Dynamics and Quantum Channels

Transcription

1 Dynamics and Quantum Channels Konstantin Riedl Simon Mack 1 Dynamics and evolutions Discussing dynamics, one has to talk about time. In contrast to most other quantities, time is being treated classically. It is described by a parameter t, not by (self-adjoint) operators, like position, velocity, spin,... There is a special observable on H: the total energy of the system. It will be denoted by H and it will be called Hamiltonian of the system. Without exterior inuence it controls how the system evolves. For ψ(t) H the Schrödinger-equation describes this time development: i ψ(t) = H ψ(t) t Formally this dierential equation can be solved for time independent H by ψ(t) = e i Ht ψ(0) Since U := e i Ht is a unitary operator it is called time evolution operator (of a closed quantum system). As already noticed one has to distinguish two cases: the evolution of closed systems and open systems. With the rst kind of system the axiom of quantum theory will be reviewed and extended, to consider how to deal with an additional operation. Afterwards open systems will lead to quantum channels. 1.1 Closed quantum system Each experiment in quantum physics was divided into preparation and measurement. Now an operation T shall be inserted. preparation operation T measurement p(i M, ρ) As usual ρ is a positive trace-class operator with trace one, M a POVM and I a set of outcomes with I B(S). If no operation between preparation and measurement takes place, the probability measuring an outcome in I is given by Born's rule: p(i M, ρ) = tr(m(i)ρ) 1

2 2 1 DYNAMICS AND EVOLUTIONS Now there are three possibilities how T can be regarded in the general framework: as a part of a new preparation. (Schrödinger picture) ρ T M p(i M, ρ) = tr(m(i)ρ ) ρ := T (ρ) as a part of a new measurement. (Heisenberg picture) ρ T M p(i M, ρ) = tr(m (I)ρ) M := T M splitting it up between the two parts. (interaction picture) The Schrödinger picture is the most commonly used one in physics. Relationship between Schrödinger and Heisenberg picture Since observables A A B(H) are transformed via T : A T (A) A T (A) and states, respectively density operators ρ, via T : S 1 (H) S 1 (H) ρ T (ρ) the equation tr(t (A)ρ) = tr(at (ρ)) has to be satised for arbitrary observables and states. Closed quantum system A system, that does not exchange information, e.g. energy, with another system, is called a closed system. As described above the time evolution is given by a unitary operator. With unitaries acting on the Hilbert space H, the necessary prerequisite for the reversibility of the evolution is satised. 1.2 Open quantum systems Since most situations demand quantum systems interacting with another one 1, the belief that one is able to have access to both systems has to be abandoned. In general the second system is too large or too complicated to (be) describe(d) 2. This one will be called environment and denoted by K, whereas the rst system is named H and can be characterized completely. 1 or even more 2 Truly it is really dicult to prohibit interaction between a quantum system and environment. This leads to decoherence on the rst system.

3 3 States and observables in a joint system With the respective state spaces of two quantum systems being H and K, the state space of the joint system is the tensor product H K. If the system H is in the state ψ and K in φ, then H K is in the product state ψ φ. If X is an observable on H and Y one on K, then they can be extended to H K as X id K and id H Y. Based on this structure one can characterize a quantum channel. 2 Quantum channel A quantum channel is an important actor in both Quantum Information Theory and the theory of open quantum systems. It describes the most general physically reasonable transformation of a quantum state and it results from any kind of interaction with a quantum environment. In QIT possible transformations of a quantum system, after being transported via a nonperfect communication channel, are represented by quantum channels. 2.1 Setup Suppose two quantum systems with state spaces H and K are given. Since the Hamiltonian H tot on the joint system H K is self-adjoint, the evolution of the complete (closed) system is described by the unitary operator U = e i Htott. Assume the initial state is a product state ρ ω, where ρ is the state of the quantum system H and ω the state the environment K is in. Both systems shall evolve for a time t together and afterwards the focus will be on the state of the system H. Translating this most general transformation into mathematical language, the following formal denitions are needed: 2.2 Technicalities Partial trace with respect to a space Denition 2.1 (Ket and Bra with respect to a space) Let H and K be separable Hilbert spaces. Dene for f K the operators: and id H f K : H H K g g f id H K f : H K H u v f v u

4 4 2 QUANTUM CHANNEL These two operators are obviously adjoint 3 and bounded. Denition 2.2 (Partial trace with respect to a space) Let H and K be separable Hilbert spaces. For a trace-class operator σ S 1 (H K) the partial trace Tr K is dened for an arbitrary orthonormal basis (e i ) i I of K by Tr K : S 1 (H K) S 1 (H) σ i I id H K e i σ id H e i K This denition matches the formal denition, satisfying tr(σ(x id)) = tr(tr K (σ)x) X B(H). 2.3 Description of a quantum channel To describe a quantum channel, two dierent approaches shall follow. The rst one is more constructive. The main focus is on the evolution of the state, the joint system is in. The second one is the more elegant but more abstract one focussing on three physically meaningful demands: linearity, preservation of the trace and complete positivity. They are necessary to map states to states, even if the quantum channel only aects to a subsystem. This will lead to the denition of a quantum channel Constructive approach Joining the systems H and K, in the states ρ and ω respectively, yields the state ρ ω on H K. After the evolution U on H K has taken place, the joint system is in the state U(ρ ω)u on H K. Focussed on the state the system H is in after the evolution, the following expression ignores the environment: Tr K (U(ρ ω)u ) Thus one nally achieves a formula for a quantum channel : Formula 2.3 (Quantum channel) A quantum channel is a map T : S 1 (H) S 1 (H) of the form T (ρ) = Tr K (U(ρ ω)u ) for some environment K, some density operator ω on K and some unitary operator U on the tensor product H K. 3 For arbitrary x H and y := y 1 y 2 H K one can easily verify: ( f K x) y H K = x f y H K = x y 1 H f y 2 K and x ( K f y) H = x f y 2 K y 1 H = f y 2 K x y 1 H

5 2.4 Representation of a quantum channel Axiomatic approach Since the aim is to map states to states respectively density operators to density operators, independently of whether the quantum channel aects the whole system or only a part of it, the following three conditions are necessary to get a physically meaningful operation Linearity: T : S 1 (H) S 1 (H). ρ 1, ρ 2 S 1 (H); a, b C : T (aρ 1 + bρ 2 ) = at (ρ 1 ) + bt (ρ 2 ). This is actually a very fundamental prerequisite. Preservation of the trace: ρ S 1 (H) : tr(t (ρ)) = tr(ρ). T has to map density operators to density operators to be well-dened. Complete positivity: ρ S 1 (H) : T (ρ ρ) 0. (positivity) Positivity alone is not enough. 4 Namely: Considering a bipartite system, undergoing the transformation T id. This quantum channel should be positive again: n N : T id n is positive. (complete positivity) Denition 2.4 (Quantum channel) A map T : S 1 (H) S 1 (H), satisfying the three conditions linearity, preservation of the trace and complete positivity is called a quantum channel. Theorem 2.5 (Stinespring dilation theorem) Denition 2.4 and formula 2.3 are equivalent, i.e. for a map T : S 1 (H) S 1 (H) satisfying the three conditions above, there is a suitable Hilbert space K, a density operator ω and a unitary operator U for the formula Representation of a quantum channel Since it is unfavorable for describing a quantum channel, having to know K, ω and U in the formula 2.3, it is crucial that there is a particular representation which only uses ingredients coming from H. It is called the Kraus representation. Theorem 2.6 (Kraus representation theorem) Let H and K be separable Hilbert spaces, U a unitary operator on H K and ω a density operator on K. Dene the quantum channel 4 Transposition delivers a counterexample. T (ρ) = Tr K (U(ρ ω)u )

6 6 2 QUANTUM CHANNEL on S 1 (H). Then there exists an at most countable family (M i ) i I of bounded operators on H such that T (ρ) = i I M i ρm i for all ρ S 1 (H). Furthermore the equation i I M i M i = id is fullled. Proof. At rst assume ω being pure (on K), so there is ψ K: ω = ψ ψ. Let (f i ) i I be a countable orthonormal basis of K. Then Tr K (U(ρ ψ ψ )U ) can be written as (id f i ) U (ρ ψ ψ ) U (id f i ). i I With the identity ρ ψ ψ = (id ψ ) ρ (id ψ ) it follows that: Tr K (U(ρ ψ ψ )U ) = i I (id f i ) U (id ψ ) ρ (id ψ ) U (id f i ). By dening M i := (id f i ) U (id ψ ) and its adjoint Mi = (id ψ ) U (id f i ) we achieve two bounded operators on H for each i. Finally we compute (id ψ ) U (id f i ) (id f i ) U (id ψ ) Mi M i = i I i I = (id ψ ) U U (id ψ ) = (id ψ ) id (id ψ ) = id H, since i I (id f i ) (id f i ) = id. Now consider ω being mixed. Thus it can be decomposed into ω = k I µ k ψ k ψ k, with (ψ k ) k I being an orthonormal basis of K. Dene Mi k := µ k (id f i ) U (id ψ k ) for all i, k K. Thus the adjoint operator is: = µ k (id ψ k ) U (id f i ). M k i Hence we can compute: Tr K (U(ρ ω)u ) = (id f i ) U (ρ ω) U (id f i ) i I = ( ) (id f i ) U (ρ µ k ψ k ψ k ) U (id f i ) i I k I = ( ) (id f i ) U µ k (id ψ k ) ρ (id ψ k ) U (id f i ) i I k I = µ k (id f i ) U (id ψ k ) ρ (id ψ k ) U (id f i ) i I k I = Mi k M k i i I i I k I M i k Mi k k I can be determined in a completely analogous way like above.

7 2.4 Representation of a quantum channel 7 Example 2.7 (Depolarizing Channel) H = C 2 describes the simplest quantum system, called qubit, a two-state quantum-mechanical system. Its two states are denoted by 0 and 1. The channel that should be described is part of a so-called noisy channel. A qubit transmitted to someone else is aected by a non perfect transmission, the environment causes some perturbations. At the colloquial level, it can be spoken of noise. Typical defects, that occur to a quantum bit are bit-ip, phase-ip and both simultaneous. Let q = 1 p [0, 1] be the probability, the qubit is left unchanged and p/3 the one for each of these 3 transformations: 5 bit-ip: phase-ip: both: { { { 0 i 1 1 i 0 or equivalent ψ σ 1 ψ or equivalent ψ σ 3 ψ or equivalent ψ σ 2 ψ To represent this channel, a four-dimensional environment K is dened. Its orthonormal basis is denoted by { 0, 1, 2, 3 }. The original system H can be identied with the subspace H 0 of H K. Now the operator U acting on H K is given by: U( ψ 0 ) = 1 p ψ 0 + p 3 ( ) σ 1 ψ 1 + σ 3 ψ 2 + σ 2 ψ 3 Furthermore U is completed in an arbitrary way as a unitary operator. Hence the quantum channel T is given by T (ρ) = Tr K (U(ρ 0 0 )U ). A bit more illustrative is the Kraus representation 2.6: T (ρ) = 3 M k ρmk, k=0 with M 0 = 1 p σ 0, M 1 = p p p 3 σ 1, M 2 = 3 σ 2, M 3 = 3 σ 3. ( ) σ i describe the Pauli matrices: σ 0 = id, σ 1 =, σ = ( ) ( ) 0 i 1 0, σ i 0 3 = 0 1

8 8 3 (SEMI)GROUPS OF OPERATORS AND GENERATORS 3 (Semi)groups of Operators and Generators Denition 3.1 (Semigroup) A family of linear operators T t (t 0) on a nite Banach Space forms a one-parameter semigroup i 1. T t T s = T t+s, t, s 2. T 0 = I Denition 3.2 (Uniformly Continuous Semigroup) A one parameter semigroup T t is said to be uniformly continuous if the map t T t is continuous, this is, lim t s T t T s = 0, s. This kind of continuity is normally referred to as continuity in the uniform operator topology, and it is sucient to analyze the nite dimensional case. Lemma 3.3 If T t forms a uniformly continuous one-parameter semigroup, then the map t T t is dierentiable, and the derivative of T t is given by with L = dtt dt t=0 Proof. We dene the function V (t) as V (t) = dt t dt = LT t, t 0 T s ds, t 0 V (t) is dierentiable, since T t is uniformly continuous on t, and V (t) lim t t = lim t V (t) V (0) t = dv (t) dt Then there exists t ɛ 0 such that V (t ɛ ) is invertible, thus dv (t) dt = T 0 = I t=0 tɛ T t = V 1 (t ɛ )V (t ɛ )T t = V 1 (t ɛ ) T s T t ds = 0 tɛ = V 1 (t ɛ ) T s+t ds = V 1 (t ɛ ) 0 t+s implying that T t is dierentiable. The derivative is given by dt t dt = lim T t+h T t h 0 h t = T t. In particular T s ds = V 1 (t ɛ )[V (t + t ɛ ) V (t)] = T h I T t = LT t h

9 9 Lemma 3.4 The exponential operator T (t) = e Lt is the only solution of the dierential problem { dt (t) dt = LT (t), t R +, T (0) = I Proof. Clearly T (0) = I and because of the commutativity of the exponents we calculate which gives us de Lt dt = lim h 0 e L(h+t) e Lt h e Lh I lim h 0 h ( = lim h 0 = lim h 0 I + Lh + O(h 2 ) I h ) e Lh I e Lt h At this point, it just remains to prove uniqueness. Let us consider another function S(t) and assume that it also satises the dierential problem. Then we dene Q(s) = T (s)s(t s), t s 0 for some xed t, so Q(s) is dierentiable with derivative = L Q(s) ds = LT (s)s(t s) T (t)ls(t s) = LT (s)s(t s) LT (t)s(t s) = 0 Thus Q(s) is a constant function, so Q(s) = Q(0) = T (t) = T (t)s(t t) = Q(t) = Q(0) = S(t) Thus any uniformly continuous semigroup can be written in the form T t = T (t) = e Lt, whereas L is called the generator of the semigroup and is the only solution to the dierential Problem d dt T t = LT t. Given the omnipresence of dierential equations describing time evolution in physics, it is now evident why semigroups are important. If we can extend the domain of t to negative values, then T t forms a one-parameter group, whose inverses are given by T t, so that T t T t = 1 for all t R. Theorem 3.5 (Stone's Theorem) Let U t be a uniformly continuous one parameter unitary group (on a nite Hilbert Space), then there exists an unique self-adjoint operator A, such that U t = e ita t R Conversely any self-adjoint Operator A generates a strongly continuous one parameter unitary group via U t = e ita Since we are mostly interested in unitary evolution, this theorem is quiet useful in Quantum mechanics.

10 10 4 SCHRÖDINGER EQUATION AND UNITARY TIME EVOLUTION To deal with time-inhomogenous dierential problems such as { dx dt = L(t)x, x B, x(t 0 ) = x 0 one has to introduce evolution families, here given by the two parameter family of operators T (t,s), fullling the (necessary) properties T (t,s) = T (t,r) T (r,s) t r s T (s,s) = I In contrast to general semigroups, continuity doesn't imply dierentiability, and the expression in terms of the generator L(t) is more complicated: Theorem 3.6 Let the generator L(t ) be bounded in [t, s], then the (dierentiable)evolution family T (t,s), which solves the problem { dt(t,s) dt = L(t)T (t,s), t s, T (s,s) = I is given by the Dyson Series: T (t, s) = I + m=1 t t1 s s tm 1... L(t 1 )... L(t m )dt m... dt 1 s 4 Schrödinger Equation and Unitary time evolution 4.1 Schrödinger Equation The Schrödinger equation describes the behavior of an isolated or closed quantum system, that is, by denition, a system which does not interchanges information (i.e. energy and/or matter) with another system. So if our isolated system is in some pure state ψ(t) H at time t, where H denotes the Hilbert Space of the system, the evolution under the action of some (possibly timedependent) Hamiltonian H(t), is given by (we set = 1): ψ(t) = ih(t) ψ(t) t An important property of Schrödinger equation is, that it does not change the norm of the states, indeed: t ψ(t) ψ(t) = i ψ(t) H (t) ψ(t) i ψ(t) H(t) ψ(t) = 0 Thus the solutions are given by an ûnitary evolution family ψ(t) = U(t, t 0 ) ψ(t 0 )

11 4.2 Von Neumann Equation 11 where the unitary time-evolution operator is given in terms of the Dyson Series. In the case that the Hamiltonian is timeindependent, H(t) = H, it can be reduced to the far simpler form U(t, t 0 ) = exp[ ih(t t 0 )] 4.2 Von Neumann Equation We may alternatively characterise the system state through the density operator, which in the present case of a pure state is given simply by Dierentiating this equation yields ρ(t) = ψ(t) ψ(t) t ρ(t) = ( t ψ(t) ) ψ(t) + ψ(t) ( t ψ(t) ) which gives use, using the Schrödinger equation: ρ = i(hρ ρh) = i[h, ρ] t This is known as the Von Neumann Equation (or also, due to the parallelity in classical mechanics, the Quantum Liouville equation ) The equation also holds for mixed states ρ(t) = k p k ψ k (t) ψ k (t) Using the solution for the Schrödinger equation, we can write the time evolution for some densitiy operator ρ(t) as ρ(t) = k U(t, t 0 )p k ψ k (t 0 ) ψ k (t 0 ) U (t, t 0 ) = U(t, t 0 )ρ(t 0 )U (t, t 0 ) in terms of the unitary time-evolution operator U(t, t 0 ) for the closed system, thus giving as a formal solution for the Von-Neumann equation. We also see that natural time evolution forms a Quantum Channel, due to its unitarity. Lemma 4.1 (purity Preserving) Purity is invariant under unitary time evolution. Or in other words, pure states reamin pure, and mixed states remain mixed. Proof. T r(ρ 2 (t)) = T r(ρ(t)ρ(t)) = T r(u(t, t 0 )ρ(t 0 )U (t, t 0 )U(t, t 0 )ρ(t 0 )U (t, t 0 )) = = T r(u (t, t 0 )U(t, t 0 )ρ(t 0 )ρ(t 0 )) = T r(ρ 2 (t 0 ))

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