Entanglement and Quantum Logical Gates. Part I.

Transcription

1 DOI 0.007/s Entanglement and Quantum Logical Gates. Part I. H. Freytes R. Giuntini R. Leporini G. Sergioli Received: 4 December 04 / Accepted: April 05 Springer Science+Business Media New York 05 Abstract Is it possible to give a logical characterization of entanglement and of entanglement-measures in terms of the probabilistic behavior of some gates? This question admits different positive or negative answers in the case of different systems of gates and in the case of different classes of density operators. In the first part of this article we investigate possible relations between entanglement-measures and the probabilistic behavior of quantum computational conjunctions. This article is dedicated to the memory of Peter Mittelstaedt. The authors will always remember the great scientist who has represented a noble example for all scholars of the IQSA-community. Sergioli s work has been supported by the Italian Ministry of Scientific Research within the FIRB project Structures and dynamics of knowledge and cognition, Cagliari unit FJ and by RAS, within the project Modeling the uncertainty: quantum theory and imaging processing ; Leporini s work has been supported by the Italian Ministry of Scientific Research within the PRIN project Automata and Formal Languages: Mathematical Aspects and Applications. Hector Freytes is also affiliates to UNR-CONICET, Argentina. R. Leporini roberto.leporini@unibg.it H. Freytes hfreytes@gmail.com R. Giuntini giuntini@unica.it G. Sergioli giuseppe.sergioli@gmail.com Dipartimento di Pedagogia, Psicologia, Filosofia, Università di Cagliari, Via Is Mirrionis, I-093 Cagliari, Italy Dipartimento di Ingegneria Gestionale, dell Informazione e della Produzione, Universitàdi Bergamo, Viale Marconi 5, I-4044 Dalmine BG, Italy

2 Keywords Entanglement Quantum gates Quantum logics Introduction Entanglement, the characteristic feature of quantum theory often described as mysterious and potentially paradoxical, represents an essential resource in quantum information for quantum computers, teleportation, quantum cryptography, semantic applications, etc.. As is well known, quantum computations are performed by quantum logical gates: unitary quantum operations that transform pieces of quantum information mathematically represented as density operators of convenient Hilbert spaces in a reversible way. We investigate the following question: is it possible to give a logical characterization of entanglement and of entanglement-measures in terms of the probabilistic behavior of some gates? This question admits different positive or negative answers in the case of different systems of gates and in the case of different classes of density operators. We will first discuss this problem by referring to the behavior of quantum computational conjunctions, defined in terms of the Toffoli-gate. Other examples of gates will be considered in the second part of this article. States and Gates Let us first recall some basic definitions. As is well known, the general mathematical environment for quantum computation is the Hilbert space H n := C }... {{ C }. Any piece n times of quantum information is represented by a density operator ρ of a space H n.aquregister which is a pure state is represented by a unit-vector ψ of a space H n or, equivalently, by the corresponding density operator P ψ the projection-operator that projects over the closed subspace determined by ψ. A qubit or qubit-state is a quregister of the space C. A register which represents a certain piece of information is an element x,...,x n of the canonical orthonormal basis of a space H n where x i 0, ; a bit is a register of C. Following a standard convention, we assume that the bit represents the truth-value Truth, while the bit 0 represents the truth-value Falsity. On this basis, we can identify, in each space H n, two special projection-operators P n and P n 0 that represent, respectively, the truth-property and the falsity-property. The truth-property P n is the projection-operator that projects over the closed subspace spanned by the set of all registers x,...,x n, ; while the falsity-property P n 0 is the projection-operator that projects over the closed subspace spanned by the set of all registers x,...,x n, 0. In this way, truth and falsity are dealt with as mathematical representatives of possible physical properties. Accordingly, by applying the Born-rule, one can naturally define the probability-value pρ of any density operator ρ of H n as follows: pρ := trp n ρ, where tr is the trace-functional. Hence, pρ represents the probability that the information ρ is true [, 5, 6]. We will denote by DH n the set of all density operators of H n, while D will represent the union n {DHn }. Pure pieces of quantum information are processed by quantum logical gates briefly, gates: unitary operators that transform quregisters into quregisters in a reversible way. We recall here the definition of two gates that will be used in this article.

3 Definition The Toffoli-gate For any m, n, p, the Toffoli-gate defined on H m+n+p is the linear operator T m,n,p such that, for every element x,...,x m y,...,y n z,...,z p of the canonical basis, T m,n,p x,...,x m,y,...,y n,z,...,z p = x,...,x m,y,...,y n,z,...,z p x m y n +z p, where + represents the addition modulo. Definition The swap-gate For any m, n, the swap-gate defined on H m,n is the linear operator SW m,n such that, for every element x,...,x m y,...,y n of the canonical basis, SW m,n x,...,x m,y,...,y n = y,...,y n,x,...,x m. All gates can be canonically extended to the set D of all density operators. Let G be any gate defined on H n. The corresponding density-operator gate also called unitary quantum operation D G is defined as follows for any ρ DH n : D Gρ = G ρ G where G is the adjoint of G. For the sake of simplicity, also the operations D G will be briefly called gates. 3 Entanglement and Entanglement-measures Consider the product-space H m+n = H m H n.anyρ DH m+n represents a possible state for a composite physical system S = S + S consisting of two subsystems. According to the quantum formalism, ρ determines the two reduced states Red [m,n]ρ and Red [m,n] ρ that represent the states of S and of S in the context ρ. In such a case, we say that ρ is a a bipartite state with respect to the decomposition m, n. It may happen that ρ is a pure state, while Red [m,n]ρ and Red [m,n]ρ are proper mixtures. In this case the information about the whole system is more precise than the pieces of information about its parts. Definition 3 Factorizability, separability and entanglement Letρ be a bipartite state of H m+n with respect to the decomposition m, n. ρ is called a bipartite factorized state of H m+n iff ρ = ρ ρ,whereρ DH m and ρ DH n ; ρ is called a bipartite separable state of H m+n iff ρ = i w iρ i, where each ρ i is a bipartite factorized state of H m+n, w i [0, ] and i w i = ; 3 ρ is called a bipartite entangled state of H m+n iff ρ is not separable. Examples are shown at the end of this Section. Definition 4 Maximally entangled states and maximally mixed states The reduced state of ρ DH m+n with respect to the subspace H m is defined as follows: Red [m,n] ρ = Tr H nρ, wheretr H n is the partial trace i.e. the unique linear operator Tr H n : H m+n H m such that Tr H na B = ATrB,whereAand B are operators on H m and H n, respectively. The reduced state Red [m,n]ρ is defined in a similar way.

4 A pure bipartite state ρ of H m+n is called maximally entangled iff Red [m,n] = m I m or Red [m,n] = n I n,wherei m and I n are the identity operators of the spaces H m and H n, respectively. A state ρ of H m+n is called a maximally mixed state of H m+n iff ρ = m+n I m+n. How to measure the entanglement-degree of a given state? As is well known, different definitions for the concept of entanglement-measure which quantify different aspects of entanglement have been proposed in the literature [7]. Any normalized entanglementmeasure EM defined on the set of all bipartite states of H m+n is supposed to satisfy the following minimal conditions:. EMρ [0, ];. EMρ = 0, if ρ is separable; 3. EMρ =, if ρ is a maximally entangled pure state; 4. EMρ is invariant under locally unitary maps. This means that: EMρ = EM U m U n ρ U m U n, for any unitary operators U m of H m and U n of H n. We will use here a particular definition of entanglement-measure which is convenient for computational aims: the notion of concurrence. Definition 5 The concurrence of a bipartite state Let P ψ be a bipartite pure state of H m+n.theconcurrence of P ψ is defined as follows: CP ψ = λ i, i where the numbers λ i are eigenvalues of Red [m,n] P ψ or, equivalently, of Red [m,n] P ψ. Let ρ be a bipartite mixed state of H m+n.theconcurrence of ρ is defined as follows: { Cρ = inf w i CP ψi : ρ = } w i P ψi. i i Onecaneasilyshowthat theconcurrencec satisfies the minimal conditions required for the general notion of entanglement-measure. We have: CP ψ = tr Red [m,n] P ψ Red [m,n] P ψ. In the study of entanglement-phenomena it is interesting to isolate a special class of bipartite states represented by the Werner states. This notion has been introduced in [] to show that entangled bipartite states do not necessarily exhibit non-local correlations. It is true that any bipartite state must be entangled in order to produce non-local correlations; at the same time there are examples of entangled Werner states whose correlations satisfy some simple instances of Bell-inequalities. Werner states have been used in teleportation [9] and for investigating deterministic purification [0].

5 Definition 6 Werner state A Werner state is a bipartite state ρ of a space H n H n that satisfies the following condition for any unitary operator U of H n : U UρU U = ρ. Hence, any Werner state is invariant under local unitary transformations. Interestingly enough, one can prove that the class of all Werner states of H n H n can be represented as a one-parameter manifold of states. Lemma [8] Any Werner state of the space H n can be represented as follows: [ ρ w n = n w n I n + w ] n SW n,n, where w while I n and SW n,n are the identity operator and the swap-gate of the space H n. As a consequence, all examples of Werner states in the space H will have the following form: + w ρ w = 0 w w w w 0, w with w [, ]. A deep and simple correlation connects the concurrence of a Werner state ρ w n with the parameter w. Theorem [, Eq.33] Cρ n w = { w, if w<0; 0, otherwise. As a consequence, one can easily show that:. ρ w n is separable iff w 0.. ρ w n is maximally mixed iff w = n. 3. ρ n w 4. If ρ n w is pure iff n = andw = ; is pure, then ρ w n is maximally entangled. Consider now the case of two-qubit Werner states ρ w, which live in the space C C and consequently have the following form: ρ w = 3 [ w I + w ] SW,. We obtain:. ρ w is separable iff w 0;. ρ w is maximally mixed iff w = iff ρ w is factorized; 3. ρ w is a maximally entangled pure state iff w = iffρ w P 0,,0. is the Bell state

6 The concurrence of an arbitrary two-qubit Werner state in terms of the parameter w is depicted in Fig.. Another interesting subclass of the class of all two-qubit states is the class of all threeparameter qubit states ρ abc, defined as follows: + a ρ abc = 0 bic ic + b 0, a where a,b,c are three real numbers such that a andb + c. One can prove that a state ρ abc is separable iff a + c Figs., 3 and 4. 4 Entanglement and Quantum Computational Conjunctions As is well known, the Toffoli-gate allows us to define a quantum computational conjunction that behaves according to classical logic, whenever it is applied to certain pieces of information bits and registers. At the same time this conjunction may have some deeply anti-classical features when the inputs are uncertain pieces of quantum information. The most peculiar property is represented by a holistic behavior: generally the conjunction defined on a global piece of quantum information represented by a given density operator cannot be described as a function of its separate parts. This is the reason why the quantum computational conjunction is called holistic [4]. Definition 7 The holistic conjunction Foranym, n and for any density operator ρ DH m+n,theholistic conjunction AND m,n is defined as follows: AND m,n ρ = D T m,n, ρ P 0. Fig. Concurrence of a two-qubit Werner state

7 Fig. The class of all ρ abc states Clearly, the falsity-property P 0 plays, in this definition, the role of an ancilla. Generally we have: AND m,n ρ = AND m,n Red [m,n]ρ Red [m,n] ρ. Hence, the holistic conjunction defined on a global information consisting of two parts does not generally coincide with the conjunction of the two separate parts. As an example, consider the following density operator which corresponds to a maximally entangled pure state: ρ = P 0,0 +,. We have: AND, ρ = P 0,0,0 +,,, which also represents a maximally entangled pure state. Atthesametimewehave: AND, Red [,]ρ Red [,] ρ = AND, I I, which is a proper mixture. The following theorem sums up the main probabilistic properties of the holistic conjunction. Theorem [3, Section 3] For any ρ DH m+n : pand m,n ρ = tr P m P n ρ.

8 Fig. 3 The class of all separable ρ abc states Fig. 4 The class of all entangled ρ abc states

9 pand m,n ρ p Red [m,n] ρ and pand m,n ρ p Red [m,n] ρ. Consequently, for any factorized density operator ρ = ρ ρ of the space H m+n with ρ DH m and ρ DH n wehave: AND m,n ρ = Red [m,n] ρ p Red [m,n] ρ = pρ pρ. Definition 8 Probabilistic factorizability A conjunction AND m,n ρ is called probabilistically factorizable iff pand m,n ρ = p Red [m,n] ρ p Red [m,n]. ρ Is there any interesting correlation between the probabilistic factorizability of AND m,n ρ, the factorizability of ρ and the separability of ρ? Of course, we have: ρ is factorized = AND m,n ρ is probabilistically factorizable. The following questions arise:. AND m,n ρ is probabilistically factorizable ρ is factorized?. AND m,n ρ is probabilistically factorizable ρ is separable? 3. ρ is separable AND m,n ρ is probabilistically factorizable? These three questions have a negative answer. As to our third question, consider the following counterexample: take a Werner state ρ w such that 0 w andw =. Thus, ρ w is a separable and non-factorized state, since the only factorized Werner state is ρ /. At the same time, AND m,n ρ w is not probabilistically factorizable, because: pand m,n ρ w = + w = 6 4 = p Red [m,n] ρ w p Red [m,n] ρ w. Negative answers to our first and to our second question can be obtained as a consequence of the following general theorem. Theorem 3 For any ɛ 0, ] R and for any m, n, there exists a bipartite pure state P ψɛ of H m+n such that: P ψɛ is entangled; CP ψɛ = ɛ; 3 pp ψɛ = ; 4 pand m,n P ψɛ = p Red [m,n] P ψ ɛ p Red [m,n] P ψ ɛ. Proof - Let ɛ 0, ] and let k ɛ := ɛ. Define the numbers a 00 and a as follows: a 00 := kɛ and a := + kɛ. We have: k ɛ [0, and a 00,a 0,. Consider the following pure state: ϕ ɛ =a 00 0,...,0 +a }{{} 0,...,0,, 0,...,0,. }{{}}{{} m+n m n One can easily show that: CP ϕɛ = a 00 a =ɛ.

10 Since 0 <ɛ, the state ϕ ɛ turns out to be entangled. Consider now the state ψ ɛ :=I m+n I ϕ ɛ. An easy computation shows that: ψ ɛ := kɛ 0,...,0, 0,...,0 + kɛ 0,...,0, 0,...,0, }{{}}{{} }{{}}{{} m n m n + + kɛ 0,...,0,, 0,...,0 + kɛ 0,...,0,, 0,...,0,. }{{}}{{} }{{}}{{} m n m n Since I m+n I is a locally unitary map and concurrence is invariant under locally unitary maps, the state ψ ɛ is entangled and its concurrence is ɛ. 3 An easy computation shows that: 4 By Theorem, pp ψɛ = 4 k ɛ + + k ɛ =. pand m,n P ψɛ = trp m P n P ψɛ = 4 + k ɛ. It turns out that: p Red [m,n] P ψ ɛ Hence, = + k ɛ and p Red [m,n] P ψ ɛ =. pand m,n P ψɛ = p Red [m,n] P ψ ɛ p Red [,] P ψ ɛ. Consequently, there are infinitely many entangled pure states P ψɛ, whose holistic conjunction is probabilistically factorizable. On this basis we can conclude that the probabilistic behavior of holistic conjunctions cannot characterize either entanglement or entanglement-measures. In spite of these general negative results, some interesting correlations between entanglement, entanglement-measures and the probabilistic behavior of holistic conjunctions can be found in the case of Werner states and in the case of three-parameter qubit states. Theorem 4 Let ρ w n be a Werner state of H n. pρ w n = ; p AND n,n ρ n = n + n w ; 3 p Red [n,n] w ρ n w = p n Red [n,n] ρ n w =. [ Proof p ρ w n = tr P n ρ w n = n i= w n + n i= pand n,n ρ w n = tr P n+ D T n,n, ρ w P 0 = tr [ n n i= w ] n + n i= w n = n + n w. n 3 In a similar way. w P n P n ]= n. ρ w =

11 As a consequence, we obtain: AND n,n ρ w n is probabilistically factorizable iff p AND n,n ρ w n = 4 iff w = iff ρn n w = n In. Hence, the holistic conjunction of a Werner state ρ w n is probabilistically factorizable iff ρ w n is the maximally mixed state I n. Accordingly, the class of all Werner states ρ n n w for which AND n,n ρ w n is probabilistically factorizable coincides with the singleton of the maximally mixed state. Whenever ρ w n is a non-factorized, separable Werner state, its holistic conjunction AND n,n ρ w n cannot be probabilistically factorizable because I n is the only factorized Werner state of H n. n A different situation arises in the case of three-parameter two-qubit states ρ abc.aswe already know, any ρ abc satisfies the following condition: ρ abc is separable iff a + c. Furthermore we have:. pρ abc = a 4 ;. p Red [,] ρ abc 3. p Red [,] ρ abc = a+b 4 ; = a b 4. Fig. 5 The class of all probabilistically factorizable AND, ρ abc s

12 Theorem 5 AND, ρ abc is probabilistically factorizable iff a = b. If AND, ρ abc is probabilistically factorizable, then ρ abc is separable. 3 If AND, ρ abc is probabilistically factorizable and c = 0,thenρ abc is separable but not factorized. Proof AND, ρ abc is probabilistically factorizable iff a 4 = pρ abc = p Red [,] p Red [,] ρ abc = 6 + a b + a + b iff a = b. If AND, ρ abc is probabilistically factorizable, then, by, a = b. By definition of ρ abc, b + c. Thus, a + c and therefore ρ abc is separable. 3 Suppose that AND, ρ abc is probabilistically factorizable. By, ρ abc is separable. By hypothesis, c = 0. Thus, ρ abc cannot be factorized since every factorized ρ abc is a diagonal matrix with c = 0. By Theorem 5 3, we can conclude that the class of all ρ abc s for which AND, ρ abc is probabilistically factorizable contains separable states that are not factorized Fig. 5. References. Beltrametti, E., Dalla Chiara, M.L., Giuntini, R., Leporini, R., Sergioli, G.: A Quantum Computational Semantics for Epistemic Logical Operators. Part II: Semantics. Int. J. Theor. Phys. 53, Chen, K., Albeverio, S., Fei, S.M.: Concurrence-based entanglement measure for Werner States. Rep. Math. Phys. 58, Also available as arxiv: Dalla Chiara, M.L., Giuntini, R., Leporini, R., Sergioli, G.: Holistic logical arguments in quantum computation, to appear in Mathematica Slovaca 4. Dalla Chiara, M.L., Giuntini, R., Ledda, A., Leporini, R., Sergioli, G.: Entanglement as a semantic resource. Foundations of Physics 40, Freytes, H., Domenech, G.: Quantum computational logic with mixed states. Mathematical Logic Quaterly 59, Freytes, H., Sergioli, G., Arico, A.: Representing continuous t-norms in quantum computation with mixed states, Journal of Physics A Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 8, Freytes, H., Sergioli, G.: Fuzzy approach for Toffoli gate in Quantum Computation with Mixed States. Reports on Mathematical Physics 4, Lee, J., Kim, M.S.: Entanglement Teleportation via Werner States. Phys. Rev. Lett. 84, Short, A.: No deterministic purification for two copies of a noisy entangled state. Phys. Rev. Lett. 0, Werner, R.F.: Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40,