**4. Frames and fusion frames in stroboscopic tomography. Generalizations to subalgebras**

As we have seen the concepts of frames and fusion frames appear in stroboscopic tomography in natural way. The conclusion is based on the discussed above polynomial representations of semigroups which describe evolutions of open systems. The possibility to represent the semigroup Φ(*t*) = exp(*t* **L**) in the form

$$\Phi(t) = \sum\_{k=0}^{\mu - 1} a\_k(t) \mathbb{L}^k,\tag{52}$$

where *μ* stands for the degree of the minimal polynomial of the superoperator **L** and *αk*(*t*), *k* = 0..., *μ* − 1, denote some functions of the eigenvalues of **L** gives the equality (32) as a sufficient condition for stroboscopic tomography. On the other hand, this equality means that the Krylov subspaces K*μ*(**L**, *Qi*), *i* = 1, . . . ,*r*, constitute a fusion frame in the Hilbert-Schmidt space B∗(H) of all observables. Moreover, this also means that the collection of vectors

$$f\_{jk} := \mathbb{L}^k \mathbb{Q}\_{j\nu} \tag{53}$$

for *j* = 1, . . . ,*r* and *k* = 0, 1, . . . , *μ* − 1, constitute a frame in B∗(H) and the system in question is (*Q*1,..., *Qr*)-reconstructible. In this case every element *Q* of the space B∗(H) can be represented as

$$Q = \sum\_{j,k} \langle F^{-1} f\_{jk} | Q \rangle f\_{jk} = \sum\_{j,k} \langle f\_{jk} | Q \rangle F^{-1} f\_{jk} \tag{54}$$

where *F* denotes the frame operator of the collection of vectors (53). One can say even more. If *Q* ∈ B∗(H) also has another representation *Q* = ∑*j*,*<sup>k</sup> cjk fjk* for some scalar coefficients *cjk*,

Quantum Systems 15

Fusion Frames and Dynamics of Open Quantum Systems 81

whole algebra B(H). In other words, we want to determine whether every element in B(H) can be represented in the form *π*(*Q*1,..., *Qr*), where *π* is a polynomial in noncommutative

Let us observe that according to the *fundamental theorem* if A is a subalgebra of the full complex

If a set of generators of A is known, then the above inequality can be verified by a finite number of arithmetic operations. The procedures possessing such property are called *effective*. A very important example of an effective procedure can be formulated when we discuss the problem of the existence a common one-dimensional invariant subspace for a pair of operators *Q*1, *Q*2. In other words, we ask about a common eigenvector for two operators *Q*1, *Q*2. An answer to this question is given by the following procedure. Let the symbol [*Q*1, *Q*2] denote, as usual, the commutator of the operators *Q*1, *Q*2. Then a common eigenvector for *Q*<sup>1</sup> and *Q*<sup>2</sup>

dim A < dim B(H). (58)

<sup>2</sup>] , (59)

<sup>2</sup>], (61)

algebra B(H), then a nontrivial invariant subspace in H exists if and only if

K :=

*N* −1 *j*=1 *k*=1

Ker[*Q<sup>j</sup>*

where *N* = dim H, satisfies the condition dim K > 0 (this is the so-called Shemesh criterion

First of all, let us observe that if |*ψ*� is a common eigenvector of the operators *Q*<sup>1</sup> and *Q*2, i.e.,

dim K > 0 means that the gist of the Shemesh condition is in observation that the subspace K is invariant under *Q*<sup>1</sup> and *Q*2. Indeed, if |*ψ*� belongs to K, then by the definition of subspaces

and extend it to a basis in H. We then observe that there exists a nonsingular matrix *S* such that matrices *SQ*1*S*−<sup>1</sup> and *SQ*2*S*−<sup>1</sup> have block-triangular forms and the submatrices which correspond to subspace K commute. This means that these submatrices have a common eigenvector and therefore the same is true for *Q*<sup>1</sup> and *Q*2. D. Shemesh observed that the

<sup>2</sup>] one can check that *Q*1|*ψ*�∈K and *Q*2|*ψ*�∈K. Now, let us choose a basis for K

<sup>1</sup>, *<sup>Q</sup><sup>k</sup>*

*Q*1|*ψ*� = *α*|*ψ*� and *Q*2|*ψ*� = *β*|*ψ*�, (60)

<sup>2</sup>] for all *j*, *k* greater then 1. This fact and the inequality

exists if and only if the subspace K of H defined by

then <sup>|</sup>*ψ*� belongs to Ker[*Q<sup>j</sup>*

Ker[*Q<sup>j</sup>*

<sup>1</sup>, *<sup>Q</sup><sup>k</sup>*

(Shemesh, 1984)). A short proof of this condition is possible.

<sup>1</sup>, *<sup>Q</sup><sup>k</sup>*

condition dim K > 0 is equivalent to the singularity of the matrix

**M** :=

*N*−1 ∑ *j*=1 *k*=1

[*Qj* <sup>1</sup>, *<sup>Q</sup><sup>k</sup>* 2] <sup>∗</sup>[*Q<sup>j</sup>* <sup>1</sup>, *<sup>Q</sup><sup>k</sup>*

where \* denotes complex conjugate transpose. For our purposes, on the basis of Burnside's theorem, more interesting is the case when matrices *Q*1, *Q*<sup>2</sup> do not have common eigenvectors and the algebra *A*(*Q*1, *Q*2) generated by them coincides with B(H). This situation may be

variables.

*j* = 1, . . . ,*r* and *k* = 0, 1, . . . , *μ* − 1, then

$$\sum\_{j\mathbf{k}} |c\_{j\mathbf{k}}|^2 = \sum\_{j\mathbf{k}} |\langle F^{-1} f\_{j\mathbf{k}} | Q \rangle|^2 + \sum\_{j, \mathbf{k}} |c\_{j\mathbf{k}} - \langle F^{-1} f\_{j\mathbf{k}} | Q \rangle|^2. \tag{55}$$

It is obvious that every frame in finite-dimensional space contains a subset that is a basis. As a conclusion we can say that if { *fjk*} is a frame but not a basis, then there exists a set of scalars {*djk*} such that ∑*j*,*<sup>k</sup> djk fjk* = 0. Therefore, any fixed element *Q* of B∗(H) can also be represented as

$$Q = \sum\_{j,k} \left( \langle F^{-1} f\_{jk} | Q \rangle + d\_{jk} \right) f\_{jk}. \tag{56}$$

The above equality means that every *Q* ∈ B∗(H) has many representations as superpositions of elements from the set (53). But according to equality (55) among all scalar coefficients {*cjk*} for which

$$Q = \sum\_{j,k} c\_{jk} f\_{jk\prime} \tag{57}$$

the sequence {�*F*−<sup>1</sup> *fjk*|*Q*�} has minimal norm. This is a general method in frame theory (Christensen, 2008) and at the same time the main observation connected with the idea of stroboscopic tomography.

In conclusion, one can say that the Krylov subspaces K*μ*(**L**, *Qi*) in the space B∗(H) generated by the superoperator **L** can be used in an effective way for procedures of stroboscopic tomography if they constitute appropriate fusion frames in this space.

#### **4.1 Generalizations to subalgebras**

Now, we will discuss some problems of reconstruction of quantum states when the Krylov subspaces playing such important role in the stroboscopic tomography are replaced by some subalgebras of the Hilbert-Schmidt space B(H). Just as the fundamental theorem of algebra ensures that every linear operator acting on a finite dimensional complex Hilbert space has a nontrivial invariant subspace, the *fundamental theorem of noncommutative algebra* asserts the existence of invariant subspaces of H for some families of operators from B(H). It is an obvious observation that an algebra generated by any fixed operator Q and the identity on H can not be equal to B(H). This statement is based on the Hamilton-Cayley theorem. However, already for two operators *Q*1, *Q*<sup>2</sup> and the identity we can have Alg(*I*, *Q*1, *Q*2) =B(H) (for details cf. below).

In general, the famous Burnside's theorem states (cf. e.g. (Farenick, 2001)) that an operator algebra on a finite-dimensional vector space with no nontrivial subspaces must be the algebra of *all* linear operators. In the sequel we will use the following version of this theorem:

*Fundamental theorem of noncommutative algebras*. If A is a proper subalgebra of B(H) containing identity, and the dimension of the Hilbert space H is greater or equal to 2, then A has a proper nonzero invariant subspace in H (i.e., the subspace is invariant for all members Q of the algebra A).

We will apply the above theorem for the following problem. Given a set F = {*Q*1,..., *Qr*} of observables, we would like to establish conditions, when the operators *Q*1,..., *Qr* generate the 14 Will-be-set-by-IN-TECH

It is obvious that every frame in finite-dimensional space contains a subset that is a basis. As a conclusion we can say that if { *fjk*} is a frame but not a basis, then there exists a set of scalars {*djk*} such that ∑*j*,*<sup>k</sup> djk fjk* = 0. Therefore, any fixed element *Q* of B∗(H) can also be

�*F*−<sup>1</sup> *fjk*|*Q*� <sup>+</sup> *djk*

The above equality means that every *Q* ∈ B∗(H) has many representations as superpositions of elements from the set (53). But according to equality (55) among all scalar coefficients {*cjk*}

the sequence {�*F*−<sup>1</sup> *fjk*|*Q*�} has minimal norm. This is a general method in frame theory (Christensen, 2008) and at the same time the main observation connected with the idea of

In conclusion, one can say that the Krylov subspaces K*μ*(**L**, *Qi*) in the space B∗(H) generated by the superoperator **L** can be used in an effective way for procedures of stroboscopic

Now, we will discuss some problems of reconstruction of quantum states when the Krylov subspaces playing such important role in the stroboscopic tomography are replaced by some subalgebras of the Hilbert-Schmidt space B(H). Just as the fundamental theorem of algebra ensures that every linear operator acting on a finite dimensional complex Hilbert space has a nontrivial invariant subspace, the *fundamental theorem of noncommutative algebra* asserts the existence of invariant subspaces of H for some families of operators from B(H). It is an obvious observation that an algebra generated by any fixed operator Q and the identity on H can not be equal to B(H). This statement is based on the Hamilton-Cayley theorem. However, already for two operators *Q*1, *Q*<sup>2</sup> and the identity we can have Alg(*I*, *Q*1, *Q*2)

In general, the famous Burnside's theorem states (cf. e.g. (Farenick, 2001)) that an operator algebra on a finite-dimensional vector space with no nontrivial subspaces must be the algebra

*Fundamental theorem of noncommutative algebras*. If A is a proper subalgebra of B(H) containing identity, and the dimension of the Hilbert space H is greater or equal to 2, then A has a proper nonzero invariant subspace in H (i.e., the subspace is invariant for all members Q

We will apply the above theorem for the following problem. Given a set F = {*Q*1,..., *Qr*} of observables, we would like to establish conditions, when the operators *Q*1,..., *Qr* generate the

of *all* linear operators. In the sequel we will use the following version of this theorem:

*Q* = ∑ *j*,*k*

*j*,*k*

<sup>|</sup>*cjk* − �*F*−<sup>1</sup> *fjk*|*Q*�|2. (55)

*fjk*. (56)

*cjk fjk*, (57)


*j* = 1, . . . ,*r* and *k* = 0, 1, . . . , *μ* − 1, then

represented as

for which

stroboscopic tomography.

**4.1 Generalizations to subalgebras**

=B(H) (for details cf. below).

of the algebra A).

∑ *jk* |*cjk*|

<sup>2</sup> <sup>=</sup> ∑ *jk*

> *Q* = ∑ *j*,*k*

tomography if they constitute appropriate fusion frames in this space.

whole algebra B(H). In other words, we want to determine whether every element in B(H) can be represented in the form *π*(*Q*1,..., *Qr*), where *π* is a polynomial in noncommutative variables.

Let us observe that according to the *fundamental theorem* if A is a subalgebra of the full complex algebra B(H), then a nontrivial invariant subspace in H exists if and only if

$$
\dim \mathcal{A} < \dim \mathcal{B}(\mathcal{H}).\tag{58}
$$

If a set of generators of A is known, then the above inequality can be verified by a finite number of arithmetic operations. The procedures possessing such property are called *effective*. A very important example of an effective procedure can be formulated when we discuss the problem of the existence a common one-dimensional invariant subspace for a pair of operators *Q*1, *Q*2. In other words, we ask about a common eigenvector for two operators *Q*1, *Q*2. An answer to this question is given by the following procedure. Let the symbol [*Q*1, *Q*2] denote, as usual, the commutator of the operators *Q*1, *Q*2. Then a common eigenvector for *Q*<sup>1</sup> and *Q*<sup>2</sup> exists if and only if the subspace K of H defined by

$$\mathcal{K} := \bigcap\_{\substack{j=1 \\ k=1}}^{N-1} \text{Ker} [\mathbb{Q}\_{1'}^j, \mathbb{Q}\_2^k] \, , \tag{59}$$

where *N* = dim H, satisfies the condition dim K > 0 (this is the so-called Shemesh criterion (Shemesh, 1984)). A short proof of this condition is possible.

First of all, let us observe that if |*ψ*� is a common eigenvector of the operators *Q*<sup>1</sup> and *Q*2, i.e.,

$$Q\_1|\psi\rangle = \alpha|\psi\rangle \quad \text{and} \quad Q\_2|\psi\rangle = \beta|\psi\rangle,\tag{60}$$

then <sup>|</sup>*ψ*� belongs to Ker[*Q<sup>j</sup>* <sup>1</sup>, *<sup>Q</sup><sup>k</sup>* <sup>2</sup>] for all *j*, *k* greater then 1. This fact and the inequality dim K > 0 means that the gist of the Shemesh condition is in observation that the subspace K is invariant under *Q*<sup>1</sup> and *Q*2. Indeed, if |*ψ*� belongs to K, then by the definition of subspaces Ker[*Q<sup>j</sup>* <sup>1</sup>, *<sup>Q</sup><sup>k</sup>* <sup>2</sup>] one can check that *Q*1|*ψ*�∈K and *Q*2|*ψ*�∈K. Now, let us choose a basis for K and extend it to a basis in H. We then observe that there exists a nonsingular matrix *S* such that matrices *SQ*1*S*−<sup>1</sup> and *SQ*2*S*−<sup>1</sup> have block-triangular forms and the submatrices which correspond to subspace K commute. This means that these submatrices have a common eigenvector and therefore the same is true for *Q*<sup>1</sup> and *Q*2. D. Shemesh observed that the condition dim K > 0 is equivalent to the singularity of the matrix

$$\mathbf{M} := \sum\_{\substack{j=1\\k=1}}^{N-1} [\mathbb{Q}\_1^j, \mathbb{Q}\_2^k]^\* [\mathbb{Q}\_1^j, \mathbb{Q}\_2^k]\_{\prime} \tag{61}$$

where \* denotes complex conjugate transpose. For our purposes, on the basis of Burnside's theorem, more interesting is the case when matrices *Q*1, *Q*<sup>2</sup> do not have common eigenvectors and the algebra *A*(*Q*1, *Q*2) generated by them coincides with B(H). This situation may be

Quantum Systems 17

Fusion Frames and Dynamics of Open Quantum Systems 83

In the case of qutrits, that is for a three-dimensional Hilbert space, one can show by direct calculation that if [*Q*1, *Q*2] is invertible and *ω*([*Q*1, *Q*2]) �= 0, where for *Q* ∈ B(H) the symbol *ω*(*Q*) denotes the linear term in the characteristic polynomial of *Q*, then one can construct an explicit basis for B(H). Indeed, if *Q*1, *Q*<sup>2</sup> belong to B(H), and (dim H) = 3, then the determinant of the 9-dimensional matrix Ω build from vec transformations of

<sup>2</sup>, *Q*1*Q*2, *Q*2*Q*1, [*Q*1, [*Q*1, *Q*2]], [*Q*2, [*Q*2, *Q*1]] satisfies the equality

That is, if det([*Q*1, *Q*2]) �= 0 and *ω*(*Q*) �= 0, then the columns of the matrix Ω correspond to a

Of course, one can also use the Shemesh criterion to characterize pairs of generators for B(H),

Papers written by mathematicians are usually focused on characterization of various properties of discussed objects and search for necessary and sufficient conditions for desired conclusion to hold. Concrete constructions offen play a minor role. The problems of frames and fusion frames are no exceptions. The main purpose of this paper was to discuss properties of some Krylov subspaces in a given Hilbert space as a natural examples of fusion frames and

their applications in reconstruction of trajectories of open quantum systems.

Daubechies, I., Grossman. A., and Meyer, Y. (1986). *J. Math. Phys.* 27, 1271.

Farenick, D.R. (2001), *Algebras of Linear Transformations*, New York, Springer.

C. I. Byrnes and A. Lindquist, Amsterdam: Elsevier, p. 347.

Kovaˇcevi´c, J. and Chebira, A. (2008). *An Introduction to Frames*, Boston-Delft, NOW.

Jamiołkowski, A. (2010), *Journal of Physics: Conference Series* 213, 012002.

Hauseholder, A.S. (2009). *The Theory of Matrices in Numerical Analysis*, Dover Books. Heil, Ch. (Ed.) (2006). *Harmonic Analysis and Applications*, Boston, Birkhäuser.

Jamiołkowski, A. (1982). *On Observability of Classical and Quantum Stochastic Systems*, Toru ´n,

Jamiołkowski, A. (1986). *Frequency Domain and State Space Methods for Linear Systems*, Eds.

Casazza, P. G. & Kutyniok, G. (2004). Frames of subspaces, *Contemp. Math.* 345, 87.

Aslaksen, H. & Sletsjøe, A.B. (2009). *Lin. Algebra Appl.* 430, 1.

Casazza, P. G. et al. (2008). *Appl. Comput. Harmon. Anal.* 25, 114. Christensen, O. (2008). *Frames and Bases*, Boston, Birkhäuser.

Duffin, R. J. & Schaeffer, A. C. (1952). *Transactions AMS* 72, 341.

Fan, H., Hu, L. (2009) *Optics Comm.* 282, 932.

Gorini, V et al. (1976) *J. Math. Phys.* 17, 149.

Jamiołkowski, A. (1974). *Rep. Math. Phys.* 5, 415.

Jamiołkowski, A. (2000). *Rep. Math. Phys.* 46, 469.

Kossakowski, A. (1972) *Rep.Math. Phys.* 3, 247.

N. Copernicus Univ. Press, (in Polish). Jamiołkowski, A. (1983). *Internat. J. Theoret. Phys.* 22, 369.

det Ω = 9 det([*Q*1, *Q*2])*ω*([*Q*1, *Q*2]). (67)

**I**, *Q*1, *Q*2, *Q*<sup>2</sup>

basis for B(H).

where dim H = 3.

**5. Conclusions**

**6. References**

<sup>1</sup>, *<sup>Q</sup>*<sup>2</sup>

expressed by the following inequality

$$\det \mathbb{M} > 0,\tag{62}$$

which can be checked by an effective procedure, that is, by a finite number of arithmetic operations. It is obvious, that the matrix **M** is in general semipositive definite, and the above condition means the strict positivity of **M**.

#### **4.2 Examples**

In order to illustrate algebraic methods in reconstruction problems, we will discuss some algebraic procedures in low dimensional cases. For quantum systems of qubits and qutrits one can formulate an explicit form of some conditions in a matrix form which is sometimes more transparent then the general operator form. We will use the so-called *vec operator* procedure which transforms a matrix into a vector by stacking its columns one underneath the other. It is well known, that the tensor product of matrices and the vec operator are intimately connected. If *<sup>A</sup>* denotes a *<sup>N</sup>* <sup>×</sup> *<sup>N</sup>* matrix and *aj* its *<sup>j</sup>*-th column, then vec *<sup>A</sup>* is the *<sup>N</sup>*2-dimensional vector constructed from *a*1,..., *aN*. Moreover if *A*, *B*, *C* are three matrices such that the matrix product *ABC* is well defined, then

$$\text{vec}(ABC) = (\mathbb{C}^T \otimes A)\text{vec}\,B.\tag{63}$$

In the above formula *C<sup>T</sup>* denotes the transposition of the matrix *C*. In particular we have

$$\text{vec}\,A = (\mathbb{I} \otimes A)\,\text{vec}\,\mathbb{I} = (A^T \otimes \mathbb{I})\,\text{vec}\,\mathbb{I}.\tag{64}$$

Let us agree that when we say that a set of matrices generates the set B(H), we are thinking about B(H) as an algebra, while when we say that a set of matrices forms a basis for B(H), we are talking about B(H) as a vector space (here we identify B(H) with the set of all matrices on <sup>H</sup> <sup>=</sup> **<sup>C</sup>***N*).

For qubits, that is for two-dimensional Hilbert space, one can show by a direct computation that

$$\det(\text{vec}\,\mathbb{I}, \text{vec}\,Q\_1, \text{vec}\,Q\_2, \text{vec}(Q\_1Q\_2)) = \det([Q\_1, Q\_2])\tag{65}$$

and

$$\det(\text{vec}\,\mathbb{I}, \text{vec}\,Q\_1, \text{vec}\,Q\_2, \text{vec}[Q\_1, Q\_2]) = 2\det([Q\_1, Q\_2]),\tag{66}$$

where on the left hand side we have the determinants of the 4 × 4 matrices and on the right hand sides [*Q*1, *Q*2] denotes the commutator of the two 2 × 2 matrices.

From the last equality it follows, that if matrices **I**, *Q*1, *Q*<sup>2</sup> and [*Q*1, *Q*2] are linearly independent, then the algebra which is spanned by them has the dimension 4, so *Q*1, *Q*<sup>2</sup> and **I** generate B(H). In other words, two operators *Q*1, *Q*<sup>2</sup> and the identity generate B(H) if and only if the matrix [*Q*1, *Q*2] has the determinant different from zero. In a similar way one can show that the matrices *Q*1, *Q*2, *Q*3, such that *no* two of them generate B(H), can generate B(H) if and only if the double commutator [*Q*1, [*Q*2, *Q*3]] is invertible. In general, the matrices *Q*1,..., *Qr* generate B(H) iff at least one of the commutators [*Qi*, *Qj*] or double commutators [*Qi*, [*Qj*, *Qk*]] is invertible (Aslaksen & Sletsjøe, 2009).

In the case of qutrits, that is for a three-dimensional Hilbert space, one can show by direct calculation that if [*Q*1, *Q*2] is invertible and *ω*([*Q*1, *Q*2]) �= 0, where for *Q* ∈ B(H) the symbol *ω*(*Q*) denotes the linear term in the characteristic polynomial of *Q*, then one can construct an explicit basis for B(H). Indeed, if *Q*1, *Q*<sup>2</sup> belong to B(H), and (dim H) = 3, then the determinant of the 9-dimensional matrix Ω build from vec transformations of **I**, *Q*1, *Q*2, *Q*<sup>2</sup> <sup>1</sup>, *<sup>Q</sup>*<sup>2</sup> <sup>2</sup>, *Q*1*Q*2, *Q*2*Q*1, [*Q*1, [*Q*1, *Q*2]], [*Q*2, [*Q*2, *Q*1]] satisfies the equality

$$\det \Omega = 9 \det([Q\_1, Q\_2]) \omega([Q\_1, Q\_2]). \tag{67}$$

That is, if det([*Q*1, *Q*2]) �= 0 and *ω*(*Q*) �= 0, then the columns of the matrix Ω correspond to a basis for B(H).

Of course, one can also use the Shemesh criterion to characterize pairs of generators for B(H), where dim H = 3.

## **5. Conclusions**

16 Will-be-set-by-IN-TECH

which can be checked by an effective procedure, that is, by a finite number of arithmetic operations. It is obvious, that the matrix **M** is in general semipositive definite, and the above

In order to illustrate algebraic methods in reconstruction problems, we will discuss some algebraic procedures in low dimensional cases. For quantum systems of qubits and qutrits one can formulate an explicit form of some conditions in a matrix form which is sometimes more transparent then the general operator form. We will use the so-called *vec operator* procedure which transforms a matrix into a vector by stacking its columns one underneath the other. It is well known, that the tensor product of matrices and the vec operator are intimately connected. If *<sup>A</sup>* denotes a *<sup>N</sup>* <sup>×</sup> *<sup>N</sup>* matrix and *aj* its *<sup>j</sup>*-th column, then vec *<sup>A</sup>* is the *<sup>N</sup>*2-dimensional vector constructed from *a*1,..., *aN*. Moreover if *A*, *B*, *C* are three matrices such that the matrix

In the above formula *C<sup>T</sup>* denotes the transposition of the matrix *C*. In particular we have

Let us agree that when we say that a set of matrices generates the set B(H), we are thinking about B(H) as an algebra, while when we say that a set of matrices forms a basis for B(H), we are talking about B(H) as a vector space (here we identify B(H) with the set of all matrices

For qubits, that is for two-dimensional Hilbert space, one can show by a direct computation

where on the left hand side we have the determinants of the 4 × 4 matrices and on the right

From the last equality it follows, that if matrices **I**, *Q*1, *Q*<sup>2</sup> and [*Q*1, *Q*2] are linearly independent, then the algebra which is spanned by them has the dimension 4, so *Q*1, *Q*<sup>2</sup> and **I** generate B(H). In other words, two operators *Q*1, *Q*<sup>2</sup> and the identity generate B(H) if and only if the matrix [*Q*1, *Q*2] has the determinant different from zero. In a similar way one can show that the matrices *Q*1, *Q*2, *Q*3, such that *no* two of them generate B(H), can generate B(H) if and only if the double commutator [*Q*1, [*Q*2, *Q*3]] is invertible. In general, the matrices *Q*1,..., *Qr* generate B(H) iff at least one of the commutators [*Qi*, *Qj*] or double commutators

hand sides [*Q*1, *Q*2] denotes the commutator of the two 2 × 2 matrices.

[*Qi*, [*Qj*, *Qk*]] is invertible (Aslaksen & Sletsjøe, 2009).

det**M** > 0, (62)

vec(*ABC*)=(*C<sup>T</sup>* <sup>⊗</sup> *<sup>A</sup>*) vec *<sup>B</sup>*. (63)

vec *<sup>A</sup>* = (**<sup>I</sup>** <sup>⊗</sup> *<sup>A</sup>*) vec **<sup>I</sup>** = (*A<sup>T</sup>* <sup>⊗</sup> **<sup>I</sup>**) vec **<sup>I</sup>**. (64)

det(vec **I**, vec *Q*1, vec *Q*2, vec(*Q*1*Q*2)) = det([*Q*1, *Q*2]) (65)

det(vec **I**, vec *Q*1, vec *Q*2, vec[*Q*1, *Q*2]) = 2 det([*Q*1, *Q*2]), (66)

expressed by the following inequality

condition means the strict positivity of **M**.

product *ABC* is well defined, then

**4.2 Examples**

on <sup>H</sup> <sup>=</sup> **<sup>C</sup>***N*).

that

and

Papers written by mathematicians are usually focused on characterization of various properties of discussed objects and search for necessary and sufficient conditions for desired conclusion to hold. Concrete constructions offen play a minor role. The problems of frames and fusion frames are no exceptions. The main purpose of this paper was to discuss properties of some Krylov subspaces in a given Hilbert space as a natural examples of fusion frames and their applications in reconstruction of trajectories of open quantum systems.

#### **6. References**

Aslaksen, H. & Sletsjøe, A.B. (2009). *Lin. Algebra Appl.* 430, 1. Casazza, P. G. & Kutyniok, G. (2004). Frames of subspaces, *Contemp. Math.* 345, 87. Casazza, P. G. et al. (2008). *Appl. Comput. Harmon. Anal.* 25, 114. Christensen, O. (2008). *Frames and Bases*, Boston, Birkhäuser. Daubechies, I., Grossman. A., and Meyer, Y. (1986). *J. Math. Phys.* 27, 1271. Duffin, R. J. & Schaeffer, A. C. (1952). *Transactions AMS* 72, 341. Fan, H., Hu, L. (2009) *Optics Comm.* 282, 932. Farenick, D.R. (2001), *Algebras of Linear Transformations*, New York, Springer. Gorini, V et al. (1976) *J. Math. Phys.* 17, 149. Hauseholder, A.S. (2009). *The Theory of Matrices in Numerical Analysis*, Dover Books. Heil, Ch. (Ed.) (2006). *Harmonic Analysis and Applications*, Boston, Birkhäuser. Jamiołkowski, A. (1974). *Rep. Math. Phys.* 5, 415. Jamiołkowski, A. (1982). *On Observability of Classical and Quantum Stochastic Systems*, Toru ´n, N. Copernicus Univ. Press, (in Polish). Jamiołkowski, A. (1983). *Internat. J. Theoret. Phys.* 22, 369. Jamiołkowski, A. (1986). *Frequency Domain and State Space Methods for Linear Systems*, Eds. C. I. Byrnes and A. Lindquist, Amsterdam: Elsevier, p. 347. Jamiołkowski, A. (2000). *Rep. Math. Phys.* 46, 469. Jamiołkowski, A. (2010), *Journal of Physics: Conference Series* 213, 012002. Kossakowski, A. (1972) *Rep.Math. Phys.* 3, 247. Kovaˇcevi´c, J. and Chebira, A. (2008). *An Introduction to Frames*, Boston-Delft, NOW.

**Part 2** 

**Quantum Phenomena with Laser Radiation** 

