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 ρ t = L ( t ) ρ t . 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 Introduction 1 Reducible Systems 2 Quantum Operations 3 Decoherence-Free Subspaces 4 Algebraic Criterion for the Existence of a DFS 5 Summary 6
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”. Φ → ρ ′ − − − − − − − − − 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 C n : d dt x ( t ) = L ( t ) x ( t ) , where L : R + 0 × M n ( C ) → M n ( 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 1 ( t ) � L ( t ) = P − 1 P . L 2 ( t ) Then, the linear equation is reduced to two independent equations: � d d dt ( P x ( t )) 1 = L 1 ( t ) ( P x ( t )) 1 dt x ( t ) = L ( t ) x ( t ) = ⇒ . d dt ( P x ( t )) 2 = L 2 ( t ) ( P x ( 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 t 1 ≥ 0 such that L 1 = L ( t 1 ) � = O . If there is some time-dependent scalar function α 1 ( t ) such that L ( t ) = α 1 ( t ) L 1 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 t 2 ≥ 0 such that L 2 = L ( t 2 ) � = β L 1 for any scalar β ∈ C . If L ( t ) = α 1 ( t ) L 1 + α 2 ( t ) L 2 for some α 2 ( t ) , then the process ends. If no such a function exists, then we go to the next step. 3 There is some t 3 ≥ 0 such that { L 1 , L 2 , L 3 } forms a linearly independent set ( L 3 = L ( t 3 ) ). If L ( t ) = α 1 ( t ) L 1 + α 2 ( t ) L 2 + α 3 ( t ) L 3 for some α 3 ( 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 { L 1 , . . . , L s } such that s � L ( t ) = α k ( t ) L k . k = 1 5 The algebra A is generated by L 1 , . . . , L s , i.e. A = A ( L 1 , . . . , L s ) .
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 L 1 , . . . , L s , 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 L 1 , . . . , L s ? One possible reducibility criterion is the “Shemesh criterion” and its generalisation: Theorem (Shemesh Criterion). Two matrices L 1 , L 2 ∈ M n ( C ) have a common eigenvector if and only if n − 1 � � � L k 1 , L ℓ N ( L 1 , L 2 ) = ker � = { 0 } , 2 k ,ℓ = 1 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 L 1 , . . . , L s ∈ M n ( C ) have a common eigenvector if and only if n − 1 � � � L k i s , L ℓ 1 1 · · · L k s 1 · · · L ℓ s N ( L 1 , . . . , L s ) = ker � = { 0 } , s k i ,ℓ i = 0 where the summation is taken so that � i k i � = 0 and � i ℓ i � = 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 − 1 � � ∗ � � � 1 , L ℓ 1 , L ℓ L k L k N ( L 1 , L 2 ) = ker 2 2 k ,ℓ = 1 � � ∗ � � � L k i L k i 1 · · · L k s s , L ℓ 1 1 · · · L ℓ s 1 · · · L k s s , L ℓ 1 1 · · · L ℓ s N ( L 1 , . . . , L s ) = ker . s s k i ,ℓ i
Introduction Reducible Systems Quantum Operations DFS Criterion for DFS Summary Shemesh Criterion An important property of the set N ( L 1 , . . . , L s ) is the following: N ( L 1 , . . . , L s ) is the “ maximal common invariant subspace on which L 1 , . . . , L s 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: � S ℓ ( X 1 , . . . , X ℓ ) = sgn ( σ ) X σ ( 1 ) · · · X σ ( ℓ ) , σ 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.
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