Journal of Physics A 37 (2004) 3241-3257
Path Integral for Quantum Operations
arXiv:0706.2142v1 [quant-ph] 14 Jun 2007
Vasily E. Tarasov Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia E-mail:
[email protected] Abstract In this paper we consider a phase space path integral for general time-dependent quantum operations, not necessarily unitary. We obtain the path integral for a completely positive 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
1
Introduction
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 quantum operation [1, 2, 3, 4, 5]. One can describe a quantum operation for a quantum system starting from a unitary evolution of some closed system if the quantum system is a part of the closed system [6]-[14]. However, situations can arise where it is difficult or impossible to find a closed system comprising the given quantum system [15]-[19]. This would render theory of quantum operations a fundamental generalization of the unitary evolution of the closed quantum system. The usual models of a quantum computer deal only with unitary quantum operations on pure states. In these models it is difficult or impossible to deal formally with measurements, dissipation, decoherence and noise. It turns out that the restriction to pure states and unitary gates is unnecessary [20]. In [20], a model of quantum computations by quantum operations with mixed states was constructed. The computations are realized by quantum operations, not necessarily unitary. Mixed states subjected to general quantum operations could increase efficiency. This increase is connected with the increasing number of computational basis elements for the Hilbert space. A pure state of n two level quantum systems is an element of the 2n -dimensional functional Hilbert space. A mixed state of the system is an element of the 4n -dimensional operator Hilbert space. Therefore, the increased efficiency can be formalized in terms of a four-valued 1
logic replacing the conventional two-valued logic. Unitary gates and quantum operations for a quantum computer with pure states can be considered as quantum gates of a mixed state quantum computer. Quantum algorithms on a quantum computer with mixed states are expected to run on a smaller network than with pure state implementation. The path integral for quantum operations can be useful for the continuous-variable generalization of quantum computations by quantum operations with mixed states. The usual models of a quantum computer deal only with discrete variables. Many quantum variables such as position and 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 with discrete variable computing. Quantum computation using continuous variables is an alternative approach to quantum computations with discrete variables. All processes occur in time. It is naturally to consider time dependence for quantum operations. In this paper we consider the path integral approach to general time-dependent quantum operations. We use the operator space [24]-[38] and superoperators on this space. The path integral for unitary evolution from the operator (Liouville) space was derived in [33]. The quantum operation is considered as a real completely positive trace-preserving superoperator on the operator space. We derive a path integral for a completely positive quantum operation satisfied Lindblad 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 are discussed. 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 quantum operation satisfied Liouville-von Neumann equation. In section 5, we obtain a path integral for time-dependent quantum operation with an infinitesimal generator such that the adjoint generator is completely dissipative. In section 6, the continuous-variables quantum computation by quantum operations with mixed states is discussed. In the appendix, the mathematical background (Liouville space, superoperators) is considered.
2
Quantum operations as superoperators
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 is called a quantum operation. For the concept of quantum operations, see [1, 2, 3, 4, 5]. A quantum operation is a superoperator Eˆ which maps the density matrix operator |ρ) to 2
the density matrix operator |ρ′ ). For the concept of superoperators and operator space see the appendix and [24]-[38] ˆ should also be a density matrix operator. Any If |ρ) is a density matrix operator, then E|ρ) density matrix operator ρ is a self-adjoint (ρ†t = ρt ), positive (ρt > 0) operator with unit trace (T rρt = 1). Therefore, the requirements for a superoperator Eˆ to be the quantum operation are as follows: † ˆ ˆ † ) for all A. The 1. The superoperator Eˆ is a real superoperator, i.e. E(A) = E(A ˆ real superoperator Eˆ maps the self-adjoint operator ρ to the self-adjoint operator E(ρ): † ˆ ˆ (E(ρ)) = E(ρ).
2. The superoperator Eˆ is a positive superoperator, i.e. Eˆ maps positive operators to positive ˆ 2 ) > 0 for all A 6= 0 or E(ρ) ˆ ≥ 0. operators: E(A ˆ = (Eˆ† (I)|ρ) = 1 or Eˆ† (I) = 3. The superoperator Eˆ is a trace-preserving map, i.e. (I|E|ρ) I. We have to assume the superoperator Eˆ to be not merely positive but completely positive [43]. The superoperator Eˆ is a completely positive map of the operator space, if n n X X
k=1 l=1
ˆ † Al )Bl ≥ 0 Bk† E(A k
for all operators Ak , Bk and all n. Let the superoperator Eˆ be a convex linear map on the set of density matrix operators, i.e. Eˆ
X
λs ρs =
s
X s
ˆ s ), λs E(ρ
where all λs are 0 < λs < 1 and s λs = 1. Any convex linear map of density matrix operators can be uniquely extended to a linear map on Hermitian operators. Note that any linear completely positive superoperator can be represented by P
Eˆ =
m X
k=1
ˆ † : ˆA R L k A
ˆ = E(ρ)
k
m X
Ak ρA†k .
k=1
If this superoperator is a trace-preserving superoperator, then m X
A†k Ak = I.
k=1
3
The restriction to linear quantum operations is unnecessary. Let us consider a linear real ˆ = T r(E(ρ)) ˆ completely positive superoperator Eˆ which is not trace-preserving. Let (I|E|ρ) be ˆ the probability that the process represented by the superoperator E occurs. Since the probability is non-negative and never exceed 1, it follows that the superoperator Eˆ is a trace-decreasing ˆ ≤ 1 or Eˆ† (I) ≤ I. In general, any real linear completely positive superoperator: 0 ≤ (I|E|ρ) trace-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 Nˆ by ˆ ˆ −1 Nˆ |ρ) = E|ρ)(I| E|ρ)
or
Nˆ (ρ) =
ˆ E(ρ) , ˆ T r(E(ρ))
(1)
where Eˆ is a real linear completely positive trace-decreasing superoperator. All processes occur in time. It is naturally to consider time dependence for quantum opˆ t0 ). Let the linear superoperators E(t, ˆ t0 ) form a completely positive quantum erations E(t, semigroup [42] such that d ˆ ˆ t E(t, ˆ t0 ), E(t, t0 ) = Λ (2) dt ˆ † is a completely dissipative superoperator [39, 42, 19]. We would like to consider the where Λ ˆ t0 ) with infinitesimal generator Λ, ˆ where the adjoint path integral for quantum operations E(t, ˆ † is completely dissipative, i.e. superoperator Λ ˆ † (Ak Al ) − Λ ˆ † (Ak )Al − Ak Λ ˆ † (Al ) ≥ 0. Λ
3
Evolution equations
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 Neumann equation ∂̺t i = − [H, ̺t ] (3) ∂t h ¯ leaves the entropy < S >= −T r(̺t ln̺t ) unchanged. Therefore, to describe general quantum systems, one normally uses [47, 38] a generalization of (3). To describe dissipative quantum systems one usually considers [47] the following equation: ∂̺t i = − [H, ̺t ] + D(̺). ∂t h ¯
4
(4)
3.1 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 d̺t = Λt (̺t ). (5) dt Using the superoperator formalism, this equation can be rewritten in the form d ˆ t |̺t ). |̺t ) = Λ (6) dt The superoperator language allows one to use the analogy with Dirac’s notations. This leads quite simple to the derivation of the appropriate equations. ˆ t is a linear Liouville superoperator on the operator space H. For the Hamiltonian Here Λ (closed) quantum systems (3) this superoperator is defined by the Hamiltonian H: ˆ t = − i (L ˆH − R ˆ H ). Λ (7) h ¯ For equation (4) the Liouville superoperator has the form ˆH − R ˆ H ) + D. ˆ ˆ t = − i (L (8) Λ h ¯ In general, the operator |̺t ) is an unnormalized density matrix operator, i.e. T r̺t = (I|̺t ) 6= 1. Equation (6) has a formal solution ˆ t0 )|̺t ), |̺t ) = E(t, 0
(9)
ˆ t0 ) is a linear quantum operation defined by where E(t, ˆ t0 ) = T exp E(t,
Z
t
t0
ˆτ. dτ Λ
(10)
The symbol T is a Dyson’s time-ordering operator [45]. The quantum operation (10) satisfies the Liouville-von Neumann equation (2). We can define a normalized density matrix operator |ρt ) by ˆ t0 )|̺) E(t, |ρt ) = |̺t )(I|̺t )−1 , or |ρt ) = , ˆ t0 )|̺) (I|E(t, i.e. ρt = ̺t /T r̺t . The evolution equation for the normalized density matrix operator ρt can be written in the form d ˆ t |ρt ) − |ρt )(I|Λ ˆ t|ρt ). |ρt ) = Λ (11) dt In general, this equation is a nonlinear equation [19]. A formal solution of equation (11) is connected with the nonlinear quantum operation (1) by |ρt ) = Nˆ (t, t0 )|ρt0 ). 5
3.2 Quantum Markovian equation Lindblad [39] has shown that there exists a one-to-one correspondence between the completely ˆ t0 ) and superoperator Λ ˆ such that positive norm continuous semigroup of superoperators E(t, ˆ † is completely dissipative. The structural theorem of Lindblad gives the adjoint superoperator Λ ˆ the most general form of the bounded adjoint completely dissipative Liouville superoperator Λ. The Liouville-von Neumann equation (11) for a completely positive evolution is a quantum Markovian master equation (Lindblad equation) [39, 40, 41]: m i 1 X dρt [Vk ρt , Vk† ] + [Vk , ρt Vk† ] . = − [H, ρt ] + dt h ¯ 2¯ h k=1
(12)
This equation in the Liouville space can be written as d ˆ t ), |ρt ) = Λ|ρ dt ˆ is given by where the Liouville superoperator Λ m X ˆV . ˆ † −L ˆV L ˆ † −R ˆ †R ˆV R ˆH − R ˆH ) + 1 ˆ = − i (L 2L Λ k Vk Vk Vk k k h ¯ 2¯ h k=1
(13)
ˆ given by the The basic assumption is that the general form of a bounded superoperator Λ, Lindblad theorem, is also valid for an unbounded superoperator [42, 46]. Another condition imposed on the operators H, Vk , Vk† 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 Vk = ak P + bk Q, were k = 1, 2, and ak , bk are complex numbers, and the Hamiltonian operator H is H=
1 2 mω 2 2 µ P + Q + (P Q + QP ). 2m 2 2
Then with the notation [46]: dqq =
h ¯ X |ak |2 , 2 k=1,2
dpp =
h ¯ X ∗ dpq = − Re ak bk , 2 k=1,2
h ¯ X |bk |2 , 2 k=1,2 X
λ = −Im
k=1,2
a∗k bk
equation (12) in the Liouville space can be written as 1 ˆ + ˆ − mω 2 ˆ + ˆ − d ˆ−L ˆ+ ˆ− ˆ+ L L + L L − (λ − µ)L |ρt ) = P Q |ρt ) + (λ + µ)LQ LP |ρt )+ dt 2m P P 2 Q Q 6
ˆ− ˆ− ˆ− ˆ− ˆ− ˆ− +dppL Q LQ |ρt ) + dqq LP LP |ρt ) − 2dpq LP LQ |ρt ),
(14)
where L± are the multiplication superoperators defined by ˆ − = 1 (L ˆA − R ˆ A ), L A i¯ h
ˆ + = 1 (L ˆA + R ˆ A ). L A 2
The properties of these superoperators are considered in the appendix. Equation (14) is a superoperator form of the well-known phenomenological dissipative model [47, 46].
4
Path integral in the general form
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
dq0 dq0′ E(q, q ′, q0 , q0′ , t, t0 )̺(q0 , q0′ , t0 ).
The function ˆ t0 )|q0 , q ′ 0 ) E(q, q ′, q0 , q0′ , t, t0 ) = (q, q ′|E(t,
(15)
ˆ t0 ) = exp{(t − t0 )Λ}. ˆ E(t,
(16)
ˆ t0 ). Let the Liouville superoperator Λ ˆ t be time is a kernel of the linear quantum operation E(t, ˆ t0 ) is given by independent, i.e. the quantum operation E(t,
ˆ t0 ), t ≥ t0 } be a superoperator semigroup on operator space H Proposition 1. Let {E(t, ˆ 0 , t0 ) = I, ˆ E(t
ˆ t0 ) = E(t, ˆ t1 )E(t ˆ 1 , t0 ), E(t,
ˆ of this semigroup is defined by (16). where t ≥ t1 ≥ t0 such that the infinitesimal generator Λ ˆ t0 ) has the following form: Then the path integral for kernel (15) of the quantum operation E(t, E(q, q
′
, q0 , q0′ , t, t0 )
=
Z
′
′
DqDq DpDp exp
Z
t
t0
ı
dt
h ¯
[qp ˙ − q˙′ p′ ] + ΛS (q, q ′, p, p′ ) .
(17)
This form is the integral over all trajectories in the double phase space with the constraints that q(t0 ) = q0 , q(t) = q, q ′ (t0 ) = q0′ , q ′ (t) = q ′ and the measure Dq =
Y
Dp =
dq(t) ,
t
7
Y t
dp(t) . 2π¯ h
ˆ is connected with the kernel Λ(q, q ′, y, y ′) The symbol ΛS (q, q ′, p, p′ ) of the Liouville superoperator Λ by Z ı ΛS (q, q ′ , p, p′ ) = dydy ′Λ(q, q ′ , y, y ′) · exp− [(q − y)p − (q ′ − y ′)p′ ], h ¯ ˆ y ′) and where Λ(q, q ′, y, y ′) = (q, q ′|Λ|y, 1 Λ(q, q , y, y ) = (2π¯ h)2n ′
′
Z
ı dpdp′ΛS (q, q ′, p, p′ ) · exp [(q − y)p − (q ′ − y ′)p′ ]. h ¯
Proof. Let us derive path integral form (17) for quantum operation (16). 1. Let time interval [t0 , t] has n + 1 equal parts τ=
t − t0 . n+1
Using the superoperator semigroup composition rule ˆ t0 ) = E(t, ˆ tn )E(t ˆ n , tn−1 )...E(t ˆ 1 , t0 ), E(t, where t ≥ tn ≥ tn−1 ≥ ... ≥ t1 ≥ t0 , we obtain the following integral representation: E(q, q
′
, q0 , q0′ , t, t0 )
=
Z
dqn dqn′ ...dq1 dq1′ E(q, q ′, qn , qn′ , t, tn ) ... E(q1 , q1′ , q0 , q0′ , t1 , t0 ).
This representation can be written in the form E(q, q
′
, q0 , q0′ , t, t0 )
=
Z Y n
dqk dqk′
k=1
n+1 Y k=1
′ E(qk , qk′ , qk−1, qk−1 , tk , tk−1).
′ Here qn+1 = q and qn+1 = q′.
2. Let us consider the kernel ′ E(qk , qk′ , qk−1, qk−1 , tk , tk−1 ),
ˆ k , tk−1). If the time interval [tk−1 , tk ] is a small, then in the of the quantum operation E(t coordinate representation we have ̺(qk , qk′ , tk )
=
Z
′ ′ ′ dqk−1dqk−1 E(qk , qk′ , qk−1 , qk−1 , tk , tk−1 )̺(qk−1 , qk−1 , tk−1 ),
8
and ˆ k , tk−1 )|̺t ) = ̺(qk , qk′ , tk ) = (qk , qk′ |̺tk ) = (qk , qk′ |E(t k−1
∞ X
τn ˆ n |̺t ) = (qk , q ′ |̺t ) + (qk , q ′ |Λ|̺ ˆ t )τ + O(τ 2 ) = (qk , qk′ |Λ = k k k−1 k−1 k−1 n=0 n! =
=
̺(qk , qk′ , tk−1 )
Z
+τ
Z
′ ′ ′ dqk−1 dqk−1 Λ(qk , qk′ , qk−1 , qk−1 ) · ̺(qk−1 , qk−1 , tk−1 ) + O(τ 2 ) =
′ ′ ′ ′ dqk−1dqk−1 δ(qk −qk−1 )δ(qk−1 −qk′ )+τ Λ(qk , qk′ , qk−1 , qk−1 )+O(τ 2 ) ̺(qk−1 , qk−1 , tk−1 ).
Therefore, we have ′ ′ ′ E(qk , qk′ , qk−1 , qk−1 , tk , tk−1) = δ(qk −qk−1 )δ(qk−1 −qk′ ) + τ Λ(qk , qk′ , qk−1 , qk−1 ) + O(τ 2).
3. Delta-functions can be written in the form δ(qk −
′ qk−1 )δ(qk−1
−
qk′ )
=
Z
dpk dp′k i ′ exp [(qk − qk−1 )pk − (qk′ − qk−1 )p′k ]. 2n (2π¯ h) h ¯
Using the relations (q, q ′|p, p′ ) =< q|p >< p′ |q ′ >= (p, p′ |q, q ′) =< p|q >< q ′ |p′ >=
i 1 exp (qp − q ′ p′ ), (2π¯ h)n h ¯
1 i exp − (qp − q ′ p′ ), n (2π¯ h) h ¯
we obtain the symbol ΛS (qk , qk′ , pk , p′k ) of the Liouville superoperator by ′ ˆ k−1 , q ′ ) = Λ(qk , qk′ , qk−1 , qk−1 ) = (qk , qk′ |Λ|q k−1
= = =
Z
1 (2π¯ h)2n
Z
ˆ k , p′ )(pk , p′ |qk−1 , q ′ ) = dpk dp′k (qk , qk′ |Λ|p k k k−1
′ dpk dp′k ΛS (qk , qk′ , pk , p′k ) · (qk , qk′ |pk , p′k )(pk , p′k |qk−1 , qk−1 )=
Z
i ′ dpk dp′k ΛS (qk , qk′ , pk , p′k ) · exp [(qk − qk−1 )pk − (qk′ − qk−1 )p′k ]. h ¯
ˆ k , tk−1) is 4. The kernel of the quantum operation E(t ′ E(qk , qk′ , qk−1 , qk−1 , tk , tk−1) =
=
1 (2π¯ h)2n
Z
dpk dp′k 1+τ ΛS (qk , qk′ , pk , p′k )+O(τ 2 ) exp 9
i ′ (qk −qk−1 )pk −(qk′ −qk−1 )p′k . h ¯
ˆ t0 ) is 5. The kernel of the quantum operation E(t, E(q, q ′, q0 , q0′ , t, t0 ) = Z Y n
n+1 Y
i Xh dpk dp′k i n+1 ′ ′ ′ exp (q − q )p − (q − q )p · k k−1 k k k−1 k h)2n h ¯ k=1 k=1 (2π¯
dqk dqk′
k=1
n+1 Y
·
k=1
6. Using lim
n→∞
1 − τ ΛS (qk , qk′ , pk , p′k ) + O(τ 2 ) .
n Y
(1 +
k=1
n Y Λk Λk ) = lim exp( ), n→∞ n n k=1
we obtain E(q, q ′, q0 , q0′ , t, t0 ) = ·exp
n+1 X
τ
k=1
Z Y n
dqk dqk′
k=1
n+1 Y
dpk dp′k · h)2n k=1 (2π¯
′ q ′ − qk−1 qk − qk−1 pk − k p′ k ] + ΛS (qk , qk′ , pk , p′k ) . h ¯ τ τ
i
[
7. Let qk , qk′ , pk , p′k be the values of the functions q(t), q ′ (t), p(t), p′ (t) and tk = t0 + kτ , i.e. qk = q(tk ), qk′ = q ′ (tk ), pk = p(tk ) , p′k = p′ (tk ), where k = 0, 1, 2, ..., n, n + 1. Using qk − qk−1 = q(t ˙ k−1 ), τ →0 τ
lim
lim
n→∞
n+1 X
A(tk )τ =
k=1
Z
t
t0
dtA(t),
ˆ t0 ) in the path integral form (17). we obtain the kernel of the quantum operation E(t,
2
Corollary. If the dissipative quantum evolution is defined by equation (4), then the path integral for the quantum operation kernel has the form E(q, q ′, q0 , q ′ 0 , t, t0 ) =
Z
DqDpDq ′Dp′ exp
ı
h ¯
(A(q, p) − A(p′ , q ′)) + D(q, q ′ , p, p′) .
Here A(q, p) and A(p′ , q ′ ) are action functionals defined by A(q, p) =
Z
t t0
dt qp ˙ − H(q, p) ,
′
′
A(p , q ) = 10
Z
t
t0
dt q˙′ p′ − H(p′ , q ′ ) .
(18)
ˆ The functional D(q, q ′, p, p′ ) is a time integral of the symbol DS of the superoperator 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 closed Hamiltonian system, then D(q, q ′, p, p′ ) = 0 and the path integral for the quantum operation can be separated E(q, q ′, q0 , q ′ 0 , t, t0 ) = U ∗ (q, q0 , t, t0 )U(q ′ , q ′ 0 , t, t0 ), where
ı Dq ′ Dp′ exp− A(p′ , q ′), h ¯ Z ı U ∗ (q, q0 , t, t0 ) = DqDp exp A(p, q). h ¯
U(q
′
, q0′ , t, t0 )
=
Z
The path integral for the dissipative quantum systems and the corresponding quantum operations cannot be separated, i.e. this path integral is defined in the double phase space.
5
Path integral for completely positive quantum operation
Let us consider the Liouville superoperator (13) for the Lindblad equation. ˆ t0 ), t ≥ t0 } be a completely positive semigroup of linear real traceProposition 2. Let {E(t, ˆ of this semigroup are defined preserving superoperators such that the infinitesimal generators Λ by (13). Then the path integral for the kernel of the completely positive quantum operation ˆ t0 ) has the form E(t, ′
′
E(q, q , q0 , q 0 , t, t0 ) =
Z
DqDpDq ′Dp′ F (q, q ′, p, p′) exp
ı A(q, p) − A(p′ , q ′ ) , h ¯
(19)
where A(q, p) and A(p′ , q ′ ) are action functionals (18) and the functional F (q, q ′, p, p′) is defined as 1 F (q, q , p, p ) = exp− 2¯ h ′
′
Z
t
t0
dt
m X
(Vk† Vk )(q, p)+(Vk† Vk )(p′ , q ′ )−2Vk (q, p)Vk† (p′ , q ′ ) . (20)
k=1
11
Proof. The kernel of superoperator (13) is ˆ y ′) = − ı (< y ′|q ′ >< q|H|y > − < q|y >< y ′ |H|q ′ >)+ Λ(q, q ′ , y, y ′) = (q, qk′ |Λ|y, h ¯ +
m m 1X 1 X < q|Vk |y >< y ′|Vk† |q ′ > − (< y ′ |q ′ >< q|Vk† Vk |y > − < q|y >< y ′|Vk† Vk |q ′ >). h ¯ k=1 2¯ h k=1
ˆ can be derived by The symbol ΛS (q, q ′ , p, p′) of the Liouville superoperator Λ ′
′
Λ(q, q , y, y ) =
Z
ı dpdp′ − (< y ′|p′ >< p′ |q ′ >< q|H1 |p >< p|y > − h ¯
− < q|p >< p|y >< y ′|p′ >< p′ |H2 |q ′ >)+
+
m 1X < q|Vk |p >< p|y >< y ′|p′ >< p′ |Vk† |q ′ > , h ¯ k=1
were the operators H1 and H2 are defined by the relations H1 ≡ H −
m ıX V † Vk , 2 k=1 k
H2 ≡ H +
m ıX V † Vk . 2 k=1 k
Then the symbol ΛS (q, q ′, p, p′ ) of the Liouville superoperator (13) can be written in the form ΛS (q, q ′ , p, p′) = − or
m X ı H1 (q, p) − H2 (p′ , q ′ ) + i Vk (q, p)Vk† (p′ , q ′) , h ¯ k=1
ı ΛS (q, q ′ , p, p′) = − [H(q, p) − H(p′ , q ′)]− h ¯ m 1 X − (Vk† Vk )(q, p) + (Vk† Vk )(p′ , q ′ ) − 2Vk (q, p)Vk† (p′ , q ′ ) , 2¯ h k=1
where H(q, p) is a qp-symbol of the Hamilton operator H and H(p, q) is a pq-symbol of the operator H. In the Hamiltonian case (Vk = 0), the symbol is given by ı ΛS (q, q ′ , p, p′ ) = − [H(q, p) − H(p′ , q ′ )]. h ¯ The path integral for a completely positive quantum operation kernel has the form ′
′
E(q, q , q0 , q 0 , t, t0 ) =
Z
ı DqDpDq ′Dp′ F (q, q ′, p, p′ ) exp (A(q, p) − A(p′ , q ′ )). h ¯
Here A(q, p) and A(p′, q ′ ) are action functionals (18), and the functional F (q, q ′, p, p′ ) is defined by (20). 2
12
The functional F (q, q ′, p, p′ ) describes the dissipative part of the evolution and can be called a (double) phase space influence functional. The completely positive quantum operation is described by the functional (20). Corollary. For the phenomenological dissipative model (14) the double phase space path integral has the form (19) with the functional 1 Zt F (q, q , p, p ) = exp 2 dt 2dqp (q − q ′ )(p − p′ )− h ¯ t0 ′
′
−dqq (p − p′ )2 − dpp (q − q ′ )2 + i¯ hλ(pq ′ − qp′ ) + i¯ hµ(q ′p′ − qp) .
(21)
Using the well-known connection between the phase space (Hamiltonian) path integral and the configuration space (Lagrangian) path integral [51, 52], we can derive the following proposition. Proposition 3. If the symbol Λs (q, q ′, p, p′ ) of the Liouville superoperator can be represented in the form ı ΛS (q, q ′, p, p′ ) = − [H(q, p) − H(p′ , q ′ )] + DS (q, q ′ , p, p′), h ¯ where
1 H(q, p) = a−1 (q)pk pl − bk (q)pk + c(q), 2 kl DS (q, q ′, p, p′ ) = −dk (q, q ′ )pk + d′ k (q, q ′ )p′ k + e(q, q ′),
(22)
(23) (24)
then the double phase space path integral (17) can be represented as a double configuration phase space path integral ′
′
E(q, q , q0 , q 0 , t, t0 ) =
Z
ı DqDq ′ F (q, q ′) exp (A(q) − A(q ′ )), h ¯
(25)
where A(q) =
Z
t
t0
dt L(q, q), ˙
1 1 L(q, q) ˙ = akl (q)q˙k q˙l +akl (q)bk (q)q˙l + akl (q)bk (q)bl (q)−c(q). (26) 2 2
This Lagrangian L(q, q) ˙ is related to the Hamiltonian (23) by the usual relations L(q, q) ˙ = q˙k pk − H(q, p), 13
pk =
∂L . ∂ q˙
Proof. Substituting (22) in (17), we obtain the kernel of the corresponding quantum operation. Integrating (17) in p and p′ , we obtain relation (25) with the functional −1 F (q, q ) = exp 2 h ′
Z
t
t0
+d′ k (q, q ′)akl (q ′ )(bl (q ′ ) − where ∆(q, q ′ ) = −
dt dk (q, q ′ )akl (q)(bl (q) −
i dl (q, q ′))+ 2¯ h
i ′ d l (q, q ′)) + e(q, q ′ ) + δ(0)∆(q, q ′ ) , 2¯ h
h ¯2 ln[det(akl (q))] + ln[det(akl (q ′ ))] . 2
2 In equation (25) the functional F (q, q ′) can be considered as the Feynman-Vernon influence functional. It is known that this functional can be derived by eliminating the bath degrees of freedom, for example by taking a partial trace or by integrating them out. The Feynman-Vernon influence functional describes the dissipative dynamics of open systems when we assume the von Hove limit for a system-reservoir coupling. One can describe a quantum system starting from a unitary evolution of some closed system ”system-reservoir” if the quantum system is a part of this closed system. However, situations can arise where it is difficult or impossible to find a closed system comprising the given quantum system [6]-[14]. The Feynman path integral is defined for configuration space. The most general form of quantum mechanical path integral is defined for the phase space. The Feynman path integral 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 the double configuration space. Therefore, this path integral is a special form of the double phase space path integral (17). The Feynman-Vernon path integral is correct only for the Liouville superoperator (22), (23), (24). Note that the symbol Λs (q, q ′ , p, p′) for most of the dissipative and non-Hamiltonian systems (with completely positive quantum operations) cannot be represented in the form (22). Corollary. In the general case, the completely positive quantum operation cannot be represented as the double configuration space path integral (25).
14
For example, the double phase space path integral (21) for the phenomenological dissipative model (14) has the term pp′ . Therefore this model and the Liouville symbol Λs (q, q ′, p, p′ ) for this model cannot be represented in the form (22), (23), (24).
6
On the continuous-variables quantum computation by quantum operations with mixed states
The usual models of a quantum computer deal only with the discrete variables, unitary quantum operations (gates) and pure states. Many quantum variables such as position and 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 with discrete variable computing. Quantum computation using continuous variables is an alternative approach to quantum computations with discrete variables. In the models with unitary quantum operations on pure states it is difficult or impossible to deal formally with measurements, dissipation, decoherence and noise. It turns out that the restriction to pure states and unitary gates is unnecessary [20]. In [20], a model of quantum computations by quantum operations with mixed states was constructed. It is known that the measurement is described by quantum operations. The measurement quantum operations are the special case of quantum operations on mixed states. The von Neumann measurement quantum operation as a nonlinear quantum gate is realized in [20]. The continuous quantum measurement is described by the path integrals [56, 57, 58, 59]. Therefore, the path integral for quantum operations can be useful for continuous-variables quantum operations on mixed states. Quantum computation by quantum operations with mixed states is considered [20] for discrete variables only. Some points of the model of the continuous-variable quantum computations with mixed states are considered in this section. The double phase space path integral can be useful for the continuous-variables quantum gates on mixed states. The main steps of the continuous-variables generalization of quantum computations by quantum operations with mixed states are following. 1. The state |ρ(t)) of the discrete-variable quantum computation with mixed states [20] is a superposition of basis elements |ρ(t)) =
N −1 X µ=0
15
|µ)ρµ (t),
(27)
where ρµ (t) = (µ|ρ(t)) are real numbers (functions). The basis |µ) of the discrete(n) variable Liouville space H is defined [20] by 1 1 |µ) = |µ1...µn ) = √ n |σµ ) = √ n |σµ1 ⊗ ... ⊗ σµn ), 2 2
(28)
where σµ are Pauli matrices, N = 4n , each µi ∈ {0, 1, 2, 3} and N −1 X
′
(µ|µ ) = δµµ′ ,
µ=0
|µ)(µ| = Iˆ
(29)
is the discrete-variable computational basis. The state |ρ(t)) of the continuous-variable quantum computation at any point of time can be considered as a superposition of basis elements Z
|ρ(t)) =
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 the continuous-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′ | = Iˆ
(31)
can be considered as a continuous-variable computational basis. 2. In the discrete-variable computational basis |µ) any linear quantum operation Eˆ acting on n-qubits mixed (or pure) states can be represented as a quantum four-valued logic gate [20]: Eˆ on n-ququats can be given by Eˆ =
N −1 N −1 X X
Eµν |µ)(ν|,
(32)
1 ˆ ν) , T r σ E(σ µ 2n
(33)
µ=0 ν=0
where N = 4n , Eµν = and σµ = σµ1 ⊗ ... ⊗ σµn .
In the continuous-variable computational basis |x, x′ ) any linear quantum operation Eˆ acts on mixed (or pure) states can be represented as a continuous-variable quantum gate: ˆ 2 , t1 ) = E(t
Z
dx dx′ dy dy ′ E(x, x′ , y, y ′, t2 , t1 ) |x, x′ )(y, y ′|, 16
(34)
ˆ 2 , t1 )|y, y ′) is a kernel of the real trace-preserving where E(x, x′ , y, y ′, t2 , t1 ) = (x, x′ |E(t ˆ 2 , t1 ) This quantum operation can be positive (or completely positive) superoperator E(t considered 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 for the quantum operations can be useful for all continuous-variables quantum operations on mixed states. 3. Many quantum variables, such as position and 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 with discrete variable computing. Mixed states subjected to the general quantum operations could increase efficiency. This increase is connected with the increasing number of computational basis elements for operator Hilbert space. A pure state of the quantum systems is an element of functional Hilbert space H. A mixed state of the system is an element |ρ) of the operator Hilbert space H. A mixed state of the system can be considered as an element ρ(x, x′ , t) of the double functional Hilbert space H ⊗ H. The use of continuous-variable quantum computation by quantum operations with mixed states can increase efficiency compared with discrete variable computing.
7
Conclusion
The usual quantum computer model deal only with the discrete variables, unitary quantum operations and pure states. It is known that many of quantum variables, such as position and momentum 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-variable quantum computer. Quantum computation using continuous variables is an alternative approach to 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 quantum operations are the special case of quantum operations on mixed states. The von Neumann measurement quantum operation as a nonlinear quantum gate is realized in Ref. [20]. The con-
17
tinuous quantum measurement is described by the path integrals [56, 57, 58, 59]. Therefore, the path integral for quantum operations can be useful for continuous-variables quantum operations on mixed states. Quantum computation by quantum operations with mixed states is considered [20] only for discrete variables. The model of continuous-variable quantum computations with mixed states will be suggested in the next publication. The double phase space path integral can be useful for continuous-variables quantum gates on mixed states. Let us note the second application of double phase space path integral. The path integral formulation 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-Hamiltonian systems 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 statistical mechanics 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. We denote an element A of H by a ket-vector |A). The inner q product of two elements |A) and |B) † of H is defined as (A|B) = T r(A B). The norm kAk = (A|A) is the Hilbert-Schmidt norm of operator A. A new Hilbert space H with the inner product is called Liouville space attached to 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 leads us to consider rigged Hilbert space [48, 49, 19, 38] (Gelfand triplet) B ⊂ H = H∗ ⊂ B∗ and associated operator space. The rigged operator Hilbert space can be considered as the usual rigged Hilbert space for the operator kernels.
18
Let the set {|x >} satisfy the following conditions: ′
Z
′
< x|x >= δ(x − x ) ,
dx|x >< x| = I.
Then |x, x′ ) = ||x >< x′ |) satisfies (x, x′ |y, y ′) = δ(x − x′ )δ(y − y ′),
Z
dx
Z
ˆ dx′ |x, x′ )(x, x′ | = 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 (HilbertSchmidt) 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. ˆ ˆ ˆ on H, which is defined by Λ|A) = |Λ(A)), we have For an arbitrary superoperator Λ ˆ (x, x′ |Λ|A) =
Z
dy
Z
ˆ y ′)(y, y ′|A) = dy ′(x, x′ |Λ|y,
Z
dy
Z
dy ′Λ(x, x′ , y, y ′)A(y, y ′),
ˆ y ′) is a kernel of the superoperator Λ. ˆ where Λ(x, x′ , y, y ′) = (x, x′ |Λ|y, Let A be a linear operator in the Hilbert space H. We can define the multiplication superˆ A and R ˆ A by the following equations: operators L ˆ A |B) = |AB) , L 19
ˆ A |B) = |BA). R
The superoperator kernels can be easy derived. For example, in the basis |x, x′ ) we have ˆ A |B) = (x, x′ |L
Z
dy
Z
ˆ A |y, y ′)(y, y ′|B) = dy ′(x, x′ |L
Z
dy
Z
dy ′LA (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 LA (x, x′ , y, y ′) =< x|A|y >< x′ |y ′ >= A(x, y)δ(x′ − y ′ ). ˆ A superoperator Eˆ† is called the adjoint superoperator for Eˆ if (Eˆ† (A)|B) = (A|E(B)) for † ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ all |A) and |B) from H. For example, if E = LA RB , then E = LA† RB† . If E = LA , then ˆ A† . Eˆ† = L ˆ± Left superoperators L A are defined as Lie and Jordan multiplication by the relations ˆ + B = 1 (AB + BA). ˆ − B = 1 (AB − BA), L L A A i¯ h 2 ˆ± ˆ± ˆ− ˆ− ˆ+ The left superoperator L A and the right superoperator RA are connected by LA = −RA , LA = ˆ + . An algebra of the superoperators L ˆ ± is defined [50] by R A A (1) the Lie relations ˆ− ˆ− ˆ− ˆ− ˆ− L A·B = LA LB − LB LA , (2) the Jordan relations ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ L (A◦B)◦C + LB LC LA + LA LC LB = LA◦B LC + LB◦C LA + LA◦C LB , ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ L (A◦B)◦C + LB LC LA + LA LC LB = LC LA◦B + LB LA◦C + LA LB◦C , ˆ +L ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ ˆ+ L C A◦B + LB LA◦C + LA LB◦C = LA◦B LC + LB◦C LA + LA◦C LB , (3) the mixed relations: ˆ+ = L ˆ−L ˆ+ ˆ+ ˆ− L A·B A B − LB LA , ¯2 ˆ− ˆ− ˆ+ = L ˆ+L ˆ+ − h L L L , A◦B A B 4 B A Here we use the notations A·B =
ˆ− = L ˆ +L ˆ− ˆ+ ˆ− L A◦B A B + LB LA , 2
h ¯ ˆ− ˆ+L ˆ+ ˆ+ ˆ+ L L . B A − LA LB = − 4 A·B
1 1 (AB − BA), A ◦ B = (AB + BA). i¯ h 2 20
References [1] Hellwing K E and K. Kraus K 1969 ”Pure operations and measurements” Commun. Math. Phys. 11 214 [2] Hellwing K E and K. Kraus K 1970 ”Operations and measurements. II.” Commun. Math. Phys. 16 142 [3] Kraus K 1971 ”General state changes in quantum theory” Ann. Phys. 64 311 [4] Kraus K 1983 States, Effects and Operations. Fundamental Notions of Quantum Theory (Berlin: Springer) [5] Schumacher B 1996 ”Sending entanglement through noisy quantum channels” Phys. Rev. A 54 2614 (quant-ph/9604023) [6] Feynman R P and Vernon F L 1963 ”The theory of a general quantum system interacting with linear dissipative system” Ann. Phys. 24 118 [7] Caldeira A O and Leggett A J 1983 ”Path integral approach to quantum Brownian motion” Physica A 121 587 [8] Grabert H, Schramm P and Ingold G L 1988 ”Quantum Brownian motion: the functional integral approach” Phys. Rep. 168 115-207 and references therein. [9] Brink D M, Neto J and Weidenmuller H A 1979 ”Transport coefficients for deeply inelastic scattering from the Feynman path integral method” Phys. Lett. B 80 170 and references therein. [10] Pechukas P 1969 ”Time-dependent semiclassical scattering theory” Phys. Rev. 181 174 [11] Ankerhold J, Graber H and Ingold G L 1995 ”Quantum dissipative systems with a potential barrier” Phys. Rev. E 51 4267 [12] Exner P 1985 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 83 203 [14] Strunz W T 1996 ”Stochastic path integral and open quantum systems” Phys. Rev. A 54 2664 [15] Tarasov V E 1994 ”Bosonic string in affine-metric curved space” Phys. Lett. B 323 296 (hep-th/0401223) [16] Tarasov V E 1994 ”Quantum dissipative systems. I. Canonical quantization and Liouville equation” Theor. Math. Phys. 100 1100 [17] Tarasov V E 1994 in ”Phase space path integral for non-Hamiltonian systems” Proceedings of the VIII International Workshop on High Energy Physics and Quantum Field Theory (Moscow: MSU) pp.205-209. 21
[18] Tarasov V E 2000 in ”Quantum theory and non-Hamiltonian systems. Path-integral approach” Proc. XIV International Workshop on High Energy Physics and Quantum Field Theory (Moscow: MSU) pp.637-640. [19] Tarasov V E 2000 Quantum Mechanics. Lectures on Theory Foundation (Moscow: Vuzovskaya kniga) [20] Tarasov V E 2002 ”Quantum computer with mixed states and four-valued logic” J. Phys. A 35 5207 (quant-ph/0312131) [21] Lloyd S and Braunstein S L 1999 ”Quantum computation over contunuous variable” Phys. Rev. Lett. 82 1784 (quant-ph/9810082) [22] Braunstein S L and Kimble H J 1998 ”Teleportation of continuous quantum variables” Phys. Rev. Lett. 80 869 [23] Reid M D 1989 ”Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification” Phys Rev A 40 913 [24] Crawford J A 1958 ”An alternative method of quantization” Nuovo Cim. 10 698 [25] Schwinger J 1960 ”The geometry of quantum states” Proc. Nat. Acad. Sci. 46 257, 570 [26] Emch G 1964 ”Coarse-graining in Liouville space and master equation” Helv. Phys. Acta 37 532 [27] Sauermann G 1966 ”On the theory of spin-spin relaxation” Phisica 32 2017 [28] Gabriel H 1969 ”Theory of the influence of environment on the angular distribution of nuclear radiation” Phys. Rev. 181 506 [29] Schwinger J 1970 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 30 97 [31] Blum K 1981 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 Liouville space” Lett. Nuovo Cim. 37 161 [34] Fick E and Sauermann G 1990 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 in non-equilibrium dissipative systems” Int. J. Mod. Phys. B 5 1821 22
[36] Tarasov V E 1997 ”Quantum non-Hamiltonian systems: Definition and algebraic structures” in 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. Superconductivity 12 707 (quant-ph/9811082) [38] Tarasov V E 2000, Mathematical Introduction to Quantum Mechanics (Moscow: MAI) ch 2,3,5 [39] Lindblad G 1976 ”On the generators of quantum dynamical semigroups” Commum. Math. Phys. 48 119 [40] Gorini V, Kossakowski A and Sudarshan E C G 1976 ”Completely positive dynamical semigroups of N-level systems” J. Math. Phys. 17 821 [41] Gorini V, Frigerio A, Verri M, Kossakowski A, Sudarshan E C G 1978 ”Properties of quantum markovian master equations” Rep. Math. Phys. 13 149 [42] Alicki R and Lendi K 1987 Quantum Dynamical Semigroups and Applications (Berlin: Springer-Verlag) [43] Arveson W 2002 ”The domain algebra of a CP-semigroup” math.OA/0005251 ch 3, and references therein [44] Tarasov V E 1997 ”Quantum dissipative systems. III. Definition and algebraic structures” Theor. Math. Phys. 110 57 [45] Dyson F J 1949 ”The radiation theories of Tomonaga, Schwinger, and Feynman” Phys. Rev. 75 486; ”The S matrix in quantum electrodynamics” Phys. Rev. 75 1736 [46] Sandulescu A and Scutaru H 1987 ”Open quantum systems and the damping of collective models in deep inelastic collisions” Ann. Phys. 173 277 [47] Dekker H 1981 ”A review of the quantization of damped oscillator” Phys. Rep. 80 1 [48] Gel’fand I M and Vilenkin N Ya 1964 Generalized Functions, Vol.4 - Applications of Harmonic Analysis (New York: Academic Press) [49] Roberts J E 1966 ”Rigged Hilbert spaces in quantum mechanics” Commun. Math. Phys. 3 98 [50] Tarasov V E 2001 ”Weyl quantization of dynamical systems with flat phase space” Moscow Univ. Phys. Bull. 56(5) 5 [51] Slavnov A and Fadeev L 1988 Introduction to Quantum Theory of Gauge Fields 2nd ed. (Moscow: Nauka) ch II.1. [52] Huang K 1992 Quarks, Leptons and Gauge Fields (New York: World Scientific) ch 7.1
23
[53] Lee T D and Yang C.N. 1962 ”Theory of charged vector mesons interacting with the electromagnetic field” Phys. Rev. 128 885 [54] Inomata A and Singh V A 1980 ”Path integrals and constraints: particle in a box” Phys. Lett A 80 105 [55] Ryder L H 1985 Quantum Field Theory (Cambridge: Cambridge University Press) ch 5.1 [56] Caves C M 1986 ”Quantum-mechanics of measurements distributed in time. A pathintegral approach” Phys. Rev. D 33 1643 [57] Mensky M B 2000 Quantum Measurements and Decoherence: Models and Phenomenology (Dordrecht, Boston, London: Kluwer Acad. Publ.) ch 5.2, 5.3 [58] Mensky M B 1994 ”Continuous quantum measurements:Restricted path integrals and master equations” Phys. Lett. A 196 159 [59] Mensky M B 1997 ”Finite resolution of time in continuous measurements: Phenomenology and the model” Phys. Lett. A 231 1 [60] Marx M and Parrinello M 1996 ”Ab initio path integral molecular dynamics: Basic ideas” J. Chem. Phys. 104 4077 [61] Tuckerman M E, Mundy C J and Martyna G J 1999 ”On the classical statistical mechanics of non-Hamiltonian systems” Europhys. Lett. 45 149 [62] Tuckerman M E, Liu Y, Ciccotti G and Martyna G J 2001 ”Non-Hamiltonian molecular dynamics: generalizing Hamiltonian phase space principles to non-Hamiltonian systems” J. Chem. Phys. 115 1678 [63] Ramshaw J D 2002 ”Remarks on non-Hamiltonian statistical mechanics” Europhys. Lett. 59 319 [64] Tarasov V E 2003 ”Classical canonical distribution for dissipative systems” Mod. Phys. Lett. B 17 1219 (cond-mat/0311536) [65] Sergi A 2003 ”Non-Hamiltonian equilibrium statistical mechanics” Phys. Rev. E 67 021101 [66] Tarasov V E 2002 ”Stationary states of dissipative quantum systems” Phys. Lett. A. 299 173 [67] Tarasov V E 2002 ”Pure stationary states of open quantum systems” Phys. Rev. E. 66 056116 (quant-ph/0311177) [68] Kapral R and Ciccotti G 1999 ”Mixed quantum-classical dynamics” J. Chem. Phys. 110 8919 [69] Nielsen S, Kapral R and Ciccotti G 2001 ”Statistical mechanics of quantum-classical systems” J. Chem. Phys. 115 5805 24
