Quantum Systems Measurement through Product Hamiltonians

45th Symposium of Mathematical Physics, Toruń, June 1-2, 2013 Quantum Systems Measurement through Product Hamiltonians Joachim DOMSTA Faculty of Applied Physics and Mathematics Gdańsk University of Technology

Contents 1 2 Basic notions 3 4 5 6

0. 0.1. Quantum systems measurement is defined as the influence of the measured subsystem onto the measuring part, with the use of partial tracing, only. 0.2. The results of a quantum measurement are defined as classes of the following equivalence relation: Two states of the measured subsystem (say, 2 ) are equivalent, if they imply the same reduced state of the measuring subsystem (say, 1 ).

0. 0.3. the case of non-selective measurement is presented for the hamiltonians of the product form (possibly with more terms) H 1,2 = H 1 I 2 + M 1 M 2 + N 1 N 2 + I 1 H 2. If it consists of multipliers (mutually commuting operators), then the maximal available in 1 information about the state of 2 is its diagonal. For noncommuting factors, the possible results of measurement are shown more informative.

0. 0.4. the case of intermediate measurements is presented by examples of three-partite systems, with the interaction hamiltonian of the form H int 1,2,3 = M 1 M 2 I 3 + I 1 N 2 N 3. If it consists of multipliers, then the results of the measurement of the state of part 3 by part 1 consists of one class (of all states of 3 ). For non-commuting M 2 and N 2, some dependence happens.

0. - PACS numbers PACS numbers: 03.65.-w Quantum mechanics: 03.65.Aa; 03.65.Ca; 03.65.Ta; 03.65.Yz 03.67.-a Quantum information 05.30.-d Quantum stat. mechanics: 05.30.Ch Keywords: Schrödinger equation, multipartite systems, quantum measurement, product hamiltonian, reduced states

1. Basics - the quantum measurement The presented notion of a quantum measurement requires the system to be divisible into at least two subsystems. Then the result of the measurement is assumed to be read from the state of the measuring part caused by interactions with the measured part. calculated by reduction of the global state with the use of partial trace with respect to the measured subsystem(s). In this fashion, the measuring part plays the rôle of the observer.

1. Basics - quantum subsystems Any k-parite system lives on the tensor product (1.1) H = k j=1 H j := H 1 H 2 H k For a subset A {1, 2,..., k}, the tensor product (1.2) H A = j A H j,, A = j A, j is the Hilbert space of the compound subsystem

1. Basics - quantum states A quantum state (simply: state) of a quantum system living on a Hilbert space H is any positive hermitian operator of trace-class defined on H with trace 1. For a bi-partite system living on H = H 1 H 2, the reduced state of subsystem 1 (...with respect to subsystem 2 ) is defined as follows: ρ 1 = Tr 2 ρ, where Tr 2 is the partial trace with respect to H 2

2. Definitions - single tool, the observed state A single tool for measurement is any unitary operator. We are interested in the actions of tools performed on a given state, which is then called the starting state. Observed state of subsystem 1 under tool U is given by (2.1) ρ (U) 1 := Tr 2 ρ (U), where ρ (U) := Uρ U.

2. Definitions - the measured state In the case of starting state ρ of the product form (2.2) ρ = ρ 1 ρ 2, the result of the reduction is called the observed state of 1 under tool U at the measured state ρ 2 of 2 and is denoted as follows (2.3) ρ (U) 1 [ρ 2 ] := ρ (U) 1 = Tr 2 ( Uρ1 ρ 2 U )

2. Definitions - the tool box A tool-box for measurements of a system living on H is any family U := {U (t) : t T } of unitary operators on H, indexed by elements of an arbitrary set T. Then the observed states are denoted simply by (2.4) ρ (t) 1 [ρ 2] := ρ (U(t) ) 1 [ρ 2 ], t T.

2. Definitions - results of quantum measurements Two states ρ 2,I and ρ 2,II of subsystem 2 measured at state ρ 1 of subsystem 1 by the tool-box U = {U (t) : t T } are said to be equivalent if the observed states are equal, i.e. satisfy the equality: ρ (t) 1 [ρ 2,I ] = ρ (t) 1 [ρ 2,II ], for every t T. The result of the measurement of the state of subsystem 2 at subsystem state ρ 1 by the tool-box U is any class of equivalent states.

3. - chosen tool-boxes Now, the tool-box is given by unitary operators (3.1) U (t) = exp ( i t 1 H 1,2 ), t T, with T R, generated by a common hamiltonian (3.2.1) H 1,2 = H 1 I 2 + H int 1,2 + I 1 H 2 (3.2.2) H1,2 int = M 1 M 2 + N 1 N 2 +... }{{}, finite sum

3. - product hamiltonians REMARKS. Elements of T are the instants of observations. Operators H 1 and H 2 describe the self-evolution of parts 1 and 2, respectively. Operator H int 1,2 describes a multi-term product interaction between the subsystems.

3. - chosen Hilbert spaces The Hilbert spaces are chosen as follows (3.3) H 1 = l 2 S and H 2 = L 2 (R), card(s) ℵ 0 where S is an at most countable set, i.e. the compound system Hilbert space equals (3.4) H = H 1 H 2 = l 2 S L 2 (R) = L 2 (S R)... Then the reduced states of 1 and 2 are given by (3.5) ρ (t) 1;j,k = ρ (t) j,k (x, x) dx, (j, k) S 2, R (3.6) ρ (t) 2 (x, x ) = j S ρ (t) j,j (x, x ), (x, x ) R 2.

Content 1 2 Basic notions 3 4 5 6

3. - commuting factors Assume, that the unitary operators (U (t), t R) are multiplications by (3.7) u (t) j (x) := exp{ i t [ω j +q j µ(x)+r j ν(x))+ε(x)]}, where q j, r j and ω j are real for j S, and µ, ν and ε denote real Borel-measurable locally bounded functions on R. Hence, the hamiltonian is the multiplication by (3.8) h j (x) := ω j + q j µ(x) + r j ν(x)) + ε(x).

3. - commuting factors Thus, entries of ρ, under action of the single tool at instant t R are equal to ρ (t) j,k (x, x ) = ρ (0) j,k (x, x ) exp( i t W j,k (x, x )), where for all (j, k) S 2 and almost everywhere in (x, x ) R 2, we have W j,k (x, x ) = (ω k ω j ) + (q k µ(x ) q j µ(x))+ +(r k ν(x ) r j ν(x)) + (ε(x ) ε(x)).

3.1. - commuting factors state of 1 ρ (t) 1 [ρ 2] = ( ρ 1;j,k c (t) 2;j,k : (j, k) S 2 ), (3.10) c (t) 2;j,k = exp[i t (ω k ω j )] χ 2;j,k (t), where χ 2;j,k (t) := exp{i t [(q k q j ) µ(x)+(r k r j ) ν(x) ]} ρ 2 (x, x) dx R is the characteristic function of the following observable of subsystem 2, treated as a random variable on R equipped with the probability distribution determined by diag(ρ 2 ) Q 2;j,k (x) := (q k q j ) µ(x) + (r k r j ) ν(x), x R,

3.1. - commuting factors state of 2 ρ (t) 2 [ρ 1] = (c (t) 1 (x, x ) ρ 2 (x, x ) : (x, x ) R 2 ), (3.11) c (t) 1 (x, x ) = exp[i t (ε(x ) ε(x))] χ 1 (t; x, x ), where χ 1 (t; x, x ) := j S exp{it [q j (µ(x ) µ(x))+r j (ν(x ) ν(x))]} ρ 1;j,j, is the characteristic function of the multiplier on H 1 Q 1;j (x, x ) = [q j (µ(x ) µ(x))) + r j (ν(x ) ν(x))]. It is a random variable on S, where atom j has probability ρ 1;j,j.

3.1. - commuting factors main corollary The Main Corollary. The information about the measured state ρ 2 [state ρ 1 ] available for subsystem 1 [subsystem 2 ] through a tool-box given by (3.7) cannot exceed the distribution determined by the diagonal diag(ρ 2 ) [ or, by the diagonal diag(ρ 1 ), respectively]. This remains true for any hamiltonian of the product form (3.2), independently of the number of terms, as long as all terms with the same index commute. The details are omitted.

Content 1 2 Basic notions 3 4 5 6

3.2. - non-commuting factors Now we assume, that the hamiltonian is of the form, 1 (H 1,2 Ψ) j (x) := (ω j + q j x) ψ j (x) + iv j ψ j (x), j S, x R. for Ψ = (ψ j : j S), with square-integrable and differentiable components ψ j H 2 = L 2 (R). Then, by [3, Lemma 1], for arbitrary initial pure state represented by ψ (t) ( Ψ (0) = ψ (0) j ) (x) : j S, x R H, j (x) = ψ (0) j (x + tv j ) exp [ iω j t iq j tx i2 ] q jv j t 2.

3.2. - non-commuting factors Accordingly, the density matrix of the system varies as follows: (3.12) ρ (t) j,k (x, x ) = exp {i [ (ω k ω j ) + (q k x q j x) ] t + + i 2 (q kv k q j v j ) t 2 } ρ (0) j,k (x + tv j, x + tv k ) and the reduced states of subsystems 1 and 2 depend on time t as follows.

3.2. - non-commuting factors state of 1 (3.13) ρ (t) 1;j,k = exp [i (ω k ω j )t + i2 ] (q kv k q j v j ) t 2 exp [i(q k q j ) x t ] ρ (0) j,k (x + tv j, x + tv k ) dx. R In particular, under the product form (2.2), we have (3.14) ρ (t) 1;j,k = exp [i(ω k ω j )t + i2 ] (q kv k q j v j )t 2 R ρ (0) 1;j,k exp [i(q k q j )xt] ρ (0) 2 (x + tv j, x + tv k ) dx.

3.2. - non-commuting factors state of 1 The information about the starting state ρ (0) 2 = ρ 2 within the reduced states ρ (t) 1 is encoded in the coefficients Φ (t) j,k [ρ 2] := exp [i(q k q j ) x t ] ρ 2 (x + tv j, x + tv k ) dx. R Thus, for sufficiently large set of values of the differences q k q j and v k v j, j, k S, the kernel ρ 2 can be completely restored, whenever it is continuous. This is a consequence of noncommutativity of multiplication by x and the momentum operator i d in the dx hamiltonian.

3.2. - non-commuting factors state of 2 ρ (t) 2 (x, x ) = exp [iq k (x x) t] ρ (0) k,k (x + tv k, x + tv k ). k S In particular, under the product form (2.2), we have (3.15) ρ (t) 2 (x, x ) = exp [iq k (x x)t] k S k S exp [iq k (x x)t] ρ 1;k,k ρ (0) 2 (x + tv k, x + tv k ).

3.2. - non-commuting factors state of 2 Thus, starting states ρ (0) 1,I and ρ (0) 1,II of 1, measured at state ρ (0) 2 of 2 with (U (t) ; t T ), are equivalent, if for all t T, x, x R we have k S ρ 1,I ;k,k χ (t) k (x, x ) = k S ρ 1,II ;k,k χ (t) k (x, x ) χ (t) k (x, x ) := exp [ i q k (x x) t ] ρ (0) 2 (x + t v k, x + t v k ) Thus, the maximal information available for 2 is the ( probability distribution given by the diagonal diag ρ (0) 1 This is due to commutativity of multiplications by ω, q and v in the hamiltonian. ).

4. Three-partite systems with product interaction In this section we assume equality (1.1) with k = 3, where the factor-spaces are chosen as follows (4.1) H 1 = l 2 S and H 2 = H 3 = L 2 (R), and S is again an at most countable set. Thus, H = H 1 H 2 H 3 = l 2 S L 2 (R 2 ) = L 2 (S R 2 ).

4. Three-partite systems with product interaction For the tool-box of a measurement we assume (4.2) U (t) = exp ( i t 1 H 1,2,3 ), t T, generated by a common hamiltonian (4.3) H 1,2,3 = H 1 I 2 I 3 +I 1 H 2 I 3 +I 1 I 2 H 3 +H int 1,2,3 with 1 (4.4) H int 1,2,3 = H 1,2 I 3 + I 1 H 2,3. 1 We apply the convention, that K A stands for an operator defined on H A, for any subset A {1, 2, 3}.

4. Three-partite systems with product interaction In our assumptions, subsystems 3 and 1 have no immediate interaction. Therefore we interpret the implied dynamics U (t) = exp( ith 1,2,3 ) as a tool-box for intermediate measurements between subsytems 1 and 3 (via subsystem 2 ). For simplicity we confine the considerations to 2 (4.5) H int 1,2,3 = M 1 M 2 I 3 + I 1 N 2 N 3. 2 The below given subclasses of interactions are suitably modified hamiltonians of Section 3. Accordingly, the solutions obtain a simply verifiable form.

4. Three-partite systems with product interaction REMARK. Now the reduced states of subsystems 1, 2 and 3 are given by the following density matrices = ρ (t) 1;j,k ρ (t) 2 (x, x ) = ρ (t) 3 (y, y ) = R 2 R R ρ (t) j,k (x, y; x, y) dx dy, j, k S, j j ρ (t) j,j (x, y; x, y) dy, x, x R, ρ (t) j,j (x, y; x, y ) dx, y, y R.

Content 1 2 Basic notions 3 4 5 6

4.1. Three-partite systems - commuting factors For this subsection the particular subclass is defined by the assumption, that the tool-box is the group of unitary operators U (t) equal to the multiplication by the functions u (t) j (x, y) equal to (4.6) exp( i t (ω j + ε 2 (x) + ε 3 (y) + m j µ(x) + ν 2 (x)ν 3 (y))), for j S, x, y R.

4.1. Three-partite systems - commuting factors Consequently, entries of ρ (t) under action of the single tool U (t) (or, at instant t R) are equal to ρ (t) j,k (x, y; x, y ) = ρ (0) j,k (x, y; x, y ) exp( i t W j,k (x, y; x, y )), where W j,k (x, y; x, y ) equals (ω k ω j ) + (ε 2 (x ) ε 2 (x)) + (ε 3 (y ) ε 3 (y))+ + (m k µ(x ) m j µ(x)) + (ν 2 (x ) ν 3 (y ) ν 2 (x) ν 3 (y)), and this holds for all (j, k) S 2 and for almost all (x, y; x, y ) in R 4.

4.1. Three-partite systems - commuting factors In the case of commuting factors, if the starting state ρ of the system is of the product form ρ 1 ρ 2 ρ 3, where ρ j acts on H j, j = 1, 2, 3, then the reduced evolution of subsystems 1, 2 and 3 are given by the multiplication of the starting density matrices term-by-term by suitable matrices c (t) 2,3, c (t) (t) 1,3 and c 1,2, respectively, as follows.

4.1. Three-partite systems - commuting factors state of 1 Observed state of system 1 equals ρ (t) 1 [ρ 2 ρ 3 ] = ρ 1 c (t) 2,3 := ( ρ 1;j,k c (t) 2,3;j,k : (j, k) S 2 ), where c (t) 2,3;j,k = exp(i t (ω k ω j )) χ 2,3;j,k (t), with (4.8) χ 2,3;j,k (t) := exp(i t [(m k m j )µ(x) ] R 2 ρ 2 (x, x) ρ 3 (y, y) dx dy = exp(i t [(m k m j )µ(x) ] ρ 2 (x, x) dx. R observed states of 1 are independent of ρ 3

4.1. Three-partite systems - commuting factors state of 2 the observed state of 2 at instant t R equals ( ) ρ (t) 2 [ρ 1 ρ 3 ] = ρ 2 c (t) 1,3 := ρ 2 (x, x ) c (t) 1,3 (x, x ) where for (x, x ) R 2, t R, c (t) 1,3 (x, x ) = exp(i t (ε 2 (x ) ε 2 (x))) χ 1 (t; x, x)), with χ 1 (t; x, x ) := exp{it [m j (µ(x ) µ(x))+ j S R (ν 2 (x ) ν 2 (x)) ν 3 (y)]} ρ 1;j,j ρ 3 (y, y) dy,

4.1. Three-partite systems - commuting factors state of 2 χ 1,3 (.; x, x ) is the characteristic function of the following multiplier on H 1 H 3 O 1,3;j,y (x, x ) = [q j (µ(x ) µ(x)))+(ν 2 (x ) ν 2 (x))ν 3 (y)], j S, y treated as the random variable on S R, with the product measure, where atom j has probability, determined by diag(ρ 1 ) and the variable y possesses p.d.f. given by diag(ρ 3 ) as follows, (4.9) F 3 (a) = ρ 3 (y, y) dy, a R. (,a]

4.1. Three-partite systems - commuting factors state of 3 Finaly, for the reduced state of subsystem 3 we obtain ( ) ρ (t) 3 [ρ 1 ρ 2 ] = ρ 3 c (t) 1,2 := ρ 3 (y, y ) c (t) 1,2 (y, y ) where for (y, y ) R 2, t R, c (t) 1,2 (y, y ) = exp(i t (ε 3 (y ) ε 3 (y))) χ 1,2 (t; y, y )), and the latter function is given by χ 1,2 (t; y, y ) = exp{it [ ν 2 (x) (ν 3 (y ) ν 3 (y) )]} ρ 2 (x, x) dx. R

4.1. Three-partite systems - commuting factors state of 3 Thus, χ 1,2 is the characteristic function of another observable of subsystem 2 O 2 (y, y ; x) := ν 2 (x) (ν 3 (y ) ν 3 (y)) x R, treated as a random variable on R equipped with the probability distribution function (p.d.f.) determined again by the diagonal diag(ρ 2 ) by F 2 (a) = ρ 2 (x, x) dx (,a] observed states of 3 are independent of ρ 1.

Content 1 2 Basic notions 3 4 5 6

4.2. Three-partite systems - non-commuting factors In this section we modify the hamiltonian of subsection 3.2: (4.10) 1 (H 1,2,3 Ψ) j (x, y) := (ω j +m j x) ψ j (x, y)+iy x ψ j(x, y), whenever Ψ = (ψ j : j S), with ψ j L 2 (R 2 ), j S, x R. For consistency with (4.3) (4.5), in particular with H int 1,2,3 = M 1 M 2 I 3 + I 1 N 2 N 3, the operators can be chosen as follows, x R,

4.2. Three-partite systems - non-commuting factors H 1 (π) = ( ω j π j : j I ), H 2 ψ(x) = H 3 ψ(x) = 0, M 1 (π) = (m j π j : j S), N 2 ψ(x) = i ψ (x), M 2 ψ(x) = x ψ(x), N 3 ψ(y) = y ψ(y) Proceeding as for Lemma 1 in [3, Section 3], one can prove that the time dependent solution Ψ (t) = (ψ (t) j : j S) to the Schrödinger equation equals

4.2. Three-partite systems - non-commuting factors ψ (t) j (x, y) := 3 := ψ (0) j (x + ty, y) exp [ iω j t i m j tx i2 m jyt 2 ], for arbitrary initial pure state represented by ( ) Ψ (0) = ψ (0) j (x, y) : j S, (x, y) R 2 H. Hence, the density matrix of the system varies as follows: 3 In order to avoid the derivation of the solution, it sufficies to check that i t ψ(t) j (x) = ( H 1,2,3 Ψ (t)) j (x, y), for differentiable Ψ(0).

4.2. Three-partite systems - non-commuting factors ρ (t) j,k (x, y; x, y ) = = exp {i [ (ω k ω j ) + (m k x m j x) ] t} { } i exp 2 (m ky m j y) t 2 ρ (0) j,k (x+ty, y; x +ty, y ), and the reduced states of subsystems 1 and 3 depend on time t as follows.

4.2. Three-partite systems - non-commuting factors: state of 1 ρ (t) 1 [ρ 2 ρ 3 ] = ρ (0) 1;j,k exp {i t (ω k ω j )} exp {i t (m R 2 k m j )(x + 12 } yt) ρ (0) 2 (x + ty, x + ty) ρ(0) 3 (y, y) dx dy = ρ (0) 1;j,k exp {i t (ω k ω j )} exp {i t (m R 2 k m j )(x 12 } yt) ρ (0) 2 (x, x) ρ(0) 3 (y, y) dx dy

4.2. Three-partite systems - non-commuting factors: state of 1 i.e. (4.12) ρ (t) 1 = ( ρ 1;j,k c (t) 2,3;j,k : (j, k) S 2 ) where for (j, k) S 2, t R, (4.13) c (t) 2,3;j,k = exp[i t (ω k ω j )] χ 2,3;j,k (t), with χ 2,3;j,k (t) given by R 2 exp{i t [(m k m j )(x 1 2 yt) ]} ρ(0) 2 (x, x)ρ(0) 3 (y, y) dx dy

4.2. Three-partite systems - non-commuting factors: state of 1 Now, χ 2,3;j,k (t) is the value of the characteristic function of the observable of the compound subsystem 2, 3 : O 2;j,k (x, y) := (m k m j ) (x 12 ) y t, x, y R, treated as a random variable on R 2 equipped with the product of the probability distributions determined by the diagonals diag(ρ (0) 2 ) and diag(ρ(0) 3 ) as given above The observed state of 1 depends on ρ 3 through diag(ρ 3 ), at most.

4.2. Three-partite systems - non-commuting factors: state of 2 One easily gets, that for (x, x ) R 2, t R, (4.14) ρ (t) 2 (x, x ) = exp{i t m j (x x)} ρ 1;j,j j S ρ 2 (x ty, x ty) ρ 3 (y, y) dy, R observed state of 2 depends on ρ 1 and ρ 3

4.2. Three-partite systems - non-commuting factors: state of 3 Finaly, for the reduced state of subsystem 3 we obtain ( ) (4.15) ρ (t) 3 [ρ 1 ρ 2 ] = ρ (0) 3 (y, y ) c (t) 1,2 (y, y ) where for (y, y ) R 2, t R, the coefficients equal c (t) 1,2 (y, y ) = j S R [ ] i exp 2 (m j(y y) t 2 ρ (0) 1;j,j ρ(0) 2 (x + ty, x + ty ) dx.

4.2. Three-partite systems - non-commuting factors: state of 3 or, equivalently ρ (t) 3 (y, y ) = ρ (0) 3 (y, y ) [ ] i exp 2 (m j(y y) t 2 ρ (0) 1;j,j j S ρ (0) 2 (x, x + t(y y) ) dx. R The observed state of 3 depends on ρ 1 through diag(ρ 1 ), at most.

In rough description, in our model the tools are observables built into the hamiltonian and we allow them to act all the time during the evolution. In this fashion we have modelled the influence of the state of the measured system upon the changes of the measuring subsystem state. This makes our considerations close to the fashion presented e.g. by Żurek [12] and Attal [2]). Our main goal was to point at the influence of commutativity of the factors in the interaction.

It is worth to stress, that in the traditional system of notions, the model of measurement is based on the averege values of observables, which goes back to the formulation of quantum mechanics by von Neumann [10]. In such cases, the problem of minimal sets of the observables is solved in the works by Jamio lkowski [5], [6] and other authors (see the survey paper [8] and the references therein). For the presented model, the set of tools (including instants of observations) sufficient for identification of the measured state becomes the main problem for investigations. Also the question of the subject of measurements is open. For instance, can we use starting states which are not of the product form.

The relation to the entropy analysis, maximal entanglement and other similar problems, considered for instance in [11] is omitted. Also questions extended over compound systems of large number of parts (as in [15]) is not even touched here, since the basic features do not require more than 3 parts. On the other hand, the proposed definition of measurement seems to be use-less in cases, when the system lives on a product of Hilbert spaces, but the set of all possible states does not cover the space of all trace class positive operators with trace 1.

- bad news EXAMPLE. Let the Hilbert space of a bi-partite system H 1,2 C 2 C 2 be spanned by 4-dimensional unit vectors ψ(σ 1, σ 2 ) = U(σ 1, σ 2 ) a σ1, (σ 1, σ 2 ) {, +} 2, U(σ 1, σ 2 ), a σ1 C, with U(σ 1, σ 2 ) = 1 σ 1 σ 2 exp(iα), σ 1, σ 2 {, +} α R. 2 H 1,2 is of complex dimension 2. The corresponding reduced density matrix is diagonal, and independently of α we have ρ 1 (+; +) = tr 2 ρ(+, +) = a + 2, ρ 1 ( ; ) = a 2.

JD: QM References 45th SMP, Toruń, June 1-2, 2013 R. Alicki, M. Fannes: Quantum Dynamical Systems, Oxford University Press, Oxford, 2000. S. Attal: Markov Chains and Dynamical Systems: the Open System Point of View, arxiv:1010.2894v1 [math.pr]. M. Bodzioch, J. Domsta: Two Examples of Quantum Dynamical Semigroups, Open Systems and Information Dynamics, 18;2 (2011), 143 155. DOI: 10.1142/S1230161211000091 H.-P. Breuer, F. Petruccione: The Theory of Open Quantum Systems, Oxford University Press, Oxford, 2002.

JD: QM References 45th SMP, Toruń, June 1-2, 2013 A. Jamio lkowski: Minimal Number of Operators for Observability of N-Level Quantum Systems I, International Journal of Theoretical Physics, 22;4 (1983), A. Jamio lkowski: On sufficient algebraic conditions for identification of quantum states, QP-PQ, 28 (2011), 185 197. A. Jamio lkowski: Effective methods in investigation of irreducible quantum operations, Int. J. Geom. Meth. Mod. Phys., 9 (2012), 1 8. DOI: 10.1142/s0219887812600146 A. Jamio lkowski: Fusion Frames and Dynamics of Open Quantum Systems; in [9], pp. 67 84.

JD: QM References 45th SMP, Toruń, June 1-2, 2013 S. Lyagushyn (Edtr.):, Quantum Optics and Laser Experiments, InTech, Rijeka, Croatia, 2012. J. von Neumann: Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932; English translation: Mathematical Foundations of Quantum Mechanics, Princeton University Press, Berlin, 1955. L. Rudnicki, P. Horodecki, K. Życzkowski: Collective Uncertainty Entanglement Test Phys. Rev. Lett. 107, 150502 (2011) [5 pages] DOI: 10.1103/PhysRevLett.107.150502

JD: QM References 45th SMP, Toruń, June 1-2, 2013 W.H. Zurek: Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?, Physical Review D, 24;6 (1981), 1516 1525. W.H. Zurek: Environment-induced superselection rules, Physical Review D, 26;8 (1982), 1862 1880. W.H. Zurek: Decoherence and the Transition from Quantum to Classical - Revisited, Quantum Decoherence, 2006 Birkhäuser Verlag Basel, Poincare Seminar 2005, 1 31 K. Życzkowski et al.: On Average Entanglement between Random Quantum Systems, 46th Karpacz Winter school, L adek, February 12, 2010.