I () = I () ; NOT () NOT () = NOT (). Both I () an NOT () can be generalize to any qubit-space H (n). Definition 0. TheHaamar-gateanthesquare-rootofnegationofH (n).. The Haamar-gate is the linear operator I (n) that is efine for every element of the canonical basis of H (n) as follows: (n) I v,...,v n := v,...,v n ) (( ) vn v n + v n.

15 The square-root of negation is the linear operator NOT (n) that is efine for every element of the canonical basis of H (n) as follows: (n) NOT v,...,v n := v,...,v n ) (+i) v n +( i) v n. ( How can I (n) an NOT (n) be generalize to quit-spaces? A natural generalizationofthehaamar-gateforthespaceh () = C isrepresenteby the Vanermone-operator, which generalizes the Walsh-Haamar matrix use for iscrete Fourier transforms. Definition. The Vanermone-gate of H (). The Vanermone-gate is the linear operator V () that is efine for every element k where ω = e πi. of the canonical basis of H() as follows: V () k = ω jk j, Lemma. The gate V () of H () satisfies the following conitions: j=0 ) V () = I (), if =. ) V () V () V () V () = I (). ) V () transforms each element of the canonical basis of H () into a superposition of all basis-elements, assigning to each basis-element the same probability-value. Conitions )- ) of Lemma clearly show that V () represents a goo generalization of the Haamar-gate for the quit-space C. In particular, conition ) explains the reason why V () is also terme the fourth root of ientity. Another possible generalization of the Haamar-gate to quit-spaces is an operator that will be calle the Haamar-gate restricte to the first two truth-values, inicate by I () []. Unlike V(), which transforms all basiselements into genuine superpositions, the gate I () [] only acts on the first ( 0 two basis-elements, ).

16 Definition. The Haamar-gate restricte to the first two truth-values of H (). The Haamar-gate restricte to the first two truth-values is the linear operator I () [] that is efine for every element v of the canonical basis of H () as follows: I () [] v = ( 0 + ( 0 v, otherwise. Of course, for the qubit-space C we have: ), if v = 0; ), if v = ; I () [] = I (). In a similar way, the (restricte) square-root of negation can be generalize to any quit-space. Definition. The square-root of negation restricte to the first two truthvalues of H (). The square-root of negation restricte to the first two truth-values is the linear operator that is efine for every element v of the canonical basis of H () as follows: NOT () [] v := ((+i) 0 +( i) (( i) 0 +(+i) v, otherwise. ), if v = 0; ), if v = ; The three gates V (), I () [] an NOT () [] can be naturally generalize to any quit-space H (n). Definition. The gates V (n), I (n) [], NOT (n) [] of H (n). Let v,...,v n be an element of the canonical basis of H (n). ) V (n) v,...,v n := v,...,v n V () v n. ) I (n) [] v,...,v n := v,...,v n I () [] v n. ) NOT (n) [] v,...,v n := v,...,v n NOT () [] v n.

17 Representing quantum information in Bloch-hyperspheres It is customary torepresent the ensity operatorsofthe qubit-space C as vectors of the three-imensional Bloch-sphere BS [] of raius. Let D(C ) represent the set of all ensity operators of C. For any ρ D(C ), the corresponing Bloch-vector b ρ is etermine as follows: b ρ = (b ρ,bρ,bρ ), where b ρ = Tr(ρσ ), b ρ = Tr(ρσ ), b ρ = Tr(ρσ ). The operators σ, σ, σ are the three Pauli-matrices that are efine as follows: [ ] [ ] [ ] 0 0 i 0 σ = ; σ 0 = ; σ i 0 =. 0 Viceversa, for any Bloch-vector b = (b,b,b ) the corresponing ensity operator ρ b is etermine by the following matrix: [ ] +b b ib. b +ib b We have: ρ bρ = ρ; b ρb = b. Via the Bloch-representation, any ensity operator ρ of C can be canonically represente as a combination of four unitary operators (of C ): the ientity operator I () an the three Pauli-matrices. For any ρ we have: ρ = (I() +b ρ σ +b ρ σ +b ρ σ ). The Pauli-representation can be generalize to any quit-space. Let us first introuce the generalize Pauli-matrices of a space C (with ). For the sake of simplicity, in the following, we will also write j instea of j. We recall that ψ ϕ enotes the linear operator A that satisfies the following conition for any vector χ : A χ = ϕ χ ψ, where ϕ χ is the inner prouct of ϕ an χ.

18 Definition. The generalize Pauli-matrices of C. Let j,k,l be three natural numbers such that: j an 0 k < l. The generalize Pauli-matrices σ j of C are efine as follows: k l + l k, if j ( ) an j = k( k) +( )k +l; i k l +i l k, σ j = l(l+) if ( ( ) < j ( ) an j = ( )+k( k) +( )k +l; l k=0 ), k k l l l if j > ( ) an j = ( )+l. As expecte, in the particular case of the space C the three generalize Pauli-matrices σ,σ,σ turn out to coincie with the three stanar Pauli-matrices. Like inthe qubit-case, any ensity operator ρ D(C ) can becanonically represente as a combination of the ientity operator an of the generalize Pauli-matrices σ,...,σ j,...,σ. We have: ρ = ( I () + ( ) j= b j σ j ), where b j = Tr(ρσ ( ) j) R. On this basis, any ensity operator ρ of C can be associate to a vector b ρ = (b ρ,...,b ρ j,...,bρ ) of the (real) Bloch-hypersphere BHS [ ] R, whose raius is an whose imension is. Notice that, unlike the case of C, not all vectors of the hypersphere BHS [ ] R correspon to ensity operators. Another interesting representation of the ensity operators of C can be obtaine in terms of the Weyl-operators (which have been use in quantum teleportation-experiments an in the stuy of the geometry of entanglement). Definition. The Weyl-operators of C. Let j be a natural number such that 0 j. The Weyl-operators W j of C are efine as follows: W j = ω km m m ˆ+ j, m=0

19 where ω = e πi, k = j (the integer part of j ). In the case of the qubit-space C we obtain: (W,W,W ) = (σ,σ,iσ ). In some applications an important role is playe by the first Weyl-operator W, which turns out to be escribe by the following matrix: W = As happens in the case of the Pauli-representation, all ensity operators ρ of C can be canonically represente as combinations of the ientity operator an of the Weyl-operators. We have: ρ = ( I () + b j W j ), where b j = Tr(ρW j ) C. On this basis, any ensity operator ρ D(C ) can be associate to a vector b ρ = (b ρ,...,b ρ j,...,bρ ) of the complex Bloch-hypersphere BHS [ ] C, whose raius is an whose imension is. In spite of their mathematical interest, both the Pauli an the Weylrepresentations turn out to be non-economical, since they essentially refer to gates, whose implementation might be highly complicate. The following theorem allows us to simplify such situation, showing that the probabilistic behavior of three gates only etermines a tomographic reconstruction of any state of C. Let B(C ) represent the set of all boune operators of the space C. Theorem. Let ρ be a ensity operator of a quit-space C. There exist unitary operators U,...,U (efine on C ) an a function f : [0,] B(C ) that satisfy the following conitions: j=

20 ) each U j (with j ) is a finite combination of the three following gates: I () [], NOT () [], W (efine on C ). ) For any sequence (p,...,p j,...,p ) [0,] such that p j = p () ( D U j (ρ)), we have: ρ = f(p,...,p j,...,p ) (where p () is the probability-function efine in Section ). Proof. (Sketch) It is expeient to istinguish the case where = from the case where >. ) Let ρ D(C ). Consier its Pauli representation an let b = b b b the Bloch-vector that correspons to ρ. Define the operators U,U,U as follows: U = I () [], U = NOT () [], U = W. Consier the following three equations: p () ( D U (ρ)) = p, p () ( D U (ρ)) = p, p () ( D U (ρ)) = p. We have: b = p, b Hence, Ab+Ac = p, where: 0 0 A = 0 0, c =, p = 0 0 = p, +b p p p. = p. Define f : [0,] B(C ) as follows (for any e,e,e [0,]): f(e,e,e ) := We obtain: ρ = f(p,p,p ). (I () +( e )σ +( e )σ ( e )σ ).

21 ) Let ρ D(C ) (with > ). For any h such that h, efine U h as follows: U h = W m I () [] Wm W n NOT () [] Wn, where W m = W...W }{{}, W n = W...W, m = h (the integer part of }{{} m times n times h ) an n = h mo (h moulo ). Consier the following equations: p () ( D U h (ρ)) = p h. We have: Ab+Ac = p, where A, b, c are matrices efine like in the C -case. Define f : [0,] B(C ) as follows (for any e [0,] ): f(e) := ( ( ) ) I () + (A e c) T σ. We obtain: ρ = f(p).. Many-value quantum computational logics Quantum computational logics are special examples of quantum logic base on the following semantic iea: linguistic formulas are suppose to enote pieces of quantum information, while logical connectives are interprete as particular gates that play an important logical role []. Accoringly, any formula of the quantum computational language can be regare as a synthetic logical escription of a quantum circuit. We will consier here a many-value version of the quantum computational semantics, where, for any choice of a truth-value number, the meaning of any formula α is ientifie with a ensity operator ρ living in a quit-space H (n), whose imension epens on the linguistic complexity of α. Let us first introuce the formal language: a minimal many-value quantum computational language L, whose alphabet contains atomic formulas (q,q,q,...) incluing two privilege formulas t an f that represent the truth-values Truth an Falsity respectively. The connectives of L are at least the following: the negation (corresponing to the gate NOT (n) ), the ternary Toffoli-connective (corresponing to the gate T (m,n,p) ), the ternary

22 Toffoli- Lukasiewicz connective L (corresponing to the gate T L (m,n,p) ), the square root of ientity i (corresponing to the gate I (n) ). The notion of formula of L is inuctively efine as follows: ) atomic formulas are formulas; ) if α, β, γ are formulas, then α, iα, (α,β,γ), L (α,β,γ) are formulas. Recalling the efinition of the two holistic conjunctions AND (m,n) an LAND (m,n) (given in Section ), two binary connectives an L can be efine in terms of the two Toffoli-connectives: α β := (α,β,f); α L β := L (α,β,f) (where f plays the role of a syntactical ancilla). On this basis two binary isjunctions ( an L ) can be efine (via e Morgan-law) in the expecte way By atomic complexity of a formula α we mean the number At(α) of occurrences of atomic subformulas in α. For instance, the atomic complexity of the formula α = q q = (q, q,f) is. The number At(α) plays an important semantic role, since it etermines, for any choice of a truth-value number, the semantic space H α = H(At(α)), where any ensity operator representing a possible meaning of α shall live. We have, for instance, H (q, q,f) = H (). Any formula α can be naturally ecompose into its parts giving rise to a special configuration, calle the syntactical tree of α (STree α ). Roughly, STree α can be represente as a sequence of levels consisting of subformulas of α. The bottom-level is (α), while all other levels are obtaine by ropping, step by step, all connectives occurring in α. Hence, the top-level is the sequence of atomic formulas occurring in α. As an example consier again the formula α = (q, q,f). In such a case, STree α is the following sequence of levels: Level α = (q,q,f) Level α = (q, q,f) Level α = ( (q, q,f)) For any α an for any choice of a truth-value number, STree α uniquely etermines the gate-tree of α: a sequence of gates all efine on the space H α. As an example, consier again the formula, α = (q, q,f). In the syntactical tree of α the secon level has been obtaine (from the thir level) by repeating the first occurrence of q, by negating the secon occurrence of q an by repeating f; while the first level has been obtaine (from the secon 0

23 level) by applying the Toffoli-connective. Accoringly, the gate-tree of α can be naturally ientifie with the following gate-sequence: ( D I () D NOT () D I (), D T (,,) ). This proceure can be naturally generalize to any α, whose gate-tree will be inicate by ( D G α n,..., D G α ) (where n is the number of levels of STree α ). We consier here a holistic version of the quantum computational semantics, where entanglement can be use as a semantic resource []. Generally, the meaning of a compoun formula etermines the contextual meanings of its parts (an not the other way aroun, as happens in the case of most compositional semantic approaches). As expecte, all basic notions of the many-value quantum computational semantics epen on the choice of the truth-value number. The concept of (-value) moel of L is base on the notion of (-value) holistic map for L. This is a map Hol that assigns to each level of the syntactical tree of any formula α a ensity operator living in the semantic space of α. We have: Hol (Level α k) D(H α ). SupposethatLevel α k = (β k,...,β kr ). Itisnaturaltoescribeρ = Hol (Level α k ) as a possible state of a composite quantum system consisting of r subsystems. Hence, the contextual meaning (Hol α (β k j )) of the occurrence β kj (in STree α ) can be ientifie with the reuce state of ρ with respect to the j-th subsystem. Accoringly, we can write: Hol α (β k j ) = Re (j) [At(β k ),...,At(β kr )] (ρ). The concepts of moel, truth an logical consequence (of the -value semantics) can be now efine as follows. Definition. Moel. A moel (or interpretation) of the language L is a holistic map Hol that satisfies the following conitions for any formula α: ) Hol assigns the same contextual meaning to ifferent occurrences of one an the same subformula of α (in STree α ). ) The contextual meanings of the true formula t an of the false formula f are the Truth P () an the Falsity P () 0, respectively.

24 ) Hol preserves the logical form of α by interpreting the connectives of α as the corresponing gates. Accoringly, if ( D G α n,..., D G α ) is the gate-tree of α, then Hol (Levelk α) = D G α k (Hol (Levelk+ α )). On this basis we put: Hol (α) := Hol (Level α ), for any formula α. Notice that any Hol (α) represents a kin of autonomous semantic context that is not necessarily correlate with the meanings of other formulas. Generally we have: Hol γ (β) Holδ (β). Thus, one an the same formula may receive ifferent contextual meanings in ifferent contexts. Definition. Truth. A formula α is calle true with respect to a moel Hol iff p () (Hol (α)) =. Definition. Logical consequence. A formula β is calle a logical consequence of a formula α iff for any formula γ such that α an β are subformulas of γ an for any moel Hol : p () (Hol γ (α)) p() (Hol γ (β)). Apparently, both Truth an Logical consequence are, in this semantics, probabilistic notions that essentially epen on the probability-function p. For any choice of, the notion of logical consequence of the -value semantics characterizes a special example of logic, calle -value holistic quantum computational logic (inicate by HQCL). All logics HQCL give rise to violations of some important logical implications (which hol for many stanar logics). For instance, both conjunctions an L are generally non-iempotent, non-commutative an non-associative, although the corresponing truth-value operations (, ) o satisfy commutativity an associativity. This can be explaine by recalling the contextual behavior of quantum meanings. It may happen that for some context γ: Hol γ (α β) Holγ (β α) an p() (Hol γ (α β)) > p() (Hol γ (β α)). Counterexamples in the framework of the qubit-semantics have been escribe in [].

25 Accoringly, ifferent forms of holistic quantum computational logics (also in a first-orer version) can be naturally applie to represent semantic phenomena (even far from microphysics), where contextuality, ambiguity an fuzziness play an essential role (as happens in the case of natural languages or in the languages of art) [, ].. Physical implementations by optical evices Physical implementations of quantum logical gates represent the basic issue for the technological realization of quantum computers. Among the ifferent choices that have been investigate in the literature we will consier here the case of optical evices, where photon-beams (possibly consisting of single photons) move in ifferent irections. Let us conventionally assume that 0 represents the state of a beam moving along the x-irection, while is the state of a beam moving along the y-irection. Intheframeworkofthis physicalsemantics, one-qubitgates(likenot (), I (), NOT () ) can be easily implemente. A natural implementation of NOT () can be obtaine by a mirror M that reflects in the y-irection any beam moving along the x-irection, an vice versa. Hence we have: 0 M ; M 0 (the mirror transforms the state 0 into the state, an vice versa). An implementation of the Haamar-gate I () can be obtaine by a symmetric 0 : 0 beam splitter BS. We have: 0 BS ( 0 + ); BS ( 0 ). Accoringly, any beam that goes through BS is split into two components: one component moves along the x-irection, while the other component moves along the y-irection. An the probability of both paths (along the x-irection or along the y-irection) is. Also the gate NOT () can be implemente in a similar way. Other apparatuses that may be useful for optical implementations of gates are the relative phase shifters along a given irection. A particular example is escribe by the following unitary operator.

26 Definition 0. The relative phase shifter along the y-irection. The relative phase shifter along the y-irection is the linear operator U PS that is efine for every element { of the canonical basis of C as follows: e U PS v = c v, where c = iπ, if v = ;, otherwise. We obtain: U PS 0 = 0 ; U PS =. Let us inicate by PS a physical apparatus that realizes the phase shift escribe by U PS. Relative phase shifters, beam splitters an mirrors are the basic physical components of the Mach-Zehner interferometer (MZI), an apparatus that has playe a very important role in the logical an philosophical ebates about the founations of quantum theory. The physical situation can be sketche as follows (Fig. ). Figure : The Mach-Zehner interferometer A beam (which may move either along the x-irection or along the y- irection) goes through the relative phase shifter PS of MZI. We have: 0 PS 0 ; PS (the phase of the beam changes only in the case where the beam is moving along the y-irection). Soon after the beam goes through the first beam

27 splitter BS. As a consequence, it is split into two components: one component moves along the interferometer s arm in the x-irection, the other component moves along the arm in the y-irection. We have: 0 BS ( 0 + ); BS ( 0 + ). Then, both components of the superpose beam (on both arms) are reflecte by the mirrors M. We have: ( 0 + ) M ( 0 + ); ( 0 + ) M ( 0 ). Finally, the superpose beam goes through the secon beam splitter BS, which re-composes the two components. We have: ( 0 + ) BS 0 ; ( 0 ) BS. Accoringly, MZI transforms the input 0 into the output 0, while the input is transforme into the output. One is ealing with a result that has for a long time been escribe as eeply counter-intuitive. In fact, accoring to a classical way of thinking we woul expect that the outcoming photons from the secon beam splitter shoul be etecte with probability either along the x-irection or along the y-irection. The Mach-Zehner interferometer clearly represents a physical implementation of the following quantum logical circuit: I () NOT () I () (calle the Mach-Zehner circuit ). In the framework of quantum computational logics such circuit can be naturally escribe by the Mach-Zehner formula : i iq, whereqisagenericatomicformula. AnyinterpretationHol (ofthequantum computational language) that assigns to q a meaning (a qubit living in the space C ) will etermine a meaning for the Mach-Zehner formula. While optical implementations of one-qubit gates are relatively simple, trying to implement many-qubit gates may be rather complicate. Consier

28 the case of a gate that plays an important logical an computational role: the Toffoli-gate T (,,) (efine on the space H () = C C C ). Mathematically we have: { T (,,) v,v,v v, if v = 0; v,v,v = v,v,(v v ), if v =, where v,v,v {0,}. The main problem is fining a evice that can realize a physical epenence of the target-bit (v v or (v v ) ) from the control-bits (v,v ). A possible strategy is base on an appropriate use of the optical Kerr-effect : a substance with an intensity-epenent refractive inex is place into a given evice, giving rise to an intensity-epenent phase shift. Let us first give the mathematical efinition of a unitary operator that escribes a particular form of conitional phase shift. Definition. The relative conitional phase shifter The relative conitional phase shifter of the space H () = C C C is the unitary operator U CPS that is efine for every element of the canonical basis as follows: U CPS v,v,v = v,v c v, { e where c = iπ, if v =, v = an v = 0;., otherwise. Let us inicate by CPS a physical apparatus that realizes the phase shift escribe by the operator U CPS. Clearly, CPS etermines a conitional phase shift. For, the phase of a three-beam system in state v,v,v is change only in the case where both control-bits ( v, v ) are the state, while the ancilla-bit v is the state 0. From a physical point of view, such a result can be obtaine by using a convenient substance that prouces the Kerr-effect. In orer to obtain an implementation of the Toffoli-gate T (,,,) we will now consier a more sophisticate version of the Mach-Zehner interferometer that will be calle Kerr-Mach-Zehner interferometer (inicate by KMZI). Besies the relative phase shifter (PS), the two beam splitters (BS, BS ) an the mirrors (M), the Kerr-Mach-Zehner interferometer also

29 Figure : The Kerr-Mach-Zehner interferometer contains a relative conitional phase shifter (CPS) that can prouce the Kerreffect (Fig. ). While the inputs of the canonical Mach-Zehner interferometer are single beams (whose states live in the space C ), the apparatus KMZI acts on composite systems consisting of three beams (S,S,S ), whose states live in the space H () = C C C. For the sake of simplicity we can assume that S,S,S are single photons that may enter into the interferometer-box either along the x-irection or along the y-irection. Let v,v,v be the input-state of the composite system S + S + S. Photons S,S (whose states v, v represent the control-bits) are suppose to enter into the box along the yz-plane, while photon S (whose state v is the ancilla-bit) will enter through the first beam-splitter BS. Mathematically, the action performe by the apparatus KMZI is escribe by the following unitary operator (of the space H () ): U KMZ := (I I I () ) (I I NOT () ) U CPS (I I I () ) (I I U PS ). In orer to see how KMZI is working from a physical point of view, it is expeient to consier a particular example. Take the input v,v,v =,, 0 an let us escribe the physical evolution etermine by the operator U KMZ for the system S +S +S, whose initial state is,,0. We have: (I I U PS ),,0 =,,0. The relative phase shifter along the y-irection (PS) oes not change the

30 state of photon S, which is moving along the x-irection. (I I I () ),,0 =, ( 0 + ). Photon S goes through the first beam splitter BS splitting into two components: one component moves along the interferometer s arm along the x-irection, the other component moves along the arm in the y-irection (like in the case of the canonical Mach-Zehner interferometer). At the same time, photons S an S (both in state ) enter into the interferometer-box along the yz-plane. U CPS (, ( 0 + )) =, ( 0 + ). The conitional phase shifter CPS etermines a phase shift for the component ofs thatismovingalongthex-irection; becausebothphotonss ans (in state ) have gone through the substance (containe in CPS) that prouces the Kerr-effect. (I I NOT () )(, ( 0 + )) =, ( 0 ). Both components of S (on both arms) are reflecte by the mirrors. (I I I () )(, ( 0 )) =,,. The secon beam splitter BS re-composes the two components of the superpose photon S. Consequently, we obtain: U KMZ,,0 =,, = T (,,),,0. In general, one can easily prove that U KMZ an T (,,) are one an the same unitary operator. Lemma. For any element v,v,v of the canonical basis of the space H (), U KMZ v,v,v = T (,,) v,v,v.

31 Although, from a mathematical point of view, U KMZ an T (,,) represent the same gate, physically it is not guarantee that the apparatus KMZI always realizes its expecte job. All ifficulties are ue to the behavior of the conitional phase shifter. In fact, the substances use to prouce the Kerreffect generally etermine only stochastic results []. As a consequence one shall conclue that the Kerr-Mach-Zehner interferometer allows us to obtain an approximate implementation of the Toffoli-gate with an accuracy that is, in some cases, very goo. So far we have consiere possible optical implementations of gates in the case of qubit-spaces. The techniques we have illustrate can be also generalize to quit-spaces. The main iea is using, instea of single beams, systems consisting of many beams (corresponing to ifferent truth-values) that may move either along the x-irection or along the y-irection. The problems concerning physical implementations for the many-value quantum computational semantics will be investigate in a future article.. Acknowlegements Sergioli s work has been supporte by the Italian Ministry of Scientic Research within the FIRB project Structures an Dynamics of Knowlege an Cognition, Cagliari: FJ an of RAS within the project Moeling the Uncertainty: Quantum Theory an Imaging Processing. References [] R. Greechie, A non-stanar quantum logic with a strong set of states, in E. Beltrametti, B. van Fraassen (es.), Current Issues in Quantum Logic, Plenum, New York,. [] A. Dvurečenskij, S. Pulmannová, New Trens in Quantum Structures, Kluwer, Dorrecht, 000. [] M.L. Dalla Chiara, R. Giuntini, R. Greechie, Reasoning in Quantum Theory, Kluwer, Dorrecht, 00. [] M.L. Dalla Chiara, R. Giuntini, R. Leporini, Logics from quantum computation, Int. J. Quantum Inf.,, 00. [] M.L. Dalla Chiara, R. Giuntini, R. Leporini, G. Sergioli, Holistic logical arguments in quantum computation, Math. Slovaca (),, 0.

32 [] M.L. Dalla Chiara, R. Giuntini, R. Luciani, E. Negri, From Quantum Information to Musical Semantics, College Publications, Lonon, 0. [] M.L. Dalla Chiara, R. Giuntini, R. Leporini, G. Sergioli, A first-orer epistemic quantum computational semantics with relativistic-like epistemic effects, Fuzzy Sets an Systems, 0, 0. [] N. Matsua, R. Shimizu, Y. Mitsumori, H. Kosaka, K. Eamatsu, Observation of optical-fibre Kerr nonlinearity at the single-photon level, Nature Photonics,, 00. 0

33 *Revision Note We mae the corrections suggeste by the referees.