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Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

The Existence of Decoherence-Free Subspaces and an Effective Criterion

Takeo Kamizawa

Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Toruń, Poland

49th Symposium on Mathematical Physics, Toruń, 17 June 2017

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Overview

In this presentation, we will study: Criteria of the decompositions of the master equation d dt ρt = L (t) ρt. Quantum operations and channels. What a decoherence-free subspace is. Application of a decomposability criterion to the existence of a decoherence-free subspace.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Outline

1

Introduction

2

Reducible Systems

3

Quantum Operations

4

Decoherence-Free Subspaces

5

Algebraic Criterion for the Existence of a DFS

6

Summary

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Introduction

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Motivations

Any physical process can be represented as a quantum operation from an initial state to a final state. Another approach is based on the so-called “master equation”. ρ Φ − − − − − − − − − → ρ′

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Motivations

If a quantum system is well-prepared (isolated from the environment), the time-evolution of a microscopic object undergoes the system unitary dynamics. ❞Closed System.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Motivations

However, perfect preparations of experiments in laboratories are usually difficult and the influence from outside can affect the time-evolution. ❞Open System.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Motivations

For instance, during the quantum information transmissions, because of the environmental noise, there is a possibility for the information to be lost.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Motivations

The noise effect can break the system coherence. ❞Decoherence

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Motivations

The perfect protection of the system is difficult... However, In some cases, we are able to protect a “part” of the system from the environmental noise! If such a protected part has a linear structure, it is called a “decoherence-free subspace” (DFS).

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Motivations

The existence of a DFS is desired for applications. However, immediately questions arise: When does a system have a DFS? How can we test the existence?

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Reducible Systems

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Reducible Linear Systems

Consider a linear differential equation on Cn: d dt x (t) = L (t) x (t) , where L : R+

0 × Mn (C) → Mn (C) is a time-dependent generator.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Reducible Linear Systems

If the generator is constant L (t) = L, the solution is x (t) = exp (Lt) x (0) . However, if the generator is time-dependent, there is no general method to compute the solution. In addition, even if the equation has the solution in a closed form, the computation of the solution can be difficult if the dimension n is large.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Reducible Linear Systems

Suppose there is a change of basis P such that the generator L (t) is brought to a block-diagonal form: L (t) = P−1 L1 (t) L2 (t)

• P.

Then, the linear equation is reduced to two independent equations: d dt x (t) = L (t) x (t) = ⇒

• d

dt (Px (t))1 = L1 (t) (Px (t))1 d dt (Px (t))2 = L2 (t) (Px (t))2

. The dimensions of those equations are smaller than n, so the dynamics becomes simpler and we may be able to compute the closed form of the solution of the reduced equations.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Reducible Linear Systems

In this way, if we can transform the system d dt x (t) = L (t) x (t) into an equivalent set of subsystems of lower dimensions, then the starting problem significantly simplifies.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Reducible Linear Systems

Here another question arises: When is it possible to reduce the generator L (t) into a block-diagonal form? The block-diagonal reducibility can be tested by analysing the algebra A generated by L (t).

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Reducible Linear Systems

The algebra A can be constructed as follows:

1 There is some t1 ≥ 0 such that L1 = L(t1) = O.

If there is some time-dependent scalar function α1(t) such that L(t) = α1(t)L1 for all t ≥ 0, then the process ends. If no such a function exists, then we go to the next step.

2 There is some t2 ≥ 0 such that L2 = L(t2) = βL1 for any

scalar β ∈ C.

If L(t) = α1(t)L1 + α2(t)L2 for some α2(t), then the process ends. If no such a function exists, then we go to the next step.

3 There is some t3 ≥ 0 such that {L1, L2, L3} forms a linearly

independent set (L3 = L(t3)).

If L(t) = α1(t)L1 + α2(t)L2 + α3(t)L3 for some α3(t), then the process ends. If no such a function exists, then we go to the next step.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Reducible Linear Systems

4 This process continues and finishes in a finite number of steps.

We obtain a linearly independent set {L1, . . . , Ls} such that L (t) =

s

• k=1

αk (t) Lk.

5 The algebra A is generated by L1, . . . , Ls, i.e.

A = A (L1, . . . , Ls).

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Shemesh Criterion

A possible invariant subspace is an eigenspace. If there is a “common eigenvector” for L1, . . . , Ls, it forms an invariant subspace for L (t) for all t ≥ 0. If such a common eigenvector exists, the linear equation reduced on the eigenspace is a one-dimensional equation: d dt y (t) = a (t) y (t) , where the structure became much simpler than the original n-dimensional equation.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Shemesh Criterion

However, how can we test the existence of a common eigenvector for L1, . . . , Ls? One possible reducibility criterion is the “Shemesh criterion” and its generalisation: Theorem (Shemesh Criterion). Two matrices L1, L2 ∈ Mn (C) have a common eigenvector if and only if N (L1, L2) =

n−1

• k,ℓ=1

ker

• Lk

1, Lℓ 2

• = {0} ,

where [X, Y ] = XY − YX is the commutator.

• D. Shemesh. Lin. Alg. Appl. 62 (1984) 11-18.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Shemesh Criterion

A generalisation of the Shemesh criterion was given as follows: Theorem (Generalised Shemesh Criterion). Matrices L1, . . . , Ls ∈ Mn (C) have a common eigenvector if and only if N (L1, . . . , Ls) =

n−1

• ki,ℓi=0

ker

• Lki

1 · · · Lks s , Lℓ1 1 · · · Lℓs s

• = {0} ,

where the summation is taken so that

i ki = 0 and i ℓi = 0.

Thus, if the condition above is satisfied, the original differential equation has a one-dimensional reduced equation.

• A. Jamiołkowski, G. Pastuszak. Lin. Multilinear Alg. 63 (2015) 314-325.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Shemesh Criterion

One of the most important property of the Shemesh criterion is that it is an “effective” method because N (L1, L2) = ker

n−1

• k,ℓ=1
• Lk

1, Lℓ 2

∗ Lk

1, Lℓ 2

• N (L1, . . . , Ls) = ker
• ki,ℓi
• Lki

1 · · · Lks s , Lℓ1 1 · · · Lℓs s

∗ Lki

1 · · · Lks s , Lℓ1 1 · · · Lℓs s

• .

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Shemesh Criterion

An important property of the set N (L1, . . . , Ls) is the following: N (L1, . . . , Ls) is the “maximal common invariant subspace on which L1, . . . , Ls commute”. This property is very important and N can be constructed in effective ways.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

ALS-Criterion

Another decomposability criterion is the Amitsur-Levitzki-Shapiro criterion (ALS-criterion). The standard polynomial is the polynomial: Sℓ (X1, . . . , Xℓ) =

• σ

sgn (σ) Xσ(1) · · · Xσ(ℓ), where σ is some permutations.

S.A. Amitsur, J.A. Levitzki. Proc. AMS 1 (1950) 449-463.

• H. Shapiro. Lin. Alg. Appl. 25 (1979) 129-137.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

ALS-Criterion

The decomposability criterion for the matrices L1, . . . , Ls is given as follows: Theorem (ALS-criterion). The algebra A = A (L1, . . . , Ls) generated by the matrices L1, . . . , Ls is block-diagonal reducible up to p-dimensional subalgebras if and only if S2p = 0 is satisfied on the algebra A.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

ALS-Criterion

In the matrix representations, if S2p = 0 is satisfied, the generator L (t) can be brought to a block-diagonal representation: L (t) = ⇒      B1 (t) B2 (t) ... Bξ (t)      , where the size of each block Bk (t) is at most p × p.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Discriminant of Algebra

The last decomposability criterion we introduce is the discriminant

• f algebra. Let A = A (L1, . . . , Ls) is the algebra generated by the

matrices L1, . . . , Ls and a basis of A is B = {B1, . . . , Bm}. Theorem (Discriminant). The algebra A is block-diagonal reducible if and

• nly if

det D = det    trB1B1 · · · trB1Bm . . . ... . . . trBmB1 · · · trBmBm    = 0.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Quantum Operations

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Kraus Representations

Our first mathematical approach for open quantum systems is the “quantum operator formalism”. Let HS be an n-dimensional Hilbert

• space. Since we require that a quantum operation

Φ : S (HS) → S (HS) is completely positive, the map Φ can be represented as Φ (ρ) =

s

• j=1

KjρK ∗

j ,

where Kj : S (HS) → S (HS) are constant operators satisfying

• j K ∗

j Kj ≤ I.

Note: The Kraus representation is NOT unique.

M.-D. Choi. Lin. Alg. Appl. 10 (1975) 285-290

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Kraus Representations

If, in addition, the condition

j K ∗ j Kj = I is satisfied, then the

quantum operation Φ (ρ) =

• j

KjρK ∗

j

also preserves the trace. In this case, Φ is said to be a quantum channel. Quantum channels are CPTP (Completely Positive and Trace-Preserving).

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Kraus Representations

If a quantum operation has an invariant subspace in B (HS), the structure of the evolution Φ becomes simpler. If the Kraus operators K1, . . . , Ks have a “common invariant subspace” K ⊂ B (HS), the Kraus operators can be represented by Kj = ⇒ Kj1 Kj2

• K

Kj3

• K⊥
• and if we assume that the initial state is from K, then the dynamics

Φ can be restricted only onto K.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Kraus Representations

In a similar manner, if there is a subspace HD ⊂ HS such that Φ (S (HD)) ⊂ S (HD) (i.e. the state space on HD are invariant under Φ), the Kraus operators are represented in block-triangular forms: Kj = ⇒

• Kj1

Kj2

• S(HD)

Kj3

• S(H⊥

D )

• .

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Kraus Representations

If HD satisfies additionally a certain condition, the Kraus operators are represented in block-diagonal forms: Kj = ⇒

• Kj1
• S(HD)

Kj3

• S(H⊥

D )

• .

This property is important for the existence of a decoherence-free subspace.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

DFS

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Definition of a DFS

A decoherence-free subspace is a subspace of the system Hilbert space HS which is free from the influence of the environment. Definition For a quantum operation Φ, a nonzero subspace HD ⊂ HS is called a decoherence-free subspace (DFS) if S (HD) is an invariant subset for Φ and the map Φ is unitary on S (HD).

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Definition of a DFS

For given Kraus operators K1, . . . , Ks of a quantum operation Φ (ρ) =

j KjρK ∗ j , a subspace HD ⊂ HS is said to be a common

invariant subspace (resp. common reducing subspace) if KjHD ⊂ HD (resp. KjHD ⊂ HD and K ∗

j HD ⊂ HD) for all

K1, . . . , Ks.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Definition of a DFS

In addition, HD is said to be a common reducing unitary subspace if HD ⊂ HS is a common reducing subspace and there is a common unitary operator U such that Kj|HD = gjU|HD for some gj ∈ C. This common reducing unitary subspace plays an important role for the DFS.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Example a DFS

Let us study a simple example: Let Φ (ρ) = K1ρK ∗

1 + K2ρK ∗ 2 ,

where K1 =   

1 2 i √ 6 1 √ 6

− i+

√ 3 2 √ 6 1−i √ 3 2 √ 6

   , K2 =   

i 2 i √ 3 1 √ 3

− i

√ 2+ √ 6 2 √ 6 √ 2−i √ 6 2 √ 6

   .

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Example of a DFS

In this case, the space HD = span      1   ,   1      forms a DFS because, for any |ϕ ∈ HD, we have Φ (|ϕ ϕ|) = U |ϕ′ ϕ′| U∗

• ,

where |ϕ′ is a restriction on HD and U =

• i

√ 2 1 √ 2

−

√ 3+i 2 √ 2 1−i √ 3 2 √ 2

• .

We can observe that any state on HD undergoes a unitary evolution.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Example of a DFS

Here, observe that HD is a common reducing unitary subspace for K1, K2 because K1 =

• 1

2 1 √ 3U

• , K2 =
• i

2

• 2

3U

• and

K ∗

1 K1 + K ∗ 2 K2 =

1

2

I

• .

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

A DFS Condition

We have the following statement: Theorem (Shabani, Lidar). A quantum operation

Φ (ρ) =

s

• j=1

KjρK ∗

j

has a DFS if and only if there are a unitary operator U and constants gj ∈ C such that the Kraus operators can be simultaneously block-diagonalised as

V ∗KjV = gjU Nj

• ,
• j

|gj|2 = 1,

where V is some constant unitary operator and Nj are some constant operators.

• A. Shabani, A. Lidar. Phys. Rev. A 72 (2005) 042303.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

A DFS Condition

However, the Kraus representations are not unique. The existence of a DFS may NOT be visible from the Kraus representations. When do we have a DFS? Is there a criterion?

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Criterion for DFS

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Criterion for the Existence of DFS

Let Φ (ρ) =

j KjρK ∗ j be a GIVEN quantum operation. We study

a condition for the quantum operation to have a DFS.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Criterion for the Existence of DFS

Theorem There is a DFS for a quantum channel Φ (ρ) =

j KjρK ∗ j if and

• nly if

N ≡ N (K1, . . . , Ks, K ∗

1 , . . . , K ∗ s )

=

• ki,ℓi

ker

• K k1

1 · · · K ks s K ∗ks+1 1

· · · K ∗k2s

s

, K ℓ1

1 · · · K ℓs s K ∗ℓs+1 1

· · · K ∗ℓ2s

s

• = {0} .

This criterion is “effective” because of the effectiveness of the (generalised) Shemesh criterion.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Criterion for the Existence of DFS

Idea of Proof: N is a (nonzero) common reducing subspace on which K1, . . . , Ks, K ∗

1 , . . . , K ∗ s commute.

N contains a common reducing eigenspace for all K ∗

j Kk, say

W. K ∗

j Kj are positive semi-definite and ∀w ∈ W, ∃αj ≥ 0 such

that K ∗

j Kjw = KjK ∗ j w = αjw.

For any rj ∈ C such that |rj|2 = 1

αj , the operator Uj = rjKj is

unitary. U1, . . . , Us are linearly dependent, so there is some common U such that Kjw = ηjUw for any w ∈ W (so W is a common reducing unitary subspace).

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Examples of the Application of the Criterion

Let us apply the criterion to the quantum channel Φ (ρ) = K1ρK ∗

1 + K2ρK ∗ 2 ,

where K1 =   

1 √ 2 i √ 6 1 √ 6

− i+

√ 3 2 √ 6 1−i √ 3 2 √ 6

   , K2 =   

i √ 2 i √ 3 1 √ 3

− i

√ 2+ √ 6 2 √ 6 √ 2−i √ 6 2 √ 6

   .

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Examples of the Application of the Criterion

First, compute K k1

1

=

• 2−k1/2

1

3

k1/2 Uk1

• , K k2

2

=  

• i

√ 2

k2 2

3

k2/2 Uk2   , where U =

• i

√ 2 1 √ 2

−

√ 3+i 2 √ 2 1−i √ 3 2 √ 2

• (unitary).

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Examples of the Application of the Criterion

Then, for any ki, ℓi, K k1

1 K k2 2 K ∗k3 1

K ∗k4

2

=  

• ik2(−i)k4

2(

ki)/2

• 2k2+k4

3

ki Uk1+k2−k3−k4

  and

• K k1

1 K k2 2 K ∗k3 1

K ∗k4

2

, K ℓ1

1 K ℓ2 2 K ∗ℓ3 1

K ∗ℓ4

2

• = O,

so N = C3. It means that any state undergoes a unitary evolution.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Examples of the Application of the Criterion

In fact, for any state ρ =   r11 r12 r13 r21 r31 R   we observe: Φ (ρ) =

• 1

√ 2 1 √ 3U

  r11 r12 r13 r21 r31 R  

• 1

√ 2 1 √ 3U∗

• +
• i

√ 2

• 2

3U

  r11 r12 r13 r21 r31 R  

• −i

√ 2

• 2

3U∗

• =

r11 URU∗

• .

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Examples of the Application of the Criterion

We will next study the depolarising channel: Φ (ρ) =

• 1 − 3p

4

• ρ + p

4

• k=x,y,z

σkρσ∗

k,

where p ∈ (0, 1) and σx, σy, σz are the Pauli matrices. In this case, since ker [σx, σy] = {0}, Ω

• I, σx, σy, σz, σ∗

x, σ∗ y, σ∗ z

• = {0} ,

so there is no DFS.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Examples of the Application of the Criterion

Next we consider another example. Let γk ∈ (0, 1) for k = 1, . . . , 5 with γ3 < γ4 < γ5 and

k γk = 1, and we consider a system such

that

1 with probability of γ1, level 1, 3 and 2, 4 are swapped, 2 with probability of γ2, level 3 decays to level 1 and 4 to 2, 3 with probability of γ3, level 5 decays to level 3 and 4, 4 with probability of γ4, level 6 decays to level 5, 5 with probability of γ5, level 7 decays to level 5.

G.I. Cirillo, T. Francesco. J. Phys. A 48 (2015) 085302.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Examples of the Application of the Criterion

In order to represent this system in matrix forms, set M1 = √γ1N1, M2 = √γ2N2, M3 = √γ3N3, M4 = √γ3N4, M5 = √γ3N5, M6 = √γ4N6 M7 = √γ4N7, M8 = √γ5N8, M9 = √γ5N9, where Nk ∈ M7 (C) are the matrices corresponding to the above

• rules. Assume the quantum channel on H = span {|j}7

j=1:

Φ (ρ) =

• k

MkρM∗

k.

In this case, we obtain N (M1, . . . , M9, M∗

1, . . . , M∗ 9) = {0} ,

so there exists a DFS.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Examples of the Application of the Criterion

In fact, we can check that HN = span                         1 . . .        ,          1 . . .                           ⊂ HS forms a DFS.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Summary

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Summary

A decoherence-free subspace (DFS) HD is an invariant subspace of HS on which the dynamics is unitary. The existence of a decoherence-free subspace requires the reducibility of HS. The reducibility of HS can be checked by several criteria: the ALS-criterion, the discriminant of algebra or the generalised Shemesh criterion.

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Summary

Using the Shemesh criterion, we are able to show the existence

• f the common reducing subspace.

The common reducing subspace for the Kraus operators contains a common reducing unitary subspace. The common reducing unitary subspace works as a DFS for a quantum channel. A necessary and sufficient condition for the existence of a DFS is that N = {0} with N = ker

• K k1

1 · · · K ks s K ∗ks+1 1

· · · K ∗k2s

s

, K ℓ1

1 · · · K ℓs s K ∗ℓs+1 1

· · · K ∗ℓ2s

s

• .

Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary

Reference

• A. Jamiołkowski, T. Kamizawa, G. Pastuszak. Int. J. Theor.
• Phys. 54 (2015) 2662-2674.
• G. Pastuszak, T. Kamizawa, A. Jamiołkowski. Open Syst.
• Infor. Dyn. 23 (2016) 1650003.