aa r X i v : . [ qu a n t - ph ] J un Journal of Physics A 37 (2004) 3241-3257
Path Integral for Quantum Operations
Vasily E. Tarasov
Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119992 Moscow, RussiaE-mail: [email protected]
Abstract
In this paper we consider a phase space path integral for general time-dependent quan-tum operations, not necessarily unitary. We obtain the path integral for a completely posi-tive quantum operation satisfied Lindblad equation (quantum Markovian master equation).We consider the path integral for quantum operation with a simple infinitesimal generator.
PACS 03.67.Lx, 03.067-a, 03.65.-w
Unitary evolution is not the most general type of state change possible for quantum systems. Themost general state change of a quantum system is a quantum operation [1, 2, 3, 4, 5]. One candescribe a quantum operation for a quantum system starting from a unitary evolution of someclosed system if the quantum system is a part of the closed system [6]-[14]. However, situationscan arise where it is difficult or impossible to find a closed system comprising the given quantumsystem [15]-[19]. This would render theory of quantum operations a fundamental generalizationof the unitary evolution of the closed quantum system.The usual models of a quantum computer deal only with unitary quantum operations on purestates. In these models it is difficult or impossible to deal formally with measurements, dissi-pation, decoherence and noise. It turns out that the restriction to pure states and unitary gatesis unnecessary [20]. In [20], a model of quantum computations by quantum operations withmixed states was constructed. The computations are realized by quantum operations, not neces-sarily unitary. Mixed states subjected to general quantum operations could increase efficiency.This increase is connected with the increasing number of computational basis elements for theHilbert space. A pure state of n two level quantum systems is an element of the n -dimensionalfunctional Hilbert space. A mixed state of the system is an element of the n -dimensional opera-tor Hilbert space. Therefore, the increased efficiency can be formalized in terms of a four-valued1ogic replacing the conventional two-valued logic. Unitary gates and quantum operations for aquantum computer with pure states can be considered as quantum gates of a mixed state quan-tum computer. Quantum algorithms on a quantum computer with mixed states are expected torun on a smaller network than with pure state implementation.The path integral for quantum operations can be useful for the continuous-variable general-ization of quantum computations by quantum operations with mixed states. The usual models ofa quantum computer deal only with discrete variables. Many quantum variables such as positionand momentum are continuous. The use of continuous-variable quantum computing [21, 22, 23]allows information to be encoded and processed much more compactly and efficiently than withdiscrete variable computing. Quantum computation using continuous variables is an alternativeapproach to quantum computations with discrete variables.All processes occur in time. It is naturally to consider time dependence for quantum opera-tions. In this paper we consider the path integral approach to general time-dependent quantumoperations. We use the operator space [24]-[38] and superoperators on this space. The path inte-gral for unitary evolution from the operator (Liouville) space was derived in [33]. The quantumoperation is considered as a real completely positive trace-preserving superoperator on the op-erator space. We derive a path integral for a completely positive quantum operation satisfiedLindblad equation (quantum Markovian master equation) [39, 40, 41, 42, 18, 38]. For example,we consider a path integral for a quantum operation with a simple infinitesimal generator.In section 2, the requirements for a superoperator to be a generalized quantum operation arediscussed. In section 3, the general Liouville-von Neumann equation and quantum Markovian(Lindblad) master equation are considered. In section 4, we derive path integral for quantumoperation satisfied Liouville-von Neumann equation. In section 5, we obtain a path integral fortime-dependent quantum operation with an infinitesimal generator such that the adjoint gener-ator is completely dissipative. In section 6, the continuous-variables quantum computation byquantum operations with mixed states is discussed. In the appendix, the mathematical back-ground (Liouville space, superoperators) is considered. Unitary evolution is not the most general type of state change possible for quantum systems.The most general state change of a quantum system is a positive trace-preserving map which iscalled a quantum operation. For the concept of quantum operations, see [1, 2, 3, 4, 5].A quantum operation is a superoperator ˆ E which maps the density matrix operator | ρ ) to2he density matrix operator | ρ ′ ) . For the concept of superoperators and operator space see theappendix and [24]-[38]If | ρ ) is a density matrix operator, then ˆ E | ρ ) should also be a density matrix operator. Anydensity matrix operator ρ is a self-adjoint ( ρ † t = ρ t ), positive ( ρ t > ) operator with unit trace( T rρ t = 1 ). Therefore, the requirements for a superoperator ˆ E to be the quantum operation areas follows:1. The superoperator ˆ E is a real superoperator, i.e. (cid:16) ˆ E ( A ) (cid:17) † = ˆ E ( A † ) for all A . Thereal superoperator ˆ E maps the self-adjoint operator ρ to the self-adjoint operator ˆ E ( ρ ) : ( ˆ E ( ρ )) † = ˆ E ( ρ ) .2. The superoperator ˆ E is a positive superoperator, i.e. ˆ E maps positive operators to positiveoperators: ˆ E ( A ) > for all A = 0 or ˆ E ( ρ ) ≥ .3. The superoperator ˆ E is a trace-preserving map, i.e. ( I | ˆ E| ρ ) = ( ˆ E † ( I ) | ρ ) = 1 or ˆ E † ( I ) = I .We have to assume the superoperator ˆ E to be not merely positive but completely positive[43]. The superoperator ˆ E is a completely positive map of the operator space, if n X k =1 n X l =1 B † k ˆ E ( A † k A l ) B l ≥ for all operators A k , B k and all n .Let the superoperator ˆ E be a convex linear map on the set of density matrix operators, i.e. ˆ E (cid:16)X s λ s ρ s (cid:17) = X s λ s ˆ E ( ρ s ) , where all λ s are < λ s < and P s λ s = 1 . Any convex linear map of density matrixoperators can be uniquely extended to a linear map on Hermitian operators. Note that anylinear completely positive superoperator can be represented by ˆ E = m X k =1 ˆ L A k ˆ R A † k : ˆ E ( ρ ) = m X k =1 A k ρA † k . If this superoperator is a trace-preserving superoperator, then m X k =1 A † k A k = I. ˆ E which is not trace-preserving. Let ( I | ˆ E | ρ ) = T r ( ˆ E ( ρ )) bethe probability that the process represented by the superoperator ˆ E occurs. Since the probabilityis non-negative and never exceed 1, it follows that the superoperator ˆ E is a trace-decreasingsuperoperator: ≤ ( I | ˆ E | ρ ) ≤ or ˆ E † ( I ) ≤ I . In general, any real linear completely positivetrace-decreasing superoperator is not a quantum operation, since it can be not trace-preserving.The quantum operation cannot be defined as a nonlinear trace-preserving operation ˆ N by ˆ N | ρ ) = ˆ E | ρ )( I | ˆ E | ρ ) − or ˆ N ( ρ ) = ˆ E ( ρ ) T r ( ˆ E ( ρ )) , (1)where ˆ E is a real linear completely positive trace-decreasing superoperator.All processes occur in time. It is naturally to consider time dependence for quantum op-erations ˆ E ( t, t ) . Let the linear superoperators ˆ E ( t, t ) form a completely positive quantumsemigroup [42] such that ddt ˆ E ( t, t ) = ˆΛ t ˆ E ( t, t ) , (2)where ˆΛ † is a completely dissipative superoperator [39, 42, 19]. We would like to consider thepath integral for quantum operations ˆ E ( t, t ) with infinitesimal generator ˆΛ , where the adjointsuperoperator ˆΛ † is completely dissipative , i.e. ˆΛ † ( A k A l ) − ˆΛ † ( A k ) A l − A k ˆΛ † ( A l ) ≥ . An important property of most open and dissipative quantum systems is the entropy variation.Nevertheless the unitary quantum evolution of a mixed state ̺ t described by von Neumannequation ∂̺ t ∂t = − i ¯ h [ H, ̺ t ] (3)leaves the entropy < S > = − T r ( ̺ t ln̺ t ) unchanged. Therefore, to describe general quantumsystems, one normally uses [47, 38] a generalization of (3).To describe dissipative quantum systems one usually considers [47] the following equation: ∂̺ t ∂t = − i ¯ h [ H, ̺ t ] + D ( ̺ ) . (4)4 .1 Liouville-von Neumann equation Let us consider a generalization of equation (3). The Liouville-von Neumann equation [16, 38,44] can be represented as the linear equation d̺ t dt = Λ t ( ̺ t ) . (5)Using the superoperator formalism, this equation can be rewritten in the form ddt | ̺ t ) = ˆΛ t | ̺ t ) . (6)The superoperator language allows one to use the analogy with Dirac’s notations. This leadsquite simple to the derivation of the appropriate equations.Here ˆΛ t is a linear Liouville superoperator on the operator space H . For the Hamiltonian(closed) quantum systems (3) this superoperator is defined by the Hamiltonian H : ˆΛ t = − i ¯ h ( ˆ L H − ˆ R H ) . (7)For equation (4) the Liouville superoperator has the form ˆΛ t = − i ¯ h ( ˆ L H − ˆ R H ) + ˆ D. (8)In general, the operator | ̺ t ) is an unnormalized density matrix operator, i.e. T r̺ t = ( I | ̺ t ) = 1 .Equation (6) has a formal solution | ̺ t ) = ˆ E ( t, t ) | ̺ t ) , (9)where ˆ E ( t, t ) is a linear quantum operation defined by ˆ E ( t, t ) = T exp Z tt dτ ˆΛ τ . (10)The symbol T is a Dyson’s time-ordering operator [45]. The quantum operation (10) satisfiesthe Liouville-von Neumann equation (2). We can define a normalized density matrix operator | ρ t ) by | ρ t ) = | ̺ t )( I | ̺ t ) − , or | ρ t ) = ˆ E ( t, t ) | ̺ )( I | ˆ E ( t, t ) | ̺ ) , i.e. ρ t = ̺ t /T r̺ t . The evolution equation for the normalized density matrix operator ρ t can bewritten in the form ddt | ρ t ) = ˆΛ t | ρ t ) − | ρ t )( I | ˆΛ t | ρ t ) . (11)In general, this equation is a nonlinear equation [19]. A formal solution of equation (11) isconnected with the nonlinear quantum operation (1) by | ρ t ) = ˆ N ( t, t ) | ρ t ) . .2 Quantum Markovian equation Lindblad [39] has shown that there exists a one-to-one correspondence between the completelypositive norm continuous semigroup of superoperators ˆ E ( t, t ) and superoperator ˆΛ such thatthe adjoint superoperator ˆΛ † is completely dissipative. The structural theorem of Lindblad givesthe most general form of the bounded adjoint completely dissipative Liouville superoperator ˆΛ .The Liouville-von Neumann equation (11) for a completely positive evolution is a quantumMarkovian master equation (Lindblad equation) [39, 40, 41]: dρ t dt = − i ¯ h [ H, ρ t ] + 12¯ h m X k =1 (cid:16) [ V k ρ t , V † k ] + [ V k , ρ t V † k ] (cid:17) . (12)This equation in the Liouville space can be written as ddt | ρ t ) = ˆΛ | ρ t ) , where the Liouville superoperator ˆΛ is given by ˆΛ = − i ¯ h ( ˆ L H − ˆ R H ) + 12¯ h m X k =1 (cid:16) L V k ˆ R V † k − ˆ L V k ˆ L V † k − ˆ R V † k ˆ R V k (cid:17) . (13)The basic assumption is that the general form of a bounded superoperator ˆΛ , given by theLindblad theorem, is also valid for an unbounded superoperator [42, 46]. Another conditionimposed on the operators H, V k , V † k is that they are functions of the observables P and Q (with [ Q, P ] = i ¯ hI ) of the one-dimensional quantum system. Let us consider V k = a k P + b k Q , were k = 1 , , and a k , b k are complex numbers, and the Hamiltonian operator H is H = 12 m P + mω Q + µ P Q + QP ) . Then with the notation [46]: d qq = ¯ h X k =1 , | a k | , d pp = ¯ h X k =1 , | b k | ,d pq = − ¯ h Re (cid:16) X k =1 , a ∗ k b k (cid:17) , λ = − Im (cid:16) X k =1 , a ∗ k b k (cid:17) equation (12) in the Liouville space can be written as ddt | ρ t ) = 12 m ˆ L + P ˆ L − P + mω L + Q ˆ L − Q − ( λ − µ ) ˆ L − P ˆ L + Q | ρ t ) + ( λ + µ ) ˆ L − Q ˆ L + P | ρ t )+ d pp ˆ L − Q ˆ L − Q | ρ t ) + d qq ˆ L − P ˆ L − P | ρ t ) − d pq ˆ L − P ˆ L − Q | ρ t ) , (14)where L ± are the multiplication superoperators defined by ˆ L − A = 1 i ¯ h ( ˆ L A − ˆ R A ) , ˆ L + A = 12 ( ˆ L A + ˆ R A ) . The properties of these superoperators are considered in the appendix. Equation (14) is a super-operator form of the well-known phenomenological dissipative model [47, 46].
In the coordinate representation the kernel ̺ ( q, q ′ , t ) = ( q, q ′ | ̺ t ) of the density operator | ̺ t ) evolves according to the equation ̺ ( q, q ′ , t ) = Z dq dq ′ E ( q, q ′ , q , q ′ , t, t ) ̺ ( q , q ′ , t ) . The function E ( q, q ′ , q , q ′ , t, t ) = ( q, q ′ | ˆ E ( t, t ) | q , q ′ ) (15)is a kernel of the linear quantum operation ˆ E ( t, t ) . Let the Liouville superoperator ˆΛ t be timeindependent, i.e. the quantum operation ˆ E ( t, t ) is given by ˆ E ( t, t ) = exp { ( t − t ) ˆΛ } . (16) Proposition 1.
Let { ˆ E ( t, t ) , t ≥ t } be a superoperator semigroup on operator space H ˆ E ( t , t ) = ˆ I, ˆ E ( t, t ) = ˆ E ( t, t ) ˆ E ( t , t ) , where t ≥ t ≥ t such that the infinitesimal generator ˆΛ of this semigroup is defined by (16).Then the path integral for kernel (15) of the quantum operation ˆ E ( t, t ) has the following form: E ( q, q ′ , q , q ′ , t, t ) = Z D q D q ′ D p D p ′ exp Z tt dt (cid:16) ı ¯ h [ ˙ qp − ˙ q ′ p ′ ] + Λ S ( q, q ′ , p, p ′ ) (cid:17) . (17) This form is the integral over all trajectories in the double phase space with the constraints that q ( t ) = q , q ( t ) = q , q ′ ( t ) = q ′ , q ′ ( t ) = q ′ and the measure D q = Y t dq ( t ) , D p = Y t dp ( t )2 π ¯ h . he symbol Λ S ( q, q ′ , p, p ′ ) of the Liouville superoperator ˆΛ is connected with the kernel Λ( q, q ′ , y, y ′ ) by Λ S ( q, q ′ , p, p ′ ) = Z dydy ′ Λ( q, q ′ , y, y ′ ) · exp − ı ¯ h [( q − y ) p − ( q ′ − y ′ ) p ′ ] , where Λ( q, q ′ , y, y ′ ) = ( q, q ′ | ˆΛ | y, y ′ ) and Λ( q, q ′ , y, y ′ ) = 1(2 π ¯ h ) n Z dpdp ′ Λ S ( q, q ′ , p, p ′ ) · exp ı ¯ h [( q − y ) p − ( q ′ − y ′ ) p ′ ] . Proof.
Let us derive path integral form (17) for quantum operation (16).1. Let time interval [ t , t ] has n + 1 equal parts τ = t − t n + 1 . Using the superoperator semigroup composition rule ˆ E ( t, t ) = ˆ E ( t, t n ) ˆ E ( t n , t n − ) ... ˆ E ( t , t ) , where t ≥ t n ≥ t n − ≥ ... ≥ t ≥ t , we obtain the following integral representation: E ( q, q ′ , q , q ′ , t, t ) = Z dq n dq ′ n ...dq dq ′ E ( q, q ′ , q n , q ′ n , t, t n ) ... E ( q , q ′ , q , q ′ , t , t ) . This representation can be written in the form E ( q, q ′ , q , q ′ , t, t ) = Z n Y k =1 dq k dq ′ k n +1 Y k =1 E ( q k , q ′ k , q k − , q ′ k − , t k , t k − ) . Here q n +1 = q and q ′ n +1 = q ′ .2. Let us consider the kernel E ( q k , q ′ k , q k − , q ′ k − , t k , t k − ) , of the quantum operation ˆ E ( t k , t k − ) . If the time interval [ t k − , t k ] is a small, then in thecoordinate representation we have ̺ ( q k , q ′ k , t k ) = Z dq k − dq ′ k − E ( q k , q ′ k , q k − , q ′ k − , t k , t k − ) ̺ ( q k − , q ′ k − , t k − ) , ̺ ( q k , q ′ k , t k ) = ( q k , q ′ k | ̺ t k ) = ( q k , q ′ k | ˆ E ( t k , t k − ) | ̺ t k − ) == ∞ X n =0 τ n n ! ( q k , q ′ k | ˆΛ n | ̺ t k − ) = ( q k , q ′ k | ̺ t k − ) + ( q k , q ′ k | ˆΛ | ̺ t k − ) τ + O ( τ ) == ̺ ( q k , q ′ k , t k − ) + τ Z dq k − dq ′ k − Λ( q k , q ′ k , q k − , q ′ k − ) · ̺ ( q k − , q ′ k − , t k − ) + O ( τ ) == Z dq k − dq ′ k − (cid:16) δ ( q k − q k − ) δ ( q ′ k − − q ′ k )+ τ Λ( q k , q ′ k , q k − , q ′ k − )+ O ( τ ) (cid:17) ̺ ( q k − , q ′ k − , t k − ) . Therefore, we have E ( q k , q ′ k , q k − , q ′ k − , t k , t k − ) = δ ( q k − q k − ) δ ( q ′ k − − q ′ k ) + τ Λ( q k , q ′ k , q k − , q ′ k − ) + O ( τ ) .
3. Delta-functions can be written in the form δ ( q k − q k − ) δ ( q ′ k − − q ′ k ) = Z dp k dp ′ k (2 π ¯ h ) n exp i ¯ h [( q k − q k − ) p k − ( q ′ k − q ′ k − ) p ′ k ] . Using the relations ( q, q ′ | p, p ′ ) = < q | p >< p ′ | q ′ > = 1(2 π ¯ h ) n exp i ¯ h ( qp − q ′ p ′ ) , ( p, p ′ | q, q ′ ) = < p | q >< q ′ | p ′ > = 1(2 π ¯ h ) n exp − i ¯ h ( qp − q ′ p ′ ) , we obtain the symbol Λ S ( q k , q ′ k , p k , p ′ k ) of the Liouville superoperator by Λ( q k , q ′ k , q k − , q ′ k − ) = ( q k , q ′ k | ˆΛ | q k − , q ′ k − ) == Z dp k dp ′ k ( q k , q ′ k | ˆΛ | p k , p ′ k )( p k , p ′ k | q k − , q ′ k − ) == Z dp k dp ′ k Λ S ( q k , q ′ k , p k , p ′ k ) · ( q k , q ′ k | p k , p ′ k )( p k , p ′ k | q k − , q ′ k − ) == 1(2 π ¯ h ) n Z dp k dp ′ k Λ S ( q k , q ′ k , p k , p ′ k ) · exp i ¯ h [( q k − q k − ) p k − ( q ′ k − q ′ k − ) p ′ k ] .
4. The kernel of the quantum operation ˆ E ( t k , t k − ) is E ( q k , q ′ k , q k − , q ′ k − , t k , t k − ) == 1(2 π ¯ h ) n Z dp k dp ′ k (cid:16) τ Λ S ( q k , q ′ k , p k , p ′ k )+ O ( τ ) (cid:17) exp i ¯ h (cid:16) ( q k − q k − ) p k − ( q ′ k − q ′ k − ) p ′ k (cid:17) .
9. The kernel of the quantum operation ˆ E ( t, t ) is E ( q, q ′ , q , q ′ , t, t ) = Z n Y k =1 dq k dq ′ k n +1 Y k =1 dp k dp ′ k (2 π ¯ h ) n exp i ¯ h n +1 X k =1 h ( q k − q k − ) p k − ( q ′ k − q ′ k − ) p ′ k i ·· n +1 Y k =1 (cid:16) − τ Λ S ( q k , q ′ k , p k , p ′ k ) + O ( τ ) (cid:17) .
6. Using lim n →∞ n Y k =1 (1 + Λ k n ) = lim n →∞ n Y k =1 exp ( Λ k n ) , we obtain E ( q, q ′ , q , q ′ , t, t ) = Z n Y k =1 dq k dq ′ k n +1 Y k =1 dp k dp ′ k (2 π ¯ h ) n ·· exp n +1 X k =1 τ (cid:16) i ¯ h [ q k − q k − τ p k − q ′ k − q ′ k − τ p ′ k ] + Λ S ( q k , q ′ k , p k , p ′ k ) (cid:17) .
7. Let q k , q ′ k , p k , p ′ k be the values of the functions q ( t ) , q ′ ( t ) , p ( t ) , p ′ ( t ) and t k = t + kτ ,i.e. q k = q ( t k ) , q ′ k = q ′ ( t k ) , p k = p ( t k ) , p ′ k = p ′ ( t k ) , where k = 0 , , , ..., n, n + 1 . Using lim τ → q k − q k − τ = ˙ q ( t k − ) , lim n →∞ n +1 X k =1 A ( t k ) τ = Z tt dtA ( t ) , we obtain the kernel of the quantum operation ˆ E ( t, t ) in the path integral form (17). ✷ Corollary . If the dissipative quantum evolution is defined by equation (4), then the pathintegral for the quantum operation kernel has the form E ( q, q ′ , q , q ′ , t, t ) = Z D q D p D q ′ D p ′ exp (cid:16) ı ¯ h ( A ( q, p ) − A ( p ′ , q ′ )) + D ( q, q ′ , p, p ′ ) (cid:17) . Here A ( q, p ) and A ( p ′ , q ′ ) are action functionals defined by A ( q, p ) = Z tt dt (cid:16) ˙ qp − H ( q, p ) (cid:17) , A ( p ′ , q ′ ) = Z tt dt (cid:16) ˙ q ′ p ′ − H ( p ′ , q ′ ) (cid:17) . (18)10 he functional D ( q, q ′ , p, p ′ ) is a time integral of the symbol D S of the superoperator ˆ D . The functional D ( q, q ′ , p, p ′ ) describes the dissipative part of evolution. Corollary . If the quantum system has no dissipation, i.e. the quantum system is a closedHamiltonian system, then D ( q, q ′ , p, p ′ ) = 0 and the path integral for the quantum operationcan be separated E ( q, q ′ , q , q ′ , t, t ) = U ∗ ( q, q , t, t ) U ( q ′ , q ′ , t, t ) , where U ( q ′ , q ′ , t, t ) = Z D q ′ D p ′ exp − ı ¯ h A ( p ′ , q ′ ) ,U ∗ ( q, q , t, t ) = Z D q D p exp ı ¯ h A ( p, q ) . The path integral for the dissipative quantum systems and the corresponding quantum oper-ations cannot be separated, i.e. this path integral is defined in the double phase space.
Let us consider the Liouville superoperator (13) for the Lindblad equation.
Proposition 2.
Let { ˆ E ( t, t ) , t ≥ t } be a completely positive semigroup of linear real trace-preserving superoperators such that the infinitesimal generators ˆΛ of this semigroup are definedby (13). Then the path integral for the kernel of the completely positive quantum operation ˆ E ( t, t ) has the form E ( q, q ′ , q , q ′ , t, t ) = Z D q D p D q ′ D p ′ F ( q, q ′ , p, p ′ ) exp ı ¯ h (cid:16) A ( q, p ) − A ( p ′ , q ′ ) (cid:17) , (19) where A ( q, p ) and A ( p ′ , q ′ ) are action functionals (18) and the functional F ( q, q ′ , p, p ′ ) is de-fined as F ( q, q ′ , p, p ′ ) = exp − h Z tt dt m X k =1 (cid:16) ( V † k V k )( q, p ) + ( V † k V k )( p ′ , q ′ ) − V k ( q, p ) V † k ( p ′ , q ′ ) (cid:17) . (20)11 roof . The kernel of superoperator (13) is Λ( q, q ′ , y, y ′ ) = ( q, q ′ k | ˆΛ | y, y ′ ) = − ı ¯ h ( < y ′ | q ′ >< q | H | y > − < q | y >< y ′ | H | q ′ > )++ 1¯ h m X k =1 < q | V k | y >< y ′ | V † k | q ′ > − h m X k =1 ( < y ′ | q ′ >< q | V † k V k | y > − < q | y >< y ′ | V † k V k | q ′ > ) . The symbol Λ S ( q, q ′ , p, p ′ ) of the Liouville superoperator ˆΛ can be derived by Λ( q, q ′ , y, y ′ ) = Z dpdp ′ (cid:16) − ı ¯ h ( < y ′ | p ′ >< p ′ | q ′ >< q | H | p >< p | y > −− < q | p >< p | y >< y ′ | p ′ >< p ′ | H | q ′ > )++ 1¯ h m X k =1 < q | V k | p >< p | y >< y ′ | p ′ >< p ′ | V † k | q ′ > (cid:17) , were the operators H and H are defined by the relations H ≡ H − ı m X k =1 V † k V k , H ≡ H + ı m X k =1 V † k V k . Then the symbol Λ S ( q, q ′ , p, p ′ ) of the Liouville superoperator (13) can be written in the form Λ S ( q, q ′ , p, p ′ ) = − ı ¯ h (cid:16) H ( q, p ) − H ( p ′ , q ′ ) + i m X k =1 V k ( q, p ) V † k ( p ′ , q ′ ) (cid:17) , or Λ S ( q, q ′ , p, p ′ ) = − ı ¯ h [ H ( q, p ) − H ( p ′ , q ′ )] −− h m X k =1 (cid:16) ( V † k V k )( q, p ) + ( V † k V k )( p ′ , q ′ ) − V k ( q, p ) V † k ( p ′ , q ′ ) (cid:17) , where H ( q, p ) is a qp -symbol of the Hamilton operator H and H ( p, q ) is a pq -symbol of theoperator H .In the Hamiltonian case ( V k = 0 ), the symbol is given by Λ S ( q, q ′ , p, p ′ ) = − ı ¯ h [ H ( q, p ) − H ( p ′ , q ′ )] . The path integral for a completely positive quantum operation kernel has the form E ( q, q ′ , q , q ′ , t, t ) = Z D q D p D q ′ D p ′ F ( q, q ′ , p, p ′ ) exp ı ¯ h ( A ( q, p ) − A ( p ′ , q ′ )) . Here A ( q, p ) and A ( p ′ , q ′ ) are action functionals (18), and the functional F ( q, q ′ , p, p ′ ) is de-fined by (20). ✷ F ( q, q ′ , p, p ′ ) describes the dissipative part of the evolution and can be calleda (double) phase space influence functional . The completely positive quantum operation is de-scribed by the functional (20). Corollary . For the phenomenological dissipative model (14) the double phase space pathintegral has the form (19) with the functional F ( q, q ′ , p, p ′ ) = exp h Z tt dt (cid:16) d qp ( q − q ′ )( p − p ′ ) −− d qq ( p − p ′ ) − d pp ( q − q ′ ) + i ¯ hλ ( pq ′ − qp ′ ) + i ¯ hµ ( q ′ p ′ − qp ) (cid:17) . (21)Using the well-known connection between the phase space (Hamiltonian) path integral andthe configuration space (Lagrangian) path integral [51, 52], we can derive the following propo-sition. Proposition 3.
If the symbol Λ s ( q, q ′ , p, p ′ ) of the Liouville superoperator can be repre-sented in the form Λ S ( q, q ′ , p, p ′ ) = − ı ¯ h [ H ( q, p ) − H ( p ′ , q ′ )] + D S ( q, q ′ , p, p ′ ) , (22) where H ( q, p ) = 12 a − kl ( q ) p k p l − b k ( q ) p k + c ( q ) , (23) D S ( q, q ′ , p, p ′ ) = − d k ( q, q ′ ) p k + d ′ k ( q, q ′ ) p ′ k + e ( q, q ′ ) , (24) then the double phase space path integral (17) can be represented as a double configurationphase space path integral E ( q, q ′ , q , q ′ , t, t ) = Z D q D q ′ F ( q, q ′ ) exp ı ¯ h ( A ( q ) − A ( q ′ )) , (25) where A ( q ) = Z tt dt L ( q, ˙ q ) , L ( q, ˙ q ) = 12 a kl ( q ) ˙ q k ˙ q l + a kl ( q ) b k ( q ) ˙ q l + 12 a kl ( q ) b k ( q ) b l ( q ) − c ( q ) . (26) This Lagrangian L ( q, ˙ q ) is related to the Hamiltonian (23) by the usual relations L ( q, ˙ q ) = ˙ q k p k − H ( q, p ) , p k = ∂ L ∂ ˙ q . roof . Substituting (22) in (17), we obtain the kernel of the corresponding quantum opera-tion. Integrating (17) in p and p ′ , we obtain relation (25) with the functional F ( q, q ′ ) = exp − h Z tt dt (cid:16) d k ( q, q ′ ) a kl ( q )( b l ( q ) − i h d l ( q, q ′ ))++ d ′ k ( q, q ′ ) a kl ( q ′ )( b l ( q ′ ) − i h d ′ l ( q, q ′ )) + e ( q, q ′ ) + δ (0)∆( q, q ′ ) (cid:17) , where ∆( q, q ′ ) = − ¯ h (cid:16) ln [ det ( a kl ( q ))] + ln [ det ( a kl ( q ′ ))] (cid:17) . ✷ In equation (25) the functional F ( q, q ′ ) can be considered as the Feynman-Vernon influencefunctional. It is known that this functional can be derived by eliminating the bath degrees offreedom, for example by taking a partial trace or by integrating them out. The Feynman-Vernoninfluence functional describes the dissipative dynamics of open systems when we assume thevon Hove limit for a system-reservoir coupling. One can describe a quantum system startingfrom a unitary evolution of some closed system ”system-reservoir” if the quantum system is apart of this closed system. However, situations can arise where it is difficult or impossible tofind a closed system comprising the given quantum system [6]-[14].The Feynman path integral is defined for configuration space. The most general form ofquantum mechanical path integral is defined for the phase space. The Feynman path inte-gral can be derived from the phase space path integral for the special form of the Hamiltonian[51, 52, 53, 54, 55]. It is known that the path integral for the configuration space is correct[51, 52] only for the Hamiltonian (23). The Feynman-Vernon path integral [6] is defined in thedouble configuration space. Therefore, this path integral is a special form of the double phasespace path integral (17). The Feynman-Vernon path integral is correct only for the Liouville su-peroperator (22), (23), (24). Note that the symbol Λ s ( q, q ′ , p, p ′ ) for most of the dissipative andnon-Hamiltonian systems (with completely positive quantum operations) cannot be representedin the form (22). Corollary . In the general case, the completely positive quantum operation cannot be rep-resented as the double configuration space path integral (25). pp ′ . Therefore this model and the Liouville symbol Λ s ( q, q ′ , p, p ′ ) forthis model cannot be represented in the form (22), (23), (24). The usual models of a quantum computer deal only with the discrete variables, unitary quantumoperations (gates) and pure states. Many quantum variables such as position and momentum arecontinuous. The use of continuous-variable quantum computing [21, 22, 23] allows informationto be encoded and processed much more compactly and efficiently than with discrete variablecomputing. Quantum computation using continuous variables is an alternative approach toquantum computations with discrete variables.In the models with unitary quantum operations on pure states it is difficult or impossibleto deal formally with measurements, dissipation, decoherence and noise. It turns out that therestriction to pure states and unitary gates is unnecessary [20]. In [20], a model of quantumcomputations by quantum operations with mixed states was constructed. It is known that themeasurement is described by quantum operations. The measurement quantum operations are thespecial case of quantum operations on mixed states. The von Neumann measurement quantumoperation as a nonlinear quantum gate is realized in [20]. The continuous quantum measurementis described by the path integrals [56, 57, 58, 59]. Therefore, the path integral for quantum op-erations can be useful for continuous-variables quantum operations on mixed states. Quantumcomputation by quantum operations with mixed states is considered [20] for discrete variablesonly. Some points of the model of the continuous-variable quantum computations with mixedstates are considered in this section. The double phase space path integral can be useful for thecontinuous-variables quantum gates on mixed states.The main steps of the continuous-variables generalization of quantum computations byquantum operations with mixed states are following.1. The state | ρ ( t )) of the discrete-variable quantum computation with mixed states [20] is asuperposition of basis elements | ρ ( t )) = N − X µ =0 | µ ) ρ µ ( t ) , (27)15here ρ µ ( t ) = ( µ | ρ ( t )) are real numbers (functions). The basis | µ ) of the discrete-variable Liouville space H ( n ) is defined [20] by | µ ) = | µ ...µ n ) = 1 √ n | σ µ ) = 1 √ n | σ µ ⊗ ... ⊗ σ µ n ) , (28)where σ µ are Pauli matrices, N = 4 n , each µ i ∈ { , , , } and ( µ | µ ′ ) = δ µµ ′ , N − X µ =0 | µ )( µ | = ˆ I (29)is the discrete-variable computational basis.The state | ρ ( t )) of the continuous-variable quantum computation at any point of time canbe considered as a superposition of basis elements | ρ ( t )) = Z dx Z dx ′ | x, x ′ ) ρ ( x, x ′ , t ) , (30)where ρ ( x, x ′ , t ) = ( x, x ′ | ρ ( t )) are the density matrix elements. The basis | x, x ′ ) of thecontinuous-variable operator space H is defined by | x, x ′ ) = || x >< x ′ | ) , where ( x, x ′ | y, y ′ ) = δ ( x − x ′ ) δ ( y − y ′ ) , Z dx Z dx ′ | x, x ′ )( x, x ′ | = ˆ I (31)can be considered as a continuous-variable computational basis.2. In the discrete-variable computational basis | µ ) any linear quantum operation ˆ E acting onn-qubits mixed (or pure) states can be represented as a quantum four-valued logic gate[20]: ˆ E on n-ququats can be given by ˆ E = N − X µ =0 N − X ν =0 E µν | µ )( ν | , (32)where N = 4 n , E µν = 12 n T r (cid:16) σ µ ˆ E ( σ ν ) (cid:17) , (33)and σ µ = σ µ ⊗ ... ⊗ σ µ n .In the continuous-variable computational basis | x, x ′ ) any linear quantum operation ˆ E acts on mixed (or pure) states can be represented as a continuous-variable quantum gate: ˆ E ( t , t ) = Z dx dx ′ dy dy ′ E ( x, x ′ , y, y ′ , t , t ) | x, x ′ )( y, y ′ | , (34)16here E ( x, x ′ , y, y ′ , t , t ) = ( x, x ′ | ˆ E ( t , t ) | y, y ′ ) is a kernel of the real trace-preservingpositive (or completely positive) superoperator ˆ E ( t , t ) This quantum operation can beconsidered as a continuous-variable quantum gate.The continuous quantum measurement which is described by the path integral [56, 57,58, 59] is the special case of the continuous-variable quantum gate. The path integral forthe quantum operations can be useful for all continuous-variables quantum operations onmixed states.3. Many quantum variables, such as position and momentum are continuous. The use ofcontinuous-variable quantum computing [21, 22, 23] allows information to be encodedand processed much more compactly and efficiently than with discrete variable comput-ing.Mixed states subjected to the general quantum operations could increase efficiency. Thisincrease is connected with the increasing number of computational basis elements foroperator Hilbert space. A pure state of the quantum systems is an element of functionalHilbert space H . A mixed state of the system is an element | ρ ) of the operator Hilbertspace H . A mixed state of the system can be considered as an element ρ ( x, x ′ , t ) of thedouble functional Hilbert space H ⊗ H .The use of continuous-variable quantum computation by quantum operations with mixedstates can increase efficiency compared with discrete variable computing.
The usual quantum computer model deal only with the discrete variables, unitary quantumoperations and pure states. It is known that many of quantum variables, such as position andmomentum are continuous. The use of continuous-variable quantum computing [21, 22, 23]allows information to be encoded and processed much more efficiently than in discrete-variablequantum computer. Quantum computation using continuous variables is an alternative approachto quantum computations with discrete variables.The quantum computation by quantum operations with mixed states is considered in [20].It is known that the measurement is described by quantum operations. The measurement quan-tum operations are the special case of quantum operations on mixed states. The von Neumannmeasurement quantum operation as a nonlinear quantum gate is realized in Ref. [20]. The con-17inuous quantum measurement is described by the path integrals [56, 57, 58, 59]. Therefore, thepath integral for quantum operations can be useful for continuous-variables quantum operationson mixed states. Quantum computation by quantum operations with mixed states is considered[20] only for discrete variables. The model of continuous-variable quantum computations withmixed states will be suggested in the next publication. The double phase space path integral canbe useful for continuous-variables quantum gates on mixed states.Let us note the second application of double phase space path integral. The path integralformulation of the quantum statistical mechanics leads to the powerful simulation scheme [60]for the molecular dynamics. In the past few years the statistical mechanics of non-Hamiltoniansystems was developed for the molecular dynamical simulation purpose [61, 62, 63, 64, 65].The suggested path integral can be useful for the application in the non-Hamiltoniam statisticalmechanics of quantum [66, 67] and quantum-classical systems [68, 69].
Acknowledgment
This work was partially supported by the RFBR grant No. 02-02-16444.
Appendix
For the concept of Liouville space and superoperators see [24]-[38].
A.1. Operator space
The space of linear operators acting on a Hilbert space H is a complex linear space H . Wedenote an element A of H by a ket-vector | A ) . The inner product of two elements | A ) and | B ) of H is defined as ( A | B ) = T r ( A † B ) . The norm k A k = q ( A | A ) is the Hilbert-Schmidt normof operator A . A new Hilbert space H with the inner product is called Liouville space attachedto H or the associated Hilbert space, or Hilbert-Schmidt space [24]-[38].The X-representation uses eigenfunctions | x > of the operator X . In general, the operator X can be an unbounded operator. This operator can have a continuous spectrum. This leadsus to consider rigged Hilbert space [48, 49, 19, 38] (Gelfand triplet) B ⊂ H = H ∗ ⊂ B ∗ andassociated operator space. The rigged operator Hilbert space can be considered as the usualrigged Hilbert space for the operator kernels. 18et the set {| x > } satisfy the following conditions: < x | x ′ > = δ ( x − x ′ ) , Z dx | x >< x | = I. Then | x, x ′ ) = || x >< x ′ | ) satisfies ( x, x ′ | y, y ′ ) = δ ( x − x ′ ) δ ( y − y ′ ) , Z dx Z dx ′ | x, x ′ )( x, x ′ | = ˆ I. For an arbitrary element | A ) of H , we have | A ) = Z dx Z dx ′ | x, x ′ )( x, x ′ | A ) , (35)where ( x, x ′ | A ) is a kernel of the operator A such that ( x, x ′ | A ) = T r (( | x >< x ′ | ) † A ) = T r ( | x ′ >< x | A ) = < x | A | x ′ > = A ( x, x ′ ) . An operator ρ of density matrix can be considered as an element | ρ ) of the Liouville (Hilbert-Schmidt) space H . Using equation (35), we obtain | ρ ) = Z dx Z dx ′ | x, x ′ )( x, x ′ | ρ ) , (36)where the trace is represented by T rρ = ( I | ρ ) = Z dx ( x, x | ρ ) = 1 . A.2. Superoperators
Operators, which act on H , are called superoperators and we denote them in general by the hat.A superoperator is a map which maps operator to operator.For an arbitrary superoperator ˆΛ on H , which is defined by ˆΛ | A ) = | ˆΛ( A )) , we have ( x, x ′ | ˆΛ | A ) = Z dy Z dy ′ ( x, x ′ | ˆΛ | y, y ′ )( y, y ′ | A ) = Z dy Z dy ′ Λ( x, x ′ , y, y ′ ) A ( y, y ′ ) , where Λ( x, x ′ , y, y ′ ) = ( x, x ′ | ˆΛ | y, y ′ ) is a kernel of the superoperator ˆΛ .Let A be a linear operator in the Hilbert space H . We can define the multiplication super-operators ˆ L A and ˆ R A by the following equations: ˆ L A | B ) = | AB ) , ˆ R A | B ) = | BA ) . | x, x ′ ) we have ( x, x ′ | ˆ L A | B ) = Z dy Z dy ′ ( x, x ′ | ˆ L A | y, y ′ )( y, y ′ | B ) = Z dy Z dy ′ L A ( x, x ′ , y, y ′ ) B ( y, y ′ ) . Using ( x, x ′ | AB ) = < x | AB | x ′ > = Z dy Z dy ′ < x | A | y >< y | B | y ′ >< y ′ | x ′ >, we obtain the kernel of the left multiplication superoperator L A ( x, x ′ , y, y ′ ) = < x | A | y >< x ′ | y ′ > = A ( x, y ) δ ( x ′ − y ′ ) . A superoperator ˆ E † is called the adjoint superoperator for ˆ E if ( ˆ E † ( A ) | B ) = ( A | ˆ E ( B )) forall | A ) and | B ) from H . For example, if ˆ E = ˆ L A ˆ R B , then ˆ E † = ˆ L A † ˆ R B † . If ˆ E = ˆ L A , then ˆ E † = ˆ L A † .Left superoperators ˆ L ± A are defined as Lie and Jordan multiplication by the relations ˆ L − A B = 1 i ¯ h ( AB − BA ) , ˆ L + A B = 12 ( AB + BA ) . The left superoperator ˆ L ± A and the right superoperator ˆ R ± A are connected by ˆ L − A = − ˆ R − A , ˆ L + A =ˆ R + A . An algebra of the superoperators ˆ L ± A is defined [50] by(1) the Lie relations ˆ L − A · B = ˆ L − A ˆ L − B − ˆ L − B ˆ L − A , (2) the Jordan relations ˆ L +( A ◦ B ) ◦ C + ˆ L + B ˆ L + C ˆ L + A + ˆ L + A ˆ L + C ˆ L + B = ˆ L + A ◦ B ˆ L + C + ˆ L + B ◦ C ˆ L + A + ˆ L + A ◦ C ˆ L + B , ˆ L +( A ◦ B ) ◦ C + ˆ L + B ˆ L + C ˆ L + A + ˆ L + A ˆ L + C ˆ L + B = ˆ L + C ˆ L + A ◦ B + ˆ L + B ˆ L + A ◦ C + ˆ L + A ˆ L + B ◦ C , ˆ L + C ˆ L + A ◦ B + ˆ L + B ˆ L + A ◦ C + ˆ L + A ˆ L + B ◦ C = ˆ L + A ◦ B ˆ L + C + ˆ L + B ◦ C ˆ L + A + ˆ L + A ◦ C ˆ L + B , (3) the mixed relations: ˆ L + A · B = ˆ L − A ˆ L + B − ˆ L + B ˆ L − A , ˆ L − A ◦ B = ˆ L + A ˆ L − B + ˆ L + B ˆ L − A , ˆ L + A ◦ B = ˆ L + A ˆ L + B − ¯ h L − B ˆ L − A , ˆ L + B ˆ L + A − ˆ L + A ˆ L + B = − ¯ h L − A · B . Here we use the notations A · B = 1 i ¯ h ( AB − BA ) , A ◦ B = 12 ( AB + BA ) . eferences [1] Hellwing K E and K. Kraus K 1969 ”Pure operations and measurements” Commun. Math.Phys. States, Effects and Operations. Fundamental Notions of Quantum Theory (Berlin: Springer)[5] Schumacher B 1996 ”Sending entanglement through noisy quantum channels” Phys. Rev.A
170 and referencestherein.[10] Pechukas P 1969 ”Time-dependent semiclassical scattering theory” Phys. Rev. Open Quantum Systems and Feynman Integrals (Reidel: Dordrecht)[13] Exner P and Kolerov G I 1981 ”Path-integral expression of dissipative dynamics” Phys.Lett. A Proceed-ings of the VIII International Workshop on High Energy Physics and Quantum Field The-ory (Moscow: MSU) pp.205-209. 2118] Tarasov V E 2000 in ”Quantum theory and non-Hamiltonian systems. Path-integral ap-proach”
Proc. XIV International Workshop on High Energy Physics and Quantum FieldTheory (Moscow: MSU) pp.637-640.[19] Tarasov V E 2000
Quantum Mechanics. Lectures on Theory Foundation (Moscow: Vu-zovskaya kniga)[20] Tarasov V E 2002 ”Quantum computer with mixed states and four-valued logic” J. Phys.A Quantum Kinematics and Dynamics (New York: W.A. Benjamin Inc.)ch 2.11, 2.12[30] Schmutz M 1978 ”Real-time Green’s functions in many body problems” Zeitsch. Phys. B Density Matrix. Theory and Applications (London: Plenum Press) ch 7.6[32] Abragam A and Goldman M 1982
Nuclear Magnetism. Order and Disorder (Oxford:Clarendon Press) ch 1B[33] Schmutz M 1983 ”Construction of functional integrals for bosons and fermions from Li-ouville space” Lett. Nuovo Cim. The Quantum Statistics of Dynamic Processes (Berlin:Springer-Verlag) ch 3.1., 8.1., 8.2.[35] Suzuki M 1991 ”Density matrix formalism, double-space and thermofield dynamics innon-equilibrium dissipative systems” Int. J. Mod. Phys. B Proc. XI International Workshop on HEP and QFT (Moscow: MSU) pp.368-371.[37] Caves C M 1999, ”Quantum errow correction and reversible operations” J. Superconduc-tivity
707 (quant-ph/9811082)[38] Tarasov V E 2000,
Mathematical Introduction to Quantum Mechanics (Moscow: MAI) ch2,3,5[39] Lindblad G 1976 ”On the generators of quantum dynamical semigroups” Commum. Math.Phys. Quantum Dynamical Semigroups and Applications (Berlin:Springer-Verlag)[43] Arveson W 2002 ”The domain algebra of a CP-semigroup” math.OA/0005251 ch 3, andreferences therein[44] Tarasov V E 1997 ”Quantum dissipative systems. III. Definition and algebraic structures”Theor. Math. Phys. Generalized Functions, Vol.4 - Applications ofHarmonic Analysis (New York: Academic Press)[49] Roberts J E 1966 ”Rigged Hilbert spaces in quantum mechanics” Commun. Math. Phys. (5) 5[51] Slavnov A and Fadeev L 1988 Introduction to Quantum Theory of Gauge Fields
Quarks, Leptons and Gauge Fields (New York: World Scientific) ch 7.12353] Lee T D and Yang C.N. 1962 ”Theory of charged vector mesons interacting with theelectromagnetic field” Phys. Rev. Quantum Field Theory (Cambridge: Cambridge University Press) ch 5.1[56] Caves C M 1986 ”Quantum-mechanics of measurements distributed in time. A path-integral approach” Phys. Rev. D Quantum Measurements and Decoherence: Models and Phenomenol-ogy (Dordrecht, Boston, London: Kluwer Acad. Publ.) ch 5.2, 5.3[58] Mensky M B 1994 ”Continuous quantum measurements:Restricted path integrals andmaster equations” Phys. Lett. A115