Error Bounds on Finite-Dimensional Approximations of Input-Output ...
Error Bounds on Finite-Dimensional Approximations of
arXiv:1509.02629v1 [quant-ph] 9 Sep 2015
Input-Output Open Quantum Systems∗ Onvaree Techakesari and Hendra I. Nurdin
†
September 10, 2015
Abstract Many physical systems of interest that are encountered in practice are input-output open quantum systems described by quantum stochastic differential equations and defined on an infinite-dimensional underlying Hilbert space. Most commonly, these systems involve coupling to a quantum harmonic oscillator as a system component. This paper is concerned with the error in the finite-dimensional approximation of input-output open quantum systems defined on an infinite-dimensional underlying Hilbert space. We present explicit error bounds between the time evolution of the state of a class of infinite-dimensional quantum systems and its approximation on a finite-dimensional subspace of the original, when both are initialized in the latter subspace. Applications to some physical examples drawn from the literature are provided to illustrate our results.
1
Introduction
Quantum stochastic differential equations (QSDEs) developed independently by Hudson and Parthasarathy [1] and Gardiner and Collett [2] (the latter in a more restricted form than the former) have been widely used to describe the input-output models of physical Markovian open quantum systems [3–5]. Such models describe the evolution of Markovian quantum systems interacting with a propagating quantum field, such as a quantum optical field, and are frequently encountered in quantum optics, optomechanics, and related fields. An example in quantum optics would be a cavity QED (quantum electrodynamics) system where a single atom is trapped inside an optical cavity that interacts with an external coherent laser beam impinging on the optical cavity. These ∗ †
This research was supported by the Australian Research Council O. Techakesari and H. I. Nurdin are with the School of Electrical Engineering and Telecommunications, UNSW
Australia, Sydney NSW 2052, Australia. Email: [email protected] and [email protected]
1
input-output models have subsequently played an important role in the modern development of quantum filtering and quantum feedback control theory [6, 7]. Many types of quantum feedback controllers have been proposed in the literature on the basis of QSDEs, using both measurementbased quantum feedback control, e.g., [3,4,6,7], and coherent feedback control, e.g., [8–10]. Besides, the QSDEs have also been applied in various developments in quantum information processing, such as in quantum computation technology; e.g., see [11]. In various physical systems of interest, one often deals with input-output systems that include coupling to a quantum harmonic oscillator. For instance, typical superconducting circuits that are of interest for quantum information processing consist of artificial two-level atoms coupled to a transmission line resonator. The former is typically described using a finite-dimensional Hilbert space and the latter is a quantum harmonic oscillator with an infinite-dimensional underlying Hilbert space (i.e., L2 (R), the space of square-integrable complex-valued functions on the real line). Another example is a proposed photonic realization of classical logic based on Kerr nonlinear optical cavities in [12], which is built around a quantum harmonic oscillator with a Kerr nonlinear medium inside it. If a mathematical model for such quantum devices is sufficiently simple, it is often possible to simulate the dynamics of the system on a digital computer to assess the predicted performance of the actual device, as carried out in [12]. The simulation carried out is typically that of a stochastic master equation that simulates the stochastic dynamics of a quantum system when one of its output is observed via laboratory procedures such as homodyne detection or photon counting, see [3, 6, 13]. However, since it is not possible to faithfully simulate a quantum system with an infinite-dimensional Hilbert space, often in simulations this space is truncated to some finite-dimensional subspace and an operator X on the infinite-dimensional space is approximated by a truncated operator of the form P XP , where P denotes an orthogonal projection projector onto the approximate finite-dimensional subspace. For instance, with quantum harmonic oscillators, a commonly used finite-dimensional space is the span of a finite number of Fock states |0i, |1i, . . . , |ni.
Despite the ubiquity of approximating infinite-dimensional Hilbert spaces of quantum systems
by finite-dimensional subspaces for simulations of input-output quantum systems, to the best of the authors’ knowledge, there does not appear to be any work that has obtained some explicit bounds on the approximation error of the joint state of the system and the quantum field it is coupled to. In this paper, we present bounds on the error between the quantum state of a quantum system described by the QSDE and the quantum state of a finite-dimensional approximation described by another QSDE, when both systems are initialized in a state in the finite-dimensional subspace. The bounds are established using the contractive property of open quantum systems, semigroup theory, and some constructions employed by Bouten, van Handel and Silberfarb in [14]. Some examples are also presented to illustrate the application of our results. 2
The rest of this paper is structured as follows. In Section 2, we present the class of open quantum systems and the associated QSDEs describing Markovian open quantum systems. Explicit error bounds for a finite-dimensional approximation of a Markovian open quantum system are established in Section 3. Some examples are provided in Section 4. Concluding remarks are then presented in Section 5.
2
Preliminaries
2.1
Notation
We use ı =
√
−1 and let (·)∗ denote the adjoint of a linear operator as well as the conjugate of
a complex number, and (·)⊤ denote matrix transposition. We denote by δij the Kronecker delta function. We define ℜ{A} = 12 (A + A∗ ) and ℑ{A} =
1 ∗ 2ı (A − A ).
For a linear operator A, we write
ker(A) to denote the kernel of A. We often write |·i to denote an element of a Hilbert space and denote by H⊗F the algebraic tensor product of Hilbert spaces H and F. For a subspace H0 of a
Hilbert space H, we write PH0 to denote an orthogonal projection operator onto H0 . For a Hilbert
space H = H0 ⊕H1 , we will write H⊖H0 to denote H1 . For a linear operator X on H, X|H0 denotes
the restriction of X to H0 . We use B(H) to denote the algebra of all bounded linear operators on
H. We write [A, B] = AB − BA. The notation || · || will be used to denote Hilbert space norms
and operator norms, h·, ·i denotes an inner product on a Hilbert space, linear in the right slot and
antilinear in the left, and |·i h·| denotes an outer product. Here, 1[0,t] (·) : [0, t] → {0, 1} denotes the
indicator function. Finally, Z+ denotes the set of all positive integers.
2.2
Open quantum systems
Consider a separable Hilbert space H and the symmetric boson Fock space (of multiplicity m) F
defined over the space L2 ([0, T ]; Cm ) = Cm ⊗ L2 ([0, T ]) with 0 < T < ∞; see [15, Ch. 4-5] for
more details. We will use e(f ) ∈ F, with f ∈ L2 ([0, T ]; Cm ), to denote exponential vectors in
m F. As in [14], let S ⊂ L2 ([0, T ]; Cm ) ∩ L∞ loc ([0, T ]; C ) be an admissible subspace in the sense of
m Hudson-Parthasarathy [1] which contains at least all simple functions, where L∞ loc ([0, T ]; C ) is the
space of locally bounded vector-valued functions. Here, we will consider a dense domain D ⊂ H
and a dense domain of exponential vectors E = span{e(f ) | f ∈ S} ⊂ F.
Consider an open Markov quantum system which can be described by a set of linear operators
defined on the Hilbert space H: (i) a self-adjoint Hamiltonian operator H, (ii) a vector of coupling operators L with the j-th element, Lj : H → H for all j = 1, 2, . . . , m, and (iii) a unitary scattering
matrix S with the ij-th element, Sij : H → H for all i, j = 1, 2, . . . , m. Moreover, the operators 3
Sij , Lj , H and their adjoints are assumed to have D as a common invariant dense domain. Under this description, we note that m corresponds to the number of external bosonic input fields driving the system. Each bosonic input field can be described by annihilation and creation field operators, ∗
∗
bit and bit , respectively, which satisfy the commutation relations [bit , bjs ] = δij δ(t − s) for all i, j =
1, 2, . . . , m and all t, s ≥ 0. We can then define the annihilation process Ait , the creation process ∗
Ait , and the gauge process Λij t as Ait
=
Z
0
t
bis ds,
∗ Ait
=
Z
t 0
∗ bis ds,
Λij t
=
Z
t 0
∗
bis bjs ds.
Note that these processes are adapted quantum stochastic processes. In the vacuum representation, ∗
∗
∗
the products of their forward differentials dAit = Ait+dt − Ait , dAit = Ait+dt − Ait , and dΛij t =
ij Λij o table t+dt − Λt satisfy the quantum Itˆ
×
dAit
∗ dAjt dΛij t
dt Here, bit =
dAit dt
dAkt
dAkt
0
∗
dΛkℓ t
dt
δik dt
δik dAℓt
0
0
0
0
0
∗ δjk dAit
δjk dΛiℓ t
0
0
0
0
0
0 .
can be interpreted as a vacuum quantum white noise, while Λii t can be interpreted
as the quantum realization of a Poisson process with zero intensity [1]. Following [14], the time evolution of a Markovian open quantum system is given by a unitary adapted process Ut satisfying the left Hudson-Parthasarathy QSDE [1]: " # m m m m X X X X 1 ∗ i∗ ∗ i ∗ ∗ + ıH − S L dA L dA − + Sji − δij )dΛij dUt = Ut (L L ) dt j i ji t i t i t 2 i,j=1
i,j=1
i=1
i=1
(1)
with U0 = I. The quantum stochastic integrals are defined relative to the domain D⊗E. With the
left QSDE, the evolution of a state vector ψ ∈ H ⊗ F is given by Ut∗ ψ.
In this paper, we are interested in the problem of approximating the system with operator
parameters (S, L, H) by a finite-dimensional open quantum system with linear operator parameters (S (k) , L(k) , H (k) ) defined on a finite-dimensional subspace H(k) ⊂ H, where the dimension of H(k)
increases with k ∈ Z+ , S (k) is unitary, and H (k) is self-adjoint. Similar to (1), the time evolution (k)
of the approximating system is given by a unitary adapted process Ut
4
satisfying the left Hudson-
Parthasarathy QSDE [1]: m m m X X X (k)∗ (k)∗ (k)∗ (k) (k) i∗ i S L dA − L dA + Sji − δij )dΛij dUt = Ut j t t t ji i i,j=1 i=1 i,j=1 " # ) m X ∗ 1 (k) (k) + ıH (k) − (Li Li ) dt , 2
(2)
i=1
(k)
with U0
= I. Here, the quantum stochastic integrals in the above equation are defined relative to (k) ∗
the domain H(k) ⊗E. Similarly, the evolution of a state vector ψ ∈ H ⊗ F is given by Ut
2.3
ψ.
Associated semigroups
Let θt : L2 ([t, T ]; Cm ) → L2 ([0, T ]; Cm ) be the canonical shift θt f (s) = f (t + s). We also let Θt :
F[t → F denote the second quantization of θt , where F[t denotes the Fock space over L2 ([t, ∞); Cm ).
Note that an adapted process Ut on H ⊗ F is called a contraction cocycle if Ut is a contraction for all t ≥ 0, t 7→ Ut is strongly continuous, and Us+t = Us (I ⊗ Θ∗s Ut Θs ).
Let us now impose an important condition on the open quantum systems, adopted from [14].
Condition 1 (Contraction cocycle solutions). For all t ≥ 0 and all k ∈ Z+ , i. the QSDE (1) possesses a unique solution Ut which extends to a unitary cocycle on H ⊗ F, (k)
ii. the QSDE (2) possesses a unique solution Ut H(k)
which extends to a contraction cocycle on
⊗ F. (k)
We note that in this paper, Condition 1(ii) on the solution of Ut
is satisfied because H(k) is (k)
finite-dimensional, S (k) is unitary, and H (k) is self-adjoint [1]. In fact, in this case Ut
is unitary.
We will assume that the Condition 1 holds throughout this paper. (αβ)
Let us define an operator Tt (αβ)
hu, Tt
1
: H → H via the identity
vi = e− 2 (||α||
2 +||β||2 )t
hu ⊗ e(α1[0,t] (·)), Ut v ⊗ e(β1[0,t] (·))i
for all u, v ∈ H and all α, β ∈ Cm . From [14, Lemma 1], under Condition 1(i), the operator (αβ)
Tt
∈ B(H) is a strongly continuous contraction semigroup on H and its generator L(αβ) satisfies
Dom(L(αβ) ) ⊃ D such that m m m X X X ∗ ∗ L∗j βj + α∗i Sji Lj + α∗i Sji βj − L(αβ) u = i,j=1
i,j=1
j=1
ıH −
m 1X
2
i=1
L∗i Li
!
−
||α||2
+ 2
||β||2
u
for all u ∈ D. We assume that D is a core for L(αβ) for all α, β ∈ Cm so that Lαβ so that the above definition completely determines L(αβ) . Here, we note that Dom(Lαβ ) is dense in H. We likewise 5
(k;αβ)
define an operator Tt
(k)
: H(k) → H(k) by replacing Ut with Ut . Since H(k) is finite-dimensional,
we note that Dom(L(k;αβ) ) = H(k) .
In the sequel, we will make use of the above semigroups associated with open quantum systems
in establishing our model approximation error bound.
3
Error bounds for finite-dimensional approximations
3.1
Assumptions and preliminary results
Assumption 1. For any k ∈ Z+ and any α, β ∈ Cm , H(k) ⊂ Dom(L(αβ) ). Let M(k) = Range L(α,β) − L(k;α,β) H(k) . Supposing that Assumption 1 holds, we also
assume the following.
(αβ)
Assumption 2. For each k ∈ Z+ and each α, β ∈ Cm , there exists γk
and a non-trivial subspace {0} ⊂ K(k) ⊆ H(k) such that (k;αβ) . i. L(αβ) − L(k;αβ) K(k) ≤ qL T ii. L(αβ) − L(k;αβ) H(k) ⊖K(k) = 0, i.e., H(k) ⊖ K(k) ⊆ α,β∈Cm ker iii. For any u ∈ H(k) ,
(k;αβ)
, qL
(k;αβ)
, qa
(k;αβ)
, qe
> 0,
L(αβ) − L(k;αβ) H(k)
ℜ{hPK(k) L(k;αβ) u, PK(k) ui} = −g(k, α, β) ||PK(k) u||2 + h(k, α, β, u) (αβ)
for some g(k, α, β) ≥ γk
(k;αβ)
and some |h(k, α, β, u)| ≤ qa
||PK(k) u|| ||u||.
iv. For any u ∈ H(k) and any t ≥ 0, (αβ)
ℜ{hTt
2 (αβ) ˆ k, α, β, u) g (k, α, β) Tt PM(k) u + h(t, PM(k) ui} = −ˆ ˆ (k;αβ) (αβ) and some h(t, k, α, β, u) ≤ qe u P Tt M(k) ||u||. (α,β)
L(αβ) PM(k) u, Tt (αβ)
for some gˆ(k, α, β) ≥ γk
Assumption 3. There exists r, s ∈ Z+ such that, for all α, β ∈ Cm , we have that −r −s ! ! (k;αβ) (1−2 ) (k;αβ) (1−2 ) q q e a (k;αβ) lim q =0 (αβ) (αβ) k→∞ L γk γk
Moreover, for any i = 0, 1, 2, . . . , r, j = 0, 1, 2, . . . , s, and ℓ = 0, 1, 2, . . . , min{r, s}, we have that −i −j ! ! (k;αβ) (k;αβ) (1−2 ) (k;αβ) (1−2 ) qL qe qa =0 lim (αβ) (αβ) k→∞ γ (αβ) γ γ k k k −ℓ ! (k;αβ) (k;αβ) (1−2 ) (αβ) q q −ℓ e a (k;αβ) lim e−2 γk t qL =0 (αβ) k→∞ (γk )2 for all α, β ∈ Cm and all t ∈ [0, T ] with 0 ≤ T < ∞. 6
Let us present some useful lemmas. Lemma 1. Suppose that Assumption 2(iii) holds. Then for any k, r ∈ Z+ , any α, β ∈ Cm , any
u ∈ H(k) , and any t ≥ 0, it holds that −r ! r−1 (k;αβ) (1−2 ) X (αβ) q a (k;αβ) t −(2−j )γk c e + u ≤ PK(k) Tt j (αβ) γk j=0 where c0 = 1 and cj =
q
(k;αβ) qa (αβ) γk
!(1−2−j )
||u||
cj−1 2j (2j − 1)−1 for j ≥ 1.
(αβ)
Proof. First note, from the definition of a strongly continuous semigroup, that T0 d (αβ) u dt Tt
(3)
= I and
= L(αβ) u [16]. From Assumption 2(iii), we have that E d d D (k;αβ) (k;αβ) (k;αβ) 2 u u, PK(k) Tt PK(k) Tt u = PK(k) Tt dt dt nD Eo (k;αβ) (k;αβ) u u, PK(k) Tt = 2ℜ PK(k) L(k;αβ) Tt (k;αβ) (k;αβ) 2 u). u + 2h(k, α, β, Tt =−2g(k, α, β) PK(k) Tt
Solving the above ODE gives us that
Z t (k;αβ) 2 2 −2g(k,α,β)t h(k, α, β, Tτ(k;α,β) u)e−2g(k,α,β)(t−τ ) dτ u|| + 2 = e ||P u T P K(k) t K(k) 0 Z t (αβ) (αβ) 2 (k;αβ) −2γk t ≤e ||PK(k) u|| + 2qa e−2γk (t−τ ) PK(k) Tτ(k;αβ) u ||u||dτ 0 " # (k;αβ) (αβ) q a ≤ e−2γk t + (αβ) ||u||2 . γk
(4)
Here, the second step follows from Assumption 2(iii). The last step follows because ||PK(k) u|| ≤ ||u|| p (k;αβ) is a contraction semigroup. Noticing that |a|2 + |b|2 ≤ |a| + |b| for any a, b ∈ R, we and Tt have that
(αβ) (k;αβ) u ≤ e−γk t + PK(k) Tt
7
(k;αβ) qa (αβ) γk
!1 2 ||u||.
(5)
Now, substituting (5) into the right-handed side of (4), we have that (k;αβ) 2 T u P K(k) t (αβ) −2γk t
≤e
||u||2 + 2qa(k;αβ)
(αβ) −2γk t
= e
(αβ) −2γk t
≤ e
(αβ) −γk t
+ 2e
(αβ) −γk t
+2e
Z
t
(αβ) −γk (2t−τ )
e
0
(αβ) 1 − e−γk t (k;αβ)
qa
(αβ) γk
!
||u||2 dτ + 2qa(k;αβ) (k;αβ)
qa
(αβ)
γk
(k;αβ)
+
qa
(αβ) γk
!
Z
t
(k;αβ)
qa
(αβ) −2γk (t−τ )
e
(αβ) γk
0
(αβ) −2γk t
+ 1−e
(k;αβ)
qa
(αβ)
γk
! 3 2
!1
2
||u||2 dτ
!3 2 ||u||2
||u||2 .
Taking the square root on both sides of the equation, we get !1 √ − 1 γ (αβ) t qa(k;αβ) 2 (αβ) (k;αβ) −γ t + + 2e 2 k u ≤ e k PK(k) Tt (αβ) γk
(k;αβ) qa (αβ) γk
!3
From repeat application of the above steps, we establish the lemma statement.
4
||u||.
Lemma 2. Suppose that Assumption 2(iv) holds. Then for any k, r ∈ Z+ , any α, β ∈ Cm , any u ∈ H(k) , and any t ≥ 0, it holds that −r ! r−1 (k;αβ) (1−2 ) X (αβ) −j q e (αβ) cj e−(2 )γk t + Tt PM(k) u ≤ (αβ) γk j=0
where c0 = 1 and cj =
q
(k;αβ)
qe
(αβ)
γk
!(1−2−j ) ||u||
cj−1 2j (2j − 1)−1 for j ≥ 1.
Proof. Similar to Lemma 1, using Assumption 2(iv), we have that 2 d (αβ) (αβ) (αβ) Tt PM(k) u = 2ℜ{hTt L(αβ) PM(k) u, Tt PM(k) ui} dt 2 (αβ) ˆ k, α, β, u). = −2ˆ g (k, α, β) Tt PM(k) u + 2h(t,
Solving the ODE, we have that
Z t 2 (αβ) (αβ) (αβ) (αβ) 2 e−2γk (t−τ ) Tt PM(k) u ||u||dτ Tt PM(k) u ≤ e−2γk t ||PM(k) u|| + 2qe(k;αβ) 0 " # (k;αβ) (αβ) qe ≤ e−2γk t + (αβ) ||u||2 . γk The lemma statement is then established by following similar arguments to Lemma 1.
8
(6)
3.2
Error bounds for finite-dimensional approximations
We begin by defining k zr,s (t, α, β)
=
(k;αβ)
qe
(k;αβ) qL t
(αβ)
γk
!(1−2−r )
(k;αβ)
qa
(αβ)
γk
r−1 q (k;αβ) X (αβ) 2i ci e −2−i γk t 1 − e + (αβ) (αβ) γk i=0 γk
s−1 q (k;αβ) X (αβ) 2i ci a −2−i γk t + 1−e (αβ) (αβ) γk i=1 γk
+
s−1 r−1 X X i=0
ci cj
j=0
j6=i
2(i+j) (αβ)
(2i − 2j ) γk
min{r,s}−1
X
+t
(αβ) −2−i γk t
c2i e p
!(1−2−i )
qa
!(1−2−i )
qe
(k;αβ) (αβ)
γk
(k;αβ) (αβ)
γk
!(1−2−s ) !(1−2−r )
q (k;αβ) −i (αβ) −j (αβ) e e−2 γk t − e−2 γk t (αβ) γk (k;αβ) (k;αβ) qa (αβ) (γk )2
qe
i=0
where c0 = 1, and cj =
!(1−2−s )
! (1−2−i )
! −j (k;αβ) (1−2 )
qa
(αβ)
γk
!(1−2−i ) ,
(7)
cj−1 2j (2j − 1)−1 for j ≥ 1. We now establish an error bound between
the two semigroups associated with the open quantum systems.
Lemma 3. Suppose Assumptions 1 and 2 hold. Then for any k, r, s ∈ Z+ , any α, β ∈ Cm , any
u ∈ H(k) , and any t ≥ 0, it holds that (αβ) (k;αβ) k (t, α, β)||u||. u ≤ zr,s − Tt Tt
(8)
Proof. First note, from the definition of a strongly continuous semigroup, that [16] (αβ)
i. T0 ii. iii.
(k;αβ)
= T0
d (αβ) u dt Tt
= I for all k ∈ Z+ ,
= L(αβ) u for all u ∈ Dom(L(αβ) ),
d (k;αβ) u dt Tt
= L(k;αβ) u for all u ∈ H(k) (since H(k) is finite-dimensional).
From the above properties and Assumption 1, we can write for all u ∈ H(k) and all t ≥ 0 that d (αβ) (k;αβ) (αβ) (k;αβ) Tt − Tt u = L(αβ) Tt − L(k;αβ) Tt u dt (αβ)
= L(αβ) Tt
with
(k;αβ)
− Tt
u + L(αβ) − L(k;αβ)
(αβ) (k;αβ) T0 − T0 u = 0. Note that by Assumption 2,
operator. Since L(αβ)
H(k)
(k;αβ)
Tt
u
(9)
L(αβ) − L(k;αβ) H(k) is a bounded (αβ) (k;αβ) is a generator of a semigroup on a Hilbert space, T0 − T0 u ∈ 9
(k;αβ) u ∈ C 1 ([0, t]; H) (the class of Dom(L(αβ) ) (due to Assumption 1), and L(αβ) − L(k;αβ) H(k) Tt continuously differentiable functions from [0, t] to H), a unique solution of (9) exists and is given by [16, Thm 3.1.3]
Z t (αβ) (k;αβ) (αβ) u= Tt−τ L(αβ) − L(k;αβ) Tτ(k;αβ) udτ − Tt Tt 0
for all t ≥ 0 and all u ∈ H(k) . From Assumption 2(ii) and the definition of M(k) , we then have for
all u ∈ H(k) and all t ≥ 0 that
Z t (αβ) (αβ) (k;αβ) u ≤ − Tt Tt−τ L(αβ) − L(k;αβ) Tτ(k;αβ) u dτ Tt 0 Z t (αβ) (k;αβ) (αβ) (k;αβ) u dτ. PK(k) Tτ = −L Tt−τ PM(k) L
(10)
0
Now using the bounds (3) and (6) (established in Lemmas 1 and 2, respectively), and applying Assumption 2(i), we have that (αβ) (k;αβ) (αβ) (k;αβ) u PK(k) Tτ −L Tt−τ PM(k) L !(1−2−r ) !(1−2−s ) (k;αβ) (k;αβ) qa (k;αβ) qe ≤ qL (αβ) (αβ) γk γk ! −s " r−1 ! −i # (k;αβ) (1−2 ) X (k;αβ) (1−2 ) (αβ) qa q −i e ci e−2 γk (t−τ ) + (αβ) (αβ) γk γk i=0 −i # ! −r " s−1 ! (k;αβ) (1−2 ) (k;αβ) (1−2 ) X (αβ) q qe −i a ci e−2 γk τ + (αβ) (αβ) γk γk i=1 −i −j # " r−1 s−1 ! ! (k;αβ) (1−2 ) (k;αβ) (1−2 ) XX (αβ) −i −i −j qe qa ||u||. ci cj eγk (−2 t+(2 −2 )τ ) + (αβ) (αβ) γ γ i=0 j=0 k k
The result (8) then follows from substitution of the above identity into (10) and integration. This establishes the lemma statement. Let S′ ⊂ L2 ([0, T ]; Cm ) denote the set of all simple functions, which is dense in L2 ([0, T ]; Cm ).
That is, for any t ∈ [0, T ) and f ∈ S′ , there exists 0 < ℓ < ∞ and a sequence 0 = t0 < t1
0 and each fj ∈ L2 ([0, T ]; Cm ), there exists fj′ ∈ S′ such that ||fj − fj′ || < ǫ. Moreover, for f1′ , f2′ ∈ S′ , there exists 0 < ℓ < ∞ and 0 = t0 < t1 < · · · < tℓ < tℓ+1 = t, such that fj′ 1[ti ,ti+1 ) = α′j (i) for all i = 0, 1, . . . , ℓ, where α′j (0), α′j (1), . . . , α′j (ℓ) ∈ Cm . Suppose that u1 , u2 6= 0 (otherwise the corollary
statement becomes trivial), then for any ǫ > 0 we may choose f1′ , f2′ ∈ S′ (choosing f1′ first followed by f2′ ) such that
ǫ 6ku1 k ||e(f2 )|| ǫ ′ e(f2 ) − e(f2 ) < 6ku2 k ||e(f1′ )||
e(f1 ) − e(f1′ )
k1 . Since the theorem statement holds trivially when ψ = 0, this completes the proof.
4
Examples
Example 1 (Kerr-nonlinear optical cavity). Consider a single-mode Kerr-nonlinear optical cavity coupled to a single external coherent field (m = 1), which is used in the construction of the photonic logic gates presented in [12]. Let H = ℓ2 (the space of infinite complex-valued sequences with P∞ 2 n=1 |xn | < ∞) which has an orthonormal Fock state basis {|ni}n≥0 . On this basis {|ni}n≥0 , the
annihilation, creation, and number operators of the cavity oscillator can be defined (see, e.g., [14]) satisfying √
a∗ |ni =
√
a∗ a |ni = n |ni , √ respectively. The Kerr-nonlinear optical cavity can be described by S = I, L = λa, and H = a |ni =
n |n − 1i ,
n + 1 |n + 1i ,
∆a∗ a + χa∗ a∗ aa, where λ, ∆, χ > 0. Consider H(k) = span {|ni | n = 0, 1, 2, . . . , k} and a system approximation of the form S (k) = I,
L(k) = PH(k) LPH(k) ,
H (k) = PH(k) HPH(k) .
(18)
For any α, β ∈ C, L(αβ) u is defined and finite for all u ∈ H(k) . Hence, Assumption 1 holds. Recall that M(k) = Range L(α,β) − L(k;α,β) H(k) . From the (S, L, H), we see that M(k) = (αβ) span{|k + 1i}. Now consider K(k) = span {|ki} and γk = 21 λk + |α − β|2 . We will now show
that Assumptions 2 and 3 hold for the Kerr-nonlinear optical cavity and the approximation. 14
Assumption 2(i)
Note that L(αβ) − L(k;αβ)
K
√ ∗ = β λ a |K(k) . (k)
(k;αβ)
Thus, we have that Assumption 2(i) holds with qL
=
p λ(k + 1)|β|.
This assumption follows for the defined K(k) because L(αβ) − L(k;αβ) H(k) = span {|ni | n = 0, 1, 2, . . . , k − 1} for all α, β ∈ C.
Assumption 2(ii) ker
∗
Assumption 2(iii) First note that L(k) PK(k) = 0, PK(k) L(k) PH(k) = 0, and ∗ ∗ PK(k) 12 L(k) L(k) − ıH PH(k) ⊖K(k) = 0. Also, for any u ∈ H(k) , PK(k) L(k) L(k) PK(k) u = λkPK(k) u.
From these identities and the fact that H (k) is self-adjoint, we have for any u ∈ H(k) that ℜ{hPK(k) L(k;αβ) u, PK(k) ui} = ℜ{hPK(k) L(k;αβ) PK(k) + PH(k) ⊖K(k) u, PK(k) ui} 1 = − λk + |α|2 + |β|2 − 2ℜ{α∗ β} ||PK(k) u||2 2 nD Eo √ λβPK(k) a∗ PH(k) ⊖K(k) u, PK(k) u +ℜ = −g(k, α, β) ||PK(k) u||2 + h(k, α, β, u).
Noticing that |α|2 + |β|2 − 2ℜ{α∗ β} = hα, αi + hβ, βi − hα, βi − hα, βi = |α − β|2 . Also, note that nD Eo √ √ ∗ ℜ λβPK(k) a PH(k) ⊖K(k) u, PK(k) u ≤ |β| λ PK(k) a∗ PH(k) ⊖K(k) u ||PK(k) u|| √ ≤ |β| λk||u|| ||PK(k) u|| .
(αβ)
Therefore, we have that Assumption 2(iii) holds for the defined K(k) and the defined γk √ (k;αβ) qa = |β| λk. Assumption 2(iv) that
For any u ∈ H, HPM(k) u = (∆(k + 1) + χk(k + 1)) PM(k) u. This implies
(αβ) (αβ) ℜ{hTt (ıH)PM(k) u, Tt PM(k) ui}
Therefore, we have that
(αβ) (αβ) ℜ{hTt L(αβ) PM(k) u, Tt PM(k) ui}
= 0. Also, L∗ LPM(k) u = λ(k + 1)PM(k) u for any u ∈ H.
1 2 2 ∗ = − λ(k + 1) + |α| + |β| − 2ℜ{α β} 2 2 (αβ) × Tt PM(k) u nD Eo √ (αβ) (αβ) + ℜ Tt λ (βa∗ − α∗ a) PM(k) u, Tt PM(k) u 2 (αβ) ˆ k, α, β, u). = −ˆ g(k, α, β) Tt PM(k) u + h(t, (αβ)
is a contraction, we see that Assumption 2(iv) p p (k;αβ) with qe = |α| λ(k + 1) + |β| λ(k + 2).
Similarly to the previous derivation, using that Tt holds for the defined
with
K(k) and
the defined
(αβ) γk
15
Assumption 3
(αβ)
From the defined γk
(k;αβ)
, qL
(k;αβ)
, qa
(k;αβ)
, qe
, we see that this assumption holds
for any r, s ∈ Z+ such that r + s ≥ 3.
Finally, because Assumptions 1-3 hold, Lemma 4, Corollary 1, and Theorem 1 can be applied
to obtain error bounds on the finite-dimensional approximations. Example 2 (Atom-cavity model). Consider a three-level atom coupled to an optical cavity, which itself is coupled to a single external coherent field (m = 1) [11]. Let H = C3 ⊗ ℓ2 . We will use
|ei = (1, 0, 0)⊤ , |+i = (0, 1, 0)⊤ , and |−i = (0, 0, 1)⊤ to denote the canonical basis vectors in Cm .
We also consider |ni, denoting the normalized n-photon Fock state of the system, as the basis of ℓ2
(as in the previous example). This atom-cavity system is then described by the following parameters: S = I,
L=
√
H = ıχ (σ+ a − σ− a∗ ) ,
λa,
where λ, χ > 0, σ+ = |ei h+|, and σ− = |+i he|.
Consider H(k) = C3 ⊗span {|ni | n = 0, 1, 2, . . . , k} and a system approximation of the form (18).
That is, we are approximating the dynamics of the harmonic oscillator. For any α, β ∈ C, L(αβ) u is defined and finite for all u ∈ H(k) . Hence, Assumption 1 holds. Similar to the previous example, (αβ) = 21 λk + |α − β|2 . we see that M(k) = C3 ⊗ |k + 1i. Now consider K(k) = C3 ⊗ |ki and γk
Using similar derivation to the previous example, we will now show that Assumptions 2 and 3 hold for the atom-cavity model and the approximation. Assumption 2(i)
Note that L(αβ) − L(k;αβ)
K
√ = β a∗ |K(k) . λ + χσ − (k) (k;αβ)
Thus, we have that Assumption 2(i) holds with qL
=
√ √ k + 1 |β| λ + χ .
This assumption follows for the defined K(k) because L(αβ) − L(k;αβ) H(k) = C3 ⊗ span {|ni | n = 0, 1, 2, . . . , k − 1} for all α, β ∈ C.
Assumption 2(ii) ker
Assumption 2(iii)
∗
∗
Note that L(k) PK(k) = 0, PK(k) L(k) PH(k) = 0, PK(k) L(k) L(k) PH(k) ⊖K(k) = 0, ∗
and PK(k) HPH(k) ⊖K(k) = −ıχPK(k) σ− a∗ PH(k) ⊖K(k) . Also, for any u ∈ H(k) , PK(k) L(k) L(k) PK(k) u = λkPK(k) u. From these identities and the fact that H (k) is self-adjoint, we have for any u ∈ H(k)
16
that ℜ{hPK(k) L(k;αβ) u, PK(k) ui} = ℜ{hPK(k) L(k;αβ) PK(k) + PH(k) ⊖K(k) u, PK(k) ui} 1 = − λk + |α|2 + |β|2 − ℜ{α∗ β} ||PK(k) u||2 2 nD Eo √ + ℜ PK(k) β λ + χσ− a∗ PH(k) ⊖K(k) u, PK(k) u = −g(k, α, β) ||PK(k) u||2 + h(k, α, β, u).
Noticing that |α|2 + |β|2 − 2ℜ{α∗ β} = hα, αi + hβ, βi − hα, βi − hα, βi = |α − β|2 . Also, note that nD √ Eo ℜ PK(k) β λ + χσ− a∗ PH(k) ⊖K(k) u, PK(k) u √ ≤ PK(k) β λ + χσ− a∗ PH(k) ⊖K(k) u ||PK(k) u|| √ √ ≤ k χ + |β| λ ||u|| ||PK(k) u|| . (αβ)
Therefore, we have that Assumption 2(iii) holds for the defined K(k) and the defined γk √ √ (k;αβ) qa = k χ + |β| λ .
with
For any u ∈ H, note that L∗ LPM(k) u = λ(k + 1)PM(k) u and ıHPM(k) u =
Assumption 2(iv)
ı (∆(k + 1) + χk(k + 1)) PM(k) u. Also, Thus, we have that 2 1 (αβ) (αβ) (αβ) (αβ) 2 2 ∗ PM(k) u, Tt PM(k) ui} = − λ(k + 1) + |α| + |β| − 2ℜ{α β} Tt PM(k) u ℜ{hTt L 2 nD Eo √ (αβ) (αβ) + ℜ Tt λ (βa∗ − α∗ a) PM(k) u, Tt PM(k) u Eo nD (αβ) (αβ) + ℜ Tt χ (σ− a∗ − σ+ a) PM(k) u, Tt PM(k) u 2 (αβ) ˆ k, α, β, u). = −ˆ g (k, α, β) Tt PM(k) u + h(t, (αβ)
is a contraction, we see that Assumption 2(iv) √ √ √ √ = k + 1 χ + |α| λ + k + 2 χ + |β| λ .
Similar to the previous derivation, using that Tt holds for the defined Assumption 3
K(k)
and
(αβ) γk
with (αβ)
From the defined γk
(k;αβ) qe (k;αβ)
, qL
(k;αβ)
, qa
(k;αβ)
, qe
, we see that this assumption holds
for any r, s ∈ Z+ such that r + s ≥ 3.
Finally, because Assumptions 1-3 hold, Lemma 4, Corollary 1, and Theorem 1 can be applied
to obtain error bounds on the finite-dimensional approximations.
5
Conclusion
This paper has presented error bounds for finite-dimensional approximations of input-output models of open quantum systems with an infinite-dimensional underlying Hilbert space (and with possibly 17
unbounded coefficients in their QSDEs). The bounds are established on the basis of the contractive property of an open quantum system and some constructions employed in [14]. Under certain conditions, we show that this bound vanishes in the limit as the dimension of the approximating finite-dimensional subspace increases. We also applied these results to some sample physical systems taken from the literature.
References [1] R. L. Hudson and K. R. Parthasarathy, “Quantum Ito’s formula and stochastic evolution,” Commun. Math. Phys., vol. 93, pp. 301–323, 1984. [2] C. Gardiner and M. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A, vol. 31, pp. 3761–3774, 1985. [3] H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control. Cambridge University Press, 2010. [4] V. P. Belavkin and S. C. Edwards, “Quantum filtering and optimal control,” in Quantum Stochastics and Information: Statistics, Filtering and Control (University of Nottingham, UK, 15 - 22 July 2006), V. P. Belavkin and M. Guta, Eds. Singapore: World Scientific, 2008, pp. 143–205. [5] H. I. Nurdin, M. R. James, and A. C. Doherty, “Network synthesis of linear dynamical quantum stochastic systems,” SIAM J. Control Optim., vol. 48, no. 4, pp. 2686–2718, 2009. [6] L. Bouten, R. van Handel, and M. R. James, “An introduction to quantum filtering,” SIAM J. Control Optim., vol. 46, pp. 2199–2241, 2007. [7] L. Bouten and R. van Handel, “On the separation principle of quantum control,” in Quantum Stochastics and Information: Statistics, Filtering and Control (University of Nottingham, UK, 15 - 22 July 2006), V. P. Belavkin and M. Guta, Eds. Singapore: World Scientific, 2008, pp. 206–238. [8] M. R. James, H. I. Nurdin, and I. R. Petersen, “H ∞ control of linear quantum stochastic systems,” IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787–1803, 2008. [9] H. I. Nurdin, M. R. James, and I. R. Petersen, “Coherent quantum LQG control,” Automatica, vol. 45, pp. 1837–1846, 2009. [10] J. Kerckhoff, H. I. Nurdin, D. Pavlichin, and H. Mabuchi, “Designing quantum memories with embedded control: photonic circuits for autonomous quantum error correction,” Phys. Rev. Lett., vol. 105, pp. 040 502–1–040 502–4, 2010. [11] L. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interaction,” Phys. Rev. Lett., vol. 92, pp. 127 902–1–127 902–4, 2004. [12] H. Mabuchi, “Nonlinear interferometry approach to photonic sequential logic,” Appl. Phys. Lett., vol. 99, pp. 153 103–1–153 103–3, 2011.
18
[13] C. W. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, 3rd ed.
Berlin and New York: Springer-
Verlag, 2004. [14] L. Bouten, R. van Handel, and A. Silberfarb, “Approximation and limit theorems for quantum stochastic models with unbounded coefficients,” Journal of Functional Analysis, vol. 254, pp. 3123–3147, 2008. [15] P. A. Meyer, Quantum Probability for Probabilists, 2nd ed. Berlin-Heidelberg: Springer-Verlag, 1995. [16] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, ser. Text in Applied Mathematics. New York: Springer-Verlag, 1995, no. 21.
19
arXiv:1509.02629v1 [quant-ph] 9 Sep 2015
Input-Output Open Quantum Systems∗ Onvaree Techakesari and Hendra I. Nurdin
†
September 10, 2015
Abstract Many physical systems of interest that are encountered in practice are input-output open quantum systems described by quantum stochastic differential equations and defined on an infinite-dimensional underlying Hilbert space. Most commonly, these systems involve coupling to a quantum harmonic oscillator as a system component. This paper is concerned with the error in the finite-dimensional approximation of input-output open quantum systems defined on an infinite-dimensional underlying Hilbert space. We present explicit error bounds between the time evolution of the state of a class of infinite-dimensional quantum systems and its approximation on a finite-dimensional subspace of the original, when both are initialized in the latter subspace. Applications to some physical examples drawn from the literature are provided to illustrate our results.
1
Introduction
Quantum stochastic differential equations (QSDEs) developed independently by Hudson and Parthasarathy [1] and Gardiner and Collett [2] (the latter in a more restricted form than the former) have been widely used to describe the input-output models of physical Markovian open quantum systems [3–5]. Such models describe the evolution of Markovian quantum systems interacting with a propagating quantum field, such as a quantum optical field, and are frequently encountered in quantum optics, optomechanics, and related fields. An example in quantum optics would be a cavity QED (quantum electrodynamics) system where a single atom is trapped inside an optical cavity that interacts with an external coherent laser beam impinging on the optical cavity. These ∗ †
This research was supported by the Australian Research Council O. Techakesari and H. I. Nurdin are with the School of Electrical Engineering and Telecommunications, UNSW
Australia, Sydney NSW 2052, Australia. Email: [email protected] and [email protected]
1
input-output models have subsequently played an important role in the modern development of quantum filtering and quantum feedback control theory [6, 7]. Many types of quantum feedback controllers have been proposed in the literature on the basis of QSDEs, using both measurementbased quantum feedback control, e.g., [3,4,6,7], and coherent feedback control, e.g., [8–10]. Besides, the QSDEs have also been applied in various developments in quantum information processing, such as in quantum computation technology; e.g., see [11]. In various physical systems of interest, one often deals with input-output systems that include coupling to a quantum harmonic oscillator. For instance, typical superconducting circuits that are of interest for quantum information processing consist of artificial two-level atoms coupled to a transmission line resonator. The former is typically described using a finite-dimensional Hilbert space and the latter is a quantum harmonic oscillator with an infinite-dimensional underlying Hilbert space (i.e., L2 (R), the space of square-integrable complex-valued functions on the real line). Another example is a proposed photonic realization of classical logic based on Kerr nonlinear optical cavities in [12], which is built around a quantum harmonic oscillator with a Kerr nonlinear medium inside it. If a mathematical model for such quantum devices is sufficiently simple, it is often possible to simulate the dynamics of the system on a digital computer to assess the predicted performance of the actual device, as carried out in [12]. The simulation carried out is typically that of a stochastic master equation that simulates the stochastic dynamics of a quantum system when one of its output is observed via laboratory procedures such as homodyne detection or photon counting, see [3, 6, 13]. However, since it is not possible to faithfully simulate a quantum system with an infinite-dimensional Hilbert space, often in simulations this space is truncated to some finite-dimensional subspace and an operator X on the infinite-dimensional space is approximated by a truncated operator of the form P XP , where P denotes an orthogonal projection projector onto the approximate finite-dimensional subspace. For instance, with quantum harmonic oscillators, a commonly used finite-dimensional space is the span of a finite number of Fock states |0i, |1i, . . . , |ni.
Despite the ubiquity of approximating infinite-dimensional Hilbert spaces of quantum systems
by finite-dimensional subspaces for simulations of input-output quantum systems, to the best of the authors’ knowledge, there does not appear to be any work that has obtained some explicit bounds on the approximation error of the joint state of the system and the quantum field it is coupled to. In this paper, we present bounds on the error between the quantum state of a quantum system described by the QSDE and the quantum state of a finite-dimensional approximation described by another QSDE, when both systems are initialized in a state in the finite-dimensional subspace. The bounds are established using the contractive property of open quantum systems, semigroup theory, and some constructions employed by Bouten, van Handel and Silberfarb in [14]. Some examples are also presented to illustrate the application of our results. 2
The rest of this paper is structured as follows. In Section 2, we present the class of open quantum systems and the associated QSDEs describing Markovian open quantum systems. Explicit error bounds for a finite-dimensional approximation of a Markovian open quantum system are established in Section 3. Some examples are provided in Section 4. Concluding remarks are then presented in Section 5.
2
Preliminaries
2.1
Notation
We use ı =
√
−1 and let (·)∗ denote the adjoint of a linear operator as well as the conjugate of
a complex number, and (·)⊤ denote matrix transposition. We denote by δij the Kronecker delta function. We define ℜ{A} = 12 (A + A∗ ) and ℑ{A} =
1 ∗ 2ı (A − A ).
For a linear operator A, we write
ker(A) to denote the kernel of A. We often write |·i to denote an element of a Hilbert space and denote by H⊗F the algebraic tensor product of Hilbert spaces H and F. For a subspace H0 of a
Hilbert space H, we write PH0 to denote an orthogonal projection operator onto H0 . For a Hilbert
space H = H0 ⊕H1 , we will write H⊖H0 to denote H1 . For a linear operator X on H, X|H0 denotes
the restriction of X to H0 . We use B(H) to denote the algebra of all bounded linear operators on
H. We write [A, B] = AB − BA. The notation || · || will be used to denote Hilbert space norms
and operator norms, h·, ·i denotes an inner product on a Hilbert space, linear in the right slot and
antilinear in the left, and |·i h·| denotes an outer product. Here, 1[0,t] (·) : [0, t] → {0, 1} denotes the
indicator function. Finally, Z+ denotes the set of all positive integers.
2.2
Open quantum systems
Consider a separable Hilbert space H and the symmetric boson Fock space (of multiplicity m) F
defined over the space L2 ([0, T ]; Cm ) = Cm ⊗ L2 ([0, T ]) with 0 < T < ∞; see [15, Ch. 4-5] for
more details. We will use e(f ) ∈ F, with f ∈ L2 ([0, T ]; Cm ), to denote exponential vectors in
m F. As in [14], let S ⊂ L2 ([0, T ]; Cm ) ∩ L∞ loc ([0, T ]; C ) be an admissible subspace in the sense of
m Hudson-Parthasarathy [1] which contains at least all simple functions, where L∞ loc ([0, T ]; C ) is the
space of locally bounded vector-valued functions. Here, we will consider a dense domain D ⊂ H
and a dense domain of exponential vectors E = span{e(f ) | f ∈ S} ⊂ F.
Consider an open Markov quantum system which can be described by a set of linear operators
defined on the Hilbert space H: (i) a self-adjoint Hamiltonian operator H, (ii) a vector of coupling operators L with the j-th element, Lj : H → H for all j = 1, 2, . . . , m, and (iii) a unitary scattering
matrix S with the ij-th element, Sij : H → H for all i, j = 1, 2, . . . , m. Moreover, the operators 3
Sij , Lj , H and their adjoints are assumed to have D as a common invariant dense domain. Under this description, we note that m corresponds to the number of external bosonic input fields driving the system. Each bosonic input field can be described by annihilation and creation field operators, ∗
∗
bit and bit , respectively, which satisfy the commutation relations [bit , bjs ] = δij δ(t − s) for all i, j =
1, 2, . . . , m and all t, s ≥ 0. We can then define the annihilation process Ait , the creation process ∗
Ait , and the gauge process Λij t as Ait
=
Z
0
t
bis ds,
∗ Ait
=
Z
t 0
∗ bis ds,
Λij t
=
Z
t 0
∗
bis bjs ds.
Note that these processes are adapted quantum stochastic processes. In the vacuum representation, ∗
∗
∗
the products of their forward differentials dAit = Ait+dt − Ait , dAit = Ait+dt − Ait , and dΛij t =
ij Λij o table t+dt − Λt satisfy the quantum Itˆ
×
dAit
∗ dAjt dΛij t
dt Here, bit =
dAit dt
dAkt
dAkt
0
∗
dΛkℓ t
dt
δik dt
δik dAℓt
0
0
0
0
0
∗ δjk dAit
δjk dΛiℓ t
0
0
0
0
0
0 .
can be interpreted as a vacuum quantum white noise, while Λii t can be interpreted
as the quantum realization of a Poisson process with zero intensity [1]. Following [14], the time evolution of a Markovian open quantum system is given by a unitary adapted process Ut satisfying the left Hudson-Parthasarathy QSDE [1]: " # m m m m X X X X 1 ∗ i∗ ∗ i ∗ ∗ + ıH − S L dA L dA − + Sji − δij )dΛij dUt = Ut (L L ) dt j i ji t i t i t 2 i,j=1
i,j=1
i=1
i=1
(1)
with U0 = I. The quantum stochastic integrals are defined relative to the domain D⊗E. With the
left QSDE, the evolution of a state vector ψ ∈ H ⊗ F is given by Ut∗ ψ.
In this paper, we are interested in the problem of approximating the system with operator
parameters (S, L, H) by a finite-dimensional open quantum system with linear operator parameters (S (k) , L(k) , H (k) ) defined on a finite-dimensional subspace H(k) ⊂ H, where the dimension of H(k)
increases with k ∈ Z+ , S (k) is unitary, and H (k) is self-adjoint. Similar to (1), the time evolution (k)
of the approximating system is given by a unitary adapted process Ut
4
satisfying the left Hudson-
Parthasarathy QSDE [1]: m m m X X X (k)∗ (k)∗ (k)∗ (k) (k) i∗ i S L dA − L dA + Sji − δij )dΛij dUt = Ut j t t t ji i i,j=1 i=1 i,j=1 " # ) m X ∗ 1 (k) (k) + ıH (k) − (Li Li ) dt , 2
(2)
i=1
(k)
with U0
= I. Here, the quantum stochastic integrals in the above equation are defined relative to (k) ∗
the domain H(k) ⊗E. Similarly, the evolution of a state vector ψ ∈ H ⊗ F is given by Ut
2.3
ψ.
Associated semigroups
Let θt : L2 ([t, T ]; Cm ) → L2 ([0, T ]; Cm ) be the canonical shift θt f (s) = f (t + s). We also let Θt :
F[t → F denote the second quantization of θt , where F[t denotes the Fock space over L2 ([t, ∞); Cm ).
Note that an adapted process Ut on H ⊗ F is called a contraction cocycle if Ut is a contraction for all t ≥ 0, t 7→ Ut is strongly continuous, and Us+t = Us (I ⊗ Θ∗s Ut Θs ).
Let us now impose an important condition on the open quantum systems, adopted from [14].
Condition 1 (Contraction cocycle solutions). For all t ≥ 0 and all k ∈ Z+ , i. the QSDE (1) possesses a unique solution Ut which extends to a unitary cocycle on H ⊗ F, (k)
ii. the QSDE (2) possesses a unique solution Ut H(k)
which extends to a contraction cocycle on
⊗ F. (k)
We note that in this paper, Condition 1(ii) on the solution of Ut
is satisfied because H(k) is (k)
finite-dimensional, S (k) is unitary, and H (k) is self-adjoint [1]. In fact, in this case Ut
is unitary.
We will assume that the Condition 1 holds throughout this paper. (αβ)
Let us define an operator Tt (αβ)
hu, Tt
1
: H → H via the identity
vi = e− 2 (||α||
2 +||β||2 )t
hu ⊗ e(α1[0,t] (·)), Ut v ⊗ e(β1[0,t] (·))i
for all u, v ∈ H and all α, β ∈ Cm . From [14, Lemma 1], under Condition 1(i), the operator (αβ)
Tt
∈ B(H) is a strongly continuous contraction semigroup on H and its generator L(αβ) satisfies
Dom(L(αβ) ) ⊃ D such that m m m X X X ∗ ∗ L∗j βj + α∗i Sji Lj + α∗i Sji βj − L(αβ) u = i,j=1
i,j=1
j=1
ıH −
m 1X
2
i=1
L∗i Li
!
−
||α||2
+ 2
||β||2
u
for all u ∈ D. We assume that D is a core for L(αβ) for all α, β ∈ Cm so that Lαβ so that the above definition completely determines L(αβ) . Here, we note that Dom(Lαβ ) is dense in H. We likewise 5
(k;αβ)
define an operator Tt
(k)
: H(k) → H(k) by replacing Ut with Ut . Since H(k) is finite-dimensional,
we note that Dom(L(k;αβ) ) = H(k) .
In the sequel, we will make use of the above semigroups associated with open quantum systems
in establishing our model approximation error bound.
3
Error bounds for finite-dimensional approximations
3.1
Assumptions and preliminary results
Assumption 1. For any k ∈ Z+ and any α, β ∈ Cm , H(k) ⊂ Dom(L(αβ) ). Let M(k) = Range L(α,β) − L(k;α,β) H(k) . Supposing that Assumption 1 holds, we also
assume the following.
(αβ)
Assumption 2. For each k ∈ Z+ and each α, β ∈ Cm , there exists γk
and a non-trivial subspace {0} ⊂ K(k) ⊆ H(k) such that (k;αβ) . i. L(αβ) − L(k;αβ) K(k) ≤ qL T ii. L(αβ) − L(k;αβ) H(k) ⊖K(k) = 0, i.e., H(k) ⊖ K(k) ⊆ α,β∈Cm ker iii. For any u ∈ H(k) ,
(k;αβ)
, qL
(k;αβ)
, qa
(k;αβ)
, qe
> 0,
L(αβ) − L(k;αβ) H(k)
ℜ{hPK(k) L(k;αβ) u, PK(k) ui} = −g(k, α, β) ||PK(k) u||2 + h(k, α, β, u) (αβ)
for some g(k, α, β) ≥ γk
(k;αβ)
and some |h(k, α, β, u)| ≤ qa
||PK(k) u|| ||u||.
iv. For any u ∈ H(k) and any t ≥ 0, (αβ)
ℜ{hTt
2 (αβ) ˆ k, α, β, u) g (k, α, β) Tt PM(k) u + h(t, PM(k) ui} = −ˆ ˆ (k;αβ) (αβ) and some h(t, k, α, β, u) ≤ qe u P Tt M(k) ||u||. (α,β)
L(αβ) PM(k) u, Tt (αβ)
for some gˆ(k, α, β) ≥ γk
Assumption 3. There exists r, s ∈ Z+ such that, for all α, β ∈ Cm , we have that −r −s ! ! (k;αβ) (1−2 ) (k;αβ) (1−2 ) q q e a (k;αβ) lim q =0 (αβ) (αβ) k→∞ L γk γk
Moreover, for any i = 0, 1, 2, . . . , r, j = 0, 1, 2, . . . , s, and ℓ = 0, 1, 2, . . . , min{r, s}, we have that −i −j ! ! (k;αβ) (k;αβ) (1−2 ) (k;αβ) (1−2 ) qL qe qa =0 lim (αβ) (αβ) k→∞ γ (αβ) γ γ k k k −ℓ ! (k;αβ) (k;αβ) (1−2 ) (αβ) q q −ℓ e a (k;αβ) lim e−2 γk t qL =0 (αβ) k→∞ (γk )2 for all α, β ∈ Cm and all t ∈ [0, T ] with 0 ≤ T < ∞. 6
Let us present some useful lemmas. Lemma 1. Suppose that Assumption 2(iii) holds. Then for any k, r ∈ Z+ , any α, β ∈ Cm , any
u ∈ H(k) , and any t ≥ 0, it holds that −r ! r−1 (k;αβ) (1−2 ) X (αβ) q a (k;αβ) t −(2−j )γk c e + u ≤ PK(k) Tt j (αβ) γk j=0 where c0 = 1 and cj =
q
(k;αβ) qa (αβ) γk
!(1−2−j )
||u||
cj−1 2j (2j − 1)−1 for j ≥ 1.
(αβ)
Proof. First note, from the definition of a strongly continuous semigroup, that T0 d (αβ) u dt Tt
(3)
= I and
= L(αβ) u [16]. From Assumption 2(iii), we have that E d d D (k;αβ) (k;αβ) (k;αβ) 2 u u, PK(k) Tt PK(k) Tt u = PK(k) Tt dt dt nD Eo (k;αβ) (k;αβ) u u, PK(k) Tt = 2ℜ PK(k) L(k;αβ) Tt (k;αβ) (k;αβ) 2 u). u + 2h(k, α, β, Tt =−2g(k, α, β) PK(k) Tt
Solving the above ODE gives us that
Z t (k;αβ) 2 2 −2g(k,α,β)t h(k, α, β, Tτ(k;α,β) u)e−2g(k,α,β)(t−τ ) dτ u|| + 2 = e ||P u T P K(k) t K(k) 0 Z t (αβ) (αβ) 2 (k;αβ) −2γk t ≤e ||PK(k) u|| + 2qa e−2γk (t−τ ) PK(k) Tτ(k;αβ) u ||u||dτ 0 " # (k;αβ) (αβ) q a ≤ e−2γk t + (αβ) ||u||2 . γk
(4)
Here, the second step follows from Assumption 2(iii). The last step follows because ||PK(k) u|| ≤ ||u|| p (k;αβ) is a contraction semigroup. Noticing that |a|2 + |b|2 ≤ |a| + |b| for any a, b ∈ R, we and Tt have that
(αβ) (k;αβ) u ≤ e−γk t + PK(k) Tt
7
(k;αβ) qa (αβ) γk
!1 2 ||u||.
(5)
Now, substituting (5) into the right-handed side of (4), we have that (k;αβ) 2 T u P K(k) t (αβ) −2γk t
≤e
||u||2 + 2qa(k;αβ)
(αβ) −2γk t
= e
(αβ) −2γk t
≤ e
(αβ) −γk t
+ 2e
(αβ) −γk t
+2e
Z
t
(αβ) −γk (2t−τ )
e
0
(αβ) 1 − e−γk t (k;αβ)
qa
(αβ) γk
!
||u||2 dτ + 2qa(k;αβ) (k;αβ)
qa
(αβ)
γk
(k;αβ)
+
qa
(αβ) γk
!
Z
t
(k;αβ)
qa
(αβ) −2γk (t−τ )
e
(αβ) γk
0
(αβ) −2γk t
+ 1−e
(k;αβ)
qa
(αβ)
γk
! 3 2
!1
2
||u||2 dτ
!3 2 ||u||2
||u||2 .
Taking the square root on both sides of the equation, we get !1 √ − 1 γ (αβ) t qa(k;αβ) 2 (αβ) (k;αβ) −γ t + + 2e 2 k u ≤ e k PK(k) Tt (αβ) γk
(k;αβ) qa (αβ) γk
!3
From repeat application of the above steps, we establish the lemma statement.
4
||u||.
Lemma 2. Suppose that Assumption 2(iv) holds. Then for any k, r ∈ Z+ , any α, β ∈ Cm , any u ∈ H(k) , and any t ≥ 0, it holds that −r ! r−1 (k;αβ) (1−2 ) X (αβ) −j q e (αβ) cj e−(2 )γk t + Tt PM(k) u ≤ (αβ) γk j=0
where c0 = 1 and cj =
q
(k;αβ)
qe
(αβ)
γk
!(1−2−j ) ||u||
cj−1 2j (2j − 1)−1 for j ≥ 1.
Proof. Similar to Lemma 1, using Assumption 2(iv), we have that 2 d (αβ) (αβ) (αβ) Tt PM(k) u = 2ℜ{hTt L(αβ) PM(k) u, Tt PM(k) ui} dt 2 (αβ) ˆ k, α, β, u). = −2ˆ g (k, α, β) Tt PM(k) u + 2h(t,
Solving the ODE, we have that
Z t 2 (αβ) (αβ) (αβ) (αβ) 2 e−2γk (t−τ ) Tt PM(k) u ||u||dτ Tt PM(k) u ≤ e−2γk t ||PM(k) u|| + 2qe(k;αβ) 0 " # (k;αβ) (αβ) qe ≤ e−2γk t + (αβ) ||u||2 . γk The lemma statement is then established by following similar arguments to Lemma 1.
8
(6)
3.2
Error bounds for finite-dimensional approximations
We begin by defining k zr,s (t, α, β)
=
(k;αβ)
qe
(k;αβ) qL t
(αβ)
γk
!(1−2−r )
(k;αβ)
qa
(αβ)
γk
r−1 q (k;αβ) X (αβ) 2i ci e −2−i γk t 1 − e + (αβ) (αβ) γk i=0 γk
s−1 q (k;αβ) X (αβ) 2i ci a −2−i γk t + 1−e (αβ) (αβ) γk i=1 γk
+
s−1 r−1 X X i=0
ci cj
j=0
j6=i
2(i+j) (αβ)
(2i − 2j ) γk
min{r,s}−1
X
+t
(αβ) −2−i γk t
c2i e p
!(1−2−i )
qa
!(1−2−i )
qe
(k;αβ) (αβ)
γk
(k;αβ) (αβ)
γk
!(1−2−s ) !(1−2−r )
q (k;αβ) −i (αβ) −j (αβ) e e−2 γk t − e−2 γk t (αβ) γk (k;αβ) (k;αβ) qa (αβ) (γk )2
qe
i=0
where c0 = 1, and cj =
!(1−2−s )
! (1−2−i )
! −j (k;αβ) (1−2 )
qa
(αβ)
γk
!(1−2−i ) ,
(7)
cj−1 2j (2j − 1)−1 for j ≥ 1. We now establish an error bound between
the two semigroups associated with the open quantum systems.
Lemma 3. Suppose Assumptions 1 and 2 hold. Then for any k, r, s ∈ Z+ , any α, β ∈ Cm , any
u ∈ H(k) , and any t ≥ 0, it holds that (αβ) (k;αβ) k (t, α, β)||u||. u ≤ zr,s − Tt Tt
(8)
Proof. First note, from the definition of a strongly continuous semigroup, that [16] (αβ)
i. T0 ii. iii.
(k;αβ)
= T0
d (αβ) u dt Tt
= I for all k ∈ Z+ ,
= L(αβ) u for all u ∈ Dom(L(αβ) ),
d (k;αβ) u dt Tt
= L(k;αβ) u for all u ∈ H(k) (since H(k) is finite-dimensional).
From the above properties and Assumption 1, we can write for all u ∈ H(k) and all t ≥ 0 that d (αβ) (k;αβ) (αβ) (k;αβ) Tt − Tt u = L(αβ) Tt − L(k;αβ) Tt u dt (αβ)
= L(αβ) Tt
with
(k;αβ)
− Tt
u + L(αβ) − L(k;αβ)
(αβ) (k;αβ) T0 − T0 u = 0. Note that by Assumption 2,
operator. Since L(αβ)
H(k)
(k;αβ)
Tt
u
(9)
L(αβ) − L(k;αβ) H(k) is a bounded (αβ) (k;αβ) is a generator of a semigroup on a Hilbert space, T0 − T0 u ∈ 9
(k;αβ) u ∈ C 1 ([0, t]; H) (the class of Dom(L(αβ) ) (due to Assumption 1), and L(αβ) − L(k;αβ) H(k) Tt continuously differentiable functions from [0, t] to H), a unique solution of (9) exists and is given by [16, Thm 3.1.3]
Z t (αβ) (k;αβ) (αβ) u= Tt−τ L(αβ) − L(k;αβ) Tτ(k;αβ) udτ − Tt Tt 0
for all t ≥ 0 and all u ∈ H(k) . From Assumption 2(ii) and the definition of M(k) , we then have for
all u ∈ H(k) and all t ≥ 0 that
Z t (αβ) (αβ) (k;αβ) u ≤ − Tt Tt−τ L(αβ) − L(k;αβ) Tτ(k;αβ) u dτ Tt 0 Z t (αβ) (k;αβ) (αβ) (k;αβ) u dτ. PK(k) Tτ = −L Tt−τ PM(k) L
(10)
0
Now using the bounds (3) and (6) (established in Lemmas 1 and 2, respectively), and applying Assumption 2(i), we have that (αβ) (k;αβ) (αβ) (k;αβ) u PK(k) Tτ −L Tt−τ PM(k) L !(1−2−r ) !(1−2−s ) (k;αβ) (k;αβ) qa (k;αβ) qe ≤ qL (αβ) (αβ) γk γk ! −s " r−1 ! −i # (k;αβ) (1−2 ) X (k;αβ) (1−2 ) (αβ) qa q −i e ci e−2 γk (t−τ ) + (αβ) (αβ) γk γk i=0 −i # ! −r " s−1 ! (k;αβ) (1−2 ) (k;αβ) (1−2 ) X (αβ) q qe −i a ci e−2 γk τ + (αβ) (αβ) γk γk i=1 −i −j # " r−1 s−1 ! ! (k;αβ) (1−2 ) (k;αβ) (1−2 ) XX (αβ) −i −i −j qe qa ||u||. ci cj eγk (−2 t+(2 −2 )τ ) + (αβ) (αβ) γ γ i=0 j=0 k k
The result (8) then follows from substitution of the above identity into (10) and integration. This establishes the lemma statement. Let S′ ⊂ L2 ([0, T ]; Cm ) denote the set of all simple functions, which is dense in L2 ([0, T ]; Cm ).
That is, for any t ∈ [0, T ) and f ∈ S′ , there exists 0 < ℓ < ∞ and a sequence 0 = t0 < t1
0 and each fj ∈ L2 ([0, T ]; Cm ), there exists fj′ ∈ S′ such that ||fj − fj′ || < ǫ. Moreover, for f1′ , f2′ ∈ S′ , there exists 0 < ℓ < ∞ and 0 = t0 < t1 < · · · < tℓ < tℓ+1 = t, such that fj′ 1[ti ,ti+1 ) = α′j (i) for all i = 0, 1, . . . , ℓ, where α′j (0), α′j (1), . . . , α′j (ℓ) ∈ Cm . Suppose that u1 , u2 6= 0 (otherwise the corollary
statement becomes trivial), then for any ǫ > 0 we may choose f1′ , f2′ ∈ S′ (choosing f1′ first followed by f2′ ) such that
ǫ 6ku1 k ||e(f2 )|| ǫ ′ e(f2 ) − e(f2 ) < 6ku2 k ||e(f1′ )||
e(f1 ) − e(f1′ )
k1 . Since the theorem statement holds trivially when ψ = 0, this completes the proof.
4
Examples
Example 1 (Kerr-nonlinear optical cavity). Consider a single-mode Kerr-nonlinear optical cavity coupled to a single external coherent field (m = 1), which is used in the construction of the photonic logic gates presented in [12]. Let H = ℓ2 (the space of infinite complex-valued sequences with P∞ 2 n=1 |xn | < ∞) which has an orthonormal Fock state basis {|ni}n≥0 . On this basis {|ni}n≥0 , the
annihilation, creation, and number operators of the cavity oscillator can be defined (see, e.g., [14]) satisfying √
a∗ |ni =
√
a∗ a |ni = n |ni , √ respectively. The Kerr-nonlinear optical cavity can be described by S = I, L = λa, and H = a |ni =
n |n − 1i ,
n + 1 |n + 1i ,
∆a∗ a + χa∗ a∗ aa, where λ, ∆, χ > 0. Consider H(k) = span {|ni | n = 0, 1, 2, . . . , k} and a system approximation of the form S (k) = I,
L(k) = PH(k) LPH(k) ,
H (k) = PH(k) HPH(k) .
(18)
For any α, β ∈ C, L(αβ) u is defined and finite for all u ∈ H(k) . Hence, Assumption 1 holds. Recall that M(k) = Range L(α,β) − L(k;α,β) H(k) . From the (S, L, H), we see that M(k) = (αβ) span{|k + 1i}. Now consider K(k) = span {|ki} and γk = 21 λk + |α − β|2 . We will now show
that Assumptions 2 and 3 hold for the Kerr-nonlinear optical cavity and the approximation. 14
Assumption 2(i)
Note that L(αβ) − L(k;αβ)
K
√ ∗ = β λ a |K(k) . (k)
(k;αβ)
Thus, we have that Assumption 2(i) holds with qL
=
p λ(k + 1)|β|.
This assumption follows for the defined K(k) because L(αβ) − L(k;αβ) H(k) = span {|ni | n = 0, 1, 2, . . . , k − 1} for all α, β ∈ C.
Assumption 2(ii) ker
∗
Assumption 2(iii) First note that L(k) PK(k) = 0, PK(k) L(k) PH(k) = 0, and ∗ ∗ PK(k) 12 L(k) L(k) − ıH PH(k) ⊖K(k) = 0. Also, for any u ∈ H(k) , PK(k) L(k) L(k) PK(k) u = λkPK(k) u.
From these identities and the fact that H (k) is self-adjoint, we have for any u ∈ H(k) that ℜ{hPK(k) L(k;αβ) u, PK(k) ui} = ℜ{hPK(k) L(k;αβ) PK(k) + PH(k) ⊖K(k) u, PK(k) ui} 1 = − λk + |α|2 + |β|2 − 2ℜ{α∗ β} ||PK(k) u||2 2 nD Eo √ λβPK(k) a∗ PH(k) ⊖K(k) u, PK(k) u +ℜ = −g(k, α, β) ||PK(k) u||2 + h(k, α, β, u).
Noticing that |α|2 + |β|2 − 2ℜ{α∗ β} = hα, αi + hβ, βi − hα, βi − hα, βi = |α − β|2 . Also, note that nD Eo √ √ ∗ ℜ λβPK(k) a PH(k) ⊖K(k) u, PK(k) u ≤ |β| λ PK(k) a∗ PH(k) ⊖K(k) u ||PK(k) u|| √ ≤ |β| λk||u|| ||PK(k) u|| .
(αβ)
Therefore, we have that Assumption 2(iii) holds for the defined K(k) and the defined γk √ (k;αβ) qa = |β| λk. Assumption 2(iv) that
For any u ∈ H, HPM(k) u = (∆(k + 1) + χk(k + 1)) PM(k) u. This implies
(αβ) (αβ) ℜ{hTt (ıH)PM(k) u, Tt PM(k) ui}
Therefore, we have that
(αβ) (αβ) ℜ{hTt L(αβ) PM(k) u, Tt PM(k) ui}
= 0. Also, L∗ LPM(k) u = λ(k + 1)PM(k) u for any u ∈ H.
1 2 2 ∗ = − λ(k + 1) + |α| + |β| − 2ℜ{α β} 2 2 (αβ) × Tt PM(k) u nD Eo √ (αβ) (αβ) + ℜ Tt λ (βa∗ − α∗ a) PM(k) u, Tt PM(k) u 2 (αβ) ˆ k, α, β, u). = −ˆ g(k, α, β) Tt PM(k) u + h(t, (αβ)
is a contraction, we see that Assumption 2(iv) p p (k;αβ) with qe = |α| λ(k + 1) + |β| λ(k + 2).
Similarly to the previous derivation, using that Tt holds for the defined
with
K(k) and
the defined
(αβ) γk
15
Assumption 3
(αβ)
From the defined γk
(k;αβ)
, qL
(k;αβ)
, qa
(k;αβ)
, qe
, we see that this assumption holds
for any r, s ∈ Z+ such that r + s ≥ 3.
Finally, because Assumptions 1-3 hold, Lemma 4, Corollary 1, and Theorem 1 can be applied
to obtain error bounds on the finite-dimensional approximations. Example 2 (Atom-cavity model). Consider a three-level atom coupled to an optical cavity, which itself is coupled to a single external coherent field (m = 1) [11]. Let H = C3 ⊗ ℓ2 . We will use
|ei = (1, 0, 0)⊤ , |+i = (0, 1, 0)⊤ , and |−i = (0, 0, 1)⊤ to denote the canonical basis vectors in Cm .
We also consider |ni, denoting the normalized n-photon Fock state of the system, as the basis of ℓ2
(as in the previous example). This atom-cavity system is then described by the following parameters: S = I,
L=
√
H = ıχ (σ+ a − σ− a∗ ) ,
λa,
where λ, χ > 0, σ+ = |ei h+|, and σ− = |+i he|.
Consider H(k) = C3 ⊗span {|ni | n = 0, 1, 2, . . . , k} and a system approximation of the form (18).
That is, we are approximating the dynamics of the harmonic oscillator. For any α, β ∈ C, L(αβ) u is defined and finite for all u ∈ H(k) . Hence, Assumption 1 holds. Similar to the previous example, (αβ) = 21 λk + |α − β|2 . we see that M(k) = C3 ⊗ |k + 1i. Now consider K(k) = C3 ⊗ |ki and γk
Using similar derivation to the previous example, we will now show that Assumptions 2 and 3 hold for the atom-cavity model and the approximation. Assumption 2(i)
Note that L(αβ) − L(k;αβ)
K
√ = β a∗ |K(k) . λ + χσ − (k) (k;αβ)
Thus, we have that Assumption 2(i) holds with qL
=
√ √ k + 1 |β| λ + χ .
This assumption follows for the defined K(k) because L(αβ) − L(k;αβ) H(k) = C3 ⊗ span {|ni | n = 0, 1, 2, . . . , k − 1} for all α, β ∈ C.
Assumption 2(ii) ker
Assumption 2(iii)
∗
∗
Note that L(k) PK(k) = 0, PK(k) L(k) PH(k) = 0, PK(k) L(k) L(k) PH(k) ⊖K(k) = 0, ∗
and PK(k) HPH(k) ⊖K(k) = −ıχPK(k) σ− a∗ PH(k) ⊖K(k) . Also, for any u ∈ H(k) , PK(k) L(k) L(k) PK(k) u = λkPK(k) u. From these identities and the fact that H (k) is self-adjoint, we have for any u ∈ H(k)
16
that ℜ{hPK(k) L(k;αβ) u, PK(k) ui} = ℜ{hPK(k) L(k;αβ) PK(k) + PH(k) ⊖K(k) u, PK(k) ui} 1 = − λk + |α|2 + |β|2 − ℜ{α∗ β} ||PK(k) u||2 2 nD Eo √ + ℜ PK(k) β λ + χσ− a∗ PH(k) ⊖K(k) u, PK(k) u = −g(k, α, β) ||PK(k) u||2 + h(k, α, β, u).
Noticing that |α|2 + |β|2 − 2ℜ{α∗ β} = hα, αi + hβ, βi − hα, βi − hα, βi = |α − β|2 . Also, note that nD √ Eo ℜ PK(k) β λ + χσ− a∗ PH(k) ⊖K(k) u, PK(k) u √ ≤ PK(k) β λ + χσ− a∗ PH(k) ⊖K(k) u ||PK(k) u|| √ √ ≤ k χ + |β| λ ||u|| ||PK(k) u|| . (αβ)
Therefore, we have that Assumption 2(iii) holds for the defined K(k) and the defined γk √ √ (k;αβ) qa = k χ + |β| λ .
with
For any u ∈ H, note that L∗ LPM(k) u = λ(k + 1)PM(k) u and ıHPM(k) u =
Assumption 2(iv)
ı (∆(k + 1) + χk(k + 1)) PM(k) u. Also, Thus, we have that 2 1 (αβ) (αβ) (αβ) (αβ) 2 2 ∗ PM(k) u, Tt PM(k) ui} = − λ(k + 1) + |α| + |β| − 2ℜ{α β} Tt PM(k) u ℜ{hTt L 2 nD Eo √ (αβ) (αβ) + ℜ Tt λ (βa∗ − α∗ a) PM(k) u, Tt PM(k) u Eo nD (αβ) (αβ) + ℜ Tt χ (σ− a∗ − σ+ a) PM(k) u, Tt PM(k) u 2 (αβ) ˆ k, α, β, u). = −ˆ g (k, α, β) Tt PM(k) u + h(t, (αβ)
is a contraction, we see that Assumption 2(iv) √ √ √ √ = k + 1 χ + |α| λ + k + 2 χ + |β| λ .
Similar to the previous derivation, using that Tt holds for the defined Assumption 3
K(k)
and
(αβ) γk
with (αβ)
From the defined γk
(k;αβ) qe (k;αβ)
, qL
(k;αβ)
, qa
(k;αβ)
, qe
, we see that this assumption holds
for any r, s ∈ Z+ such that r + s ≥ 3.
Finally, because Assumptions 1-3 hold, Lemma 4, Corollary 1, and Theorem 1 can be applied
to obtain error bounds on the finite-dimensional approximations.
5
Conclusion
This paper has presented error bounds for finite-dimensional approximations of input-output models of open quantum systems with an infinite-dimensional underlying Hilbert space (and with possibly 17
unbounded coefficients in their QSDEs). The bounds are established on the basis of the contractive property of an open quantum system and some constructions employed in [14]. Under certain conditions, we show that this bound vanishes in the limit as the dimension of the approximating finite-dimensional subspace increases. We also applied these results to some sample physical systems taken from the literature.
References [1] R. L. Hudson and K. R. Parthasarathy, “Quantum Ito’s formula and stochastic evolution,” Commun. Math. Phys., vol. 93, pp. 301–323, 1984. [2] C. Gardiner and M. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A, vol. 31, pp. 3761–3774, 1985. [3] H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control. Cambridge University Press, 2010. [4] V. P. Belavkin and S. C. Edwards, “Quantum filtering and optimal control,” in Quantum Stochastics and Information: Statistics, Filtering and Control (University of Nottingham, UK, 15 - 22 July 2006), V. P. Belavkin and M. Guta, Eds. Singapore: World Scientific, 2008, pp. 143–205. [5] H. I. Nurdin, M. R. James, and A. C. Doherty, “Network synthesis of linear dynamical quantum stochastic systems,” SIAM J. Control Optim., vol. 48, no. 4, pp. 2686–2718, 2009. [6] L. Bouten, R. van Handel, and M. R. James, “An introduction to quantum filtering,” SIAM J. Control Optim., vol. 46, pp. 2199–2241, 2007. [7] L. Bouten and R. van Handel, “On the separation principle of quantum control,” in Quantum Stochastics and Information: Statistics, Filtering and Control (University of Nottingham, UK, 15 - 22 July 2006), V. P. Belavkin and M. Guta, Eds. Singapore: World Scientific, 2008, pp. 206–238. [8] M. R. James, H. I. Nurdin, and I. R. Petersen, “H ∞ control of linear quantum stochastic systems,” IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1787–1803, 2008. [9] H. I. Nurdin, M. R. James, and I. R. Petersen, “Coherent quantum LQG control,” Automatica, vol. 45, pp. 1837–1846, 2009. [10] J. Kerckhoff, H. I. Nurdin, D. Pavlichin, and H. Mabuchi, “Designing quantum memories with embedded control: photonic circuits for autonomous quantum error correction,” Phys. Rev. Lett., vol. 105, pp. 040 502–1–040 502–4, 2010. [11] L. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interaction,” Phys. Rev. Lett., vol. 92, pp. 127 902–1–127 902–4, 2004. [12] H. Mabuchi, “Nonlinear interferometry approach to photonic sequential logic,” Appl. Phys. Lett., vol. 99, pp. 153 103–1–153 103–3, 2011.
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[13] C. W. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, 3rd ed.
Berlin and New York: Springer-
Verlag, 2004. [14] L. Bouten, R. van Handel, and A. Silberfarb, “Approximation and limit theorems for quantum stochastic models with unbounded coefficients,” Journal of Functional Analysis, vol. 254, pp. 3123–3147, 2008. [15] P. A. Meyer, Quantum Probability for Probabilists, 2nd ed. Berlin-Heidelberg: Springer-Verlag, 1995. [16] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, ser. Text in Applied Mathematics. New York: Springer-Verlag, 1995, no. 21.
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