Outline. Motivation: Numerical Ranges as Measuring Sticks for Optimising Quantum Dynamics. I Symmetry Approach to Quantum Systems Theory

Transcription

1 Some Remarks on Quantum Systems Theory as Pertaining to Numerical Ranges A Unified Lie-Geometric Viewpoint Thomas Schulte-Herbrüggen relating to (joint) work with G. Dirr, R. Zeier, C.K. Li, Y.T. Poon, F. vom Ende 14 th Workshop on Numerical Ranges and Radii WONRA 100 th anniversary of the TOEPLITZ-HAUSDORFF Theorem (1918/1919) MPQ and TUM-IAS, Munich-Garching, June 2018

2 Outline : Numerical Ranges as Measuring Sticks for Optimising Quantum Dynamics I Symmetry Approach to Quantum Systems Theory II Generalising C-Numerical Ranges III Gradient Flows

3 Outline : Numerical Ranges as Measuring Sticks for Optimising Quantum Dynamics I Symmetry Approach to Quantum Systems Theory II Generalising C-Numerical Ranges III Gradient Flows

4 Outline : Numerical Ranges as Measuring Sticks for Optimising Quantum Dynamics I Symmetry Approach to Quantum Systems Theory II Generalising C-Numerical Ranges III Gradient Flows

5 Significance of Numerical and C-Numerical Ranges Generalising Expectation Values Expectation values of observables B = B B(H): C-Numerical Ranges Approx. by Sums of Orbits pure quantum states ψ(t) = U(t) ψ 0 H: B t := ψ(t)b ψ(t) W (B) := { φb φ, φ = 1} mixed states ρ(t) Uρ 0 U : tr ( B ρ(t) ) W (B, ρ 0 ) = {tr ( B Uρ 0 U 1) U U(H)} C numerical range: generalisation to non-hermitian operators A, C W (C, A) := {tr(c UAU 1 ) U U(H)}

6 Significance of Numerical and C-Numerical Ranges Generalising Expectation Values Expectation values of observables B = B B(H): C-Numerical Ranges Approx. by Sums of Orbits pure quantum states ψ(t) = U(t) ψ 0 H: B t := ψ(t)b ψ(t) W (B) := { φb φ, φ = 1} mixed states ρ(t) Uρ 0 U : tr ( B ρ(t) ) W (B, ρ 0 ) = {tr ( B Uρ 0 U 1) U U(H)} C numerical range: generalisation to non-hermitian operators A, C W (C, A) := {tr(c UAU 1 ) U U(H)}

7 Significance of Numerical and C-Numerical Ranges Generalising Expectation Values Generalise from B = B to non-hermitian operator C: C-Numerical Ranges Approx. by Sums of Orbits pure quantum states ψ(t) = U(t) ψ 0 H: B t := ψ(t)b ψ(t) W (B) := { φb φ, φ = 1} mixed states ρ(t) Uρ 0 U : tr ( B ρ(t) ) W (B, ρ 0 ) = {tr ( B Uρ 0 U 1) U U(H)} C numerical range: generalisation to non-hermitian operators A, C W (C, A) := {tr(c UAU 1 ) U U(H)}

8 Significance of C-Numerical Radius Geometric Optimisation Problems Find points on unitary orbit of initial state A with C-Numerical Ranges Approx. by Sums of Orbits minimal Euclidean distance to target C min U C UAU max U Re tr{c UAU 1 } find max. real part of C num. range minimal angle to target C max U cos 2 A,C (U) = max U tr{c UAU 1 } 2 A 2 2 C 2 2 find: C num. radius r C (A) = max U tr{c UAU 1 } pro memoria: C UAU = A C Re tr{c UAU 1 }

9 Significance of C-Numerical Radius Geometric Optimisation Problems Find points on unitary orbit of initial state A with C-Numerical Ranges Approx. by Sums of Orbits minimal Euclidean distance to target C min U C UAU max U Re tr{c UAU 1 } find max. real part of C num. range minimal angle to target C max U cos 2 A,C (U) = max U tr{c UAU 1 } 2 A 2 2 C 2 2 find: C num. radius r C (A) = max U tr{c UAU 1 } pro memoria: C UAU = A C Re tr{c UAU 1 }

10 Examples of Quantum Control Maximising Spectroscopic Sensitivity C-Numerical Ranges Approx. by Sums of Orbits find r C (A) by gradient flow on unitary group A, C Mat 3 (C ) A, C Mat 8 (C ) Glaser, T.S.H., Sieveking, Schedletzky, Nielsen, Sørensen, Griesinger, Science 280 (1998), 421

11 Early Connections Trip from ETH to ILAS 1996 C-Numerical Ranges Approx. by Sums of Orbits

12 Early Connections Trip from ETH to ILAS 1996 C-Numerical Ranges Approx. by Sums of Orbits

13 Early Connections Trip from ETH to ILAS 1996 C-Numerical Ranges Approx. by Sums of Orbits

14 Generalisation: Sums of Orbits Least-Squares Approx. with Li & Poon, Math. Comput. 80, 1601 (2011) C-Numerical Ranges Approx. by Sums of Orbits Generalised task: approximate A 0 by elements on the sum of orbits unitary similarity: unitary equivalence: min N U j A j U j A 0 2 U j SU(n) 2 j=1 min U j,v j SU(n) N U j A j V j A 0 2 j=1 2 unitary t-congruence: min N U j A j Uj t A 0 2 U j SU(n) 2 j=1 -congruence: min N S j A j S j A 0 2 S j SL(n) 2 j=1

15 Generalisation: Sums of Orbits Least-Squares Approx. with Li & Poon, Math. Comput. 80, 1601 (2011) C-Numerical Ranges Approx. by Sums of Orbits Generalised task: approximate A 0 by elements on the sum of orbits unitary similarity: unitary equivalence: min N U j A j U j A 0 2 U j SU(n) 2 j=1 min U j,v j SU(n) N U j A j V j A 0 2 j=1 2 unitary t-congruence: min N U j A j Uj t A 0 2 U j SU(n) 2 j=1 -congruence: min N S j A j S j A 0 2 S j SL(n) 2 j=1

16 Generalisation: Sums of Orbits Least-Squares Approx. with Li & Poon, Math. Comput. 80, 1601 (2011) C-Numerical Ranges Approx. by Sums of Orbits Generalised task: approximate A 0 by elements on the sum of orbits unitary similarity: unitary equivalence: min N U j A j U j A 0 2 U j SU(n) 2 j=1 min U j,v j SU(n) N U j A j V j A 0 2 j=1 2 unitary t-congruence: min N U j A j Uj t A 0 2 U j SU(n) 2 j=1 -congruence: min N S j A j S j A 0 2 S j SL(n) 2 j=1

17 Generalisation: Sums of Orbits Least-Squares Approx. with Li & Poon, Math. Comput. 80, 1601 (2011) C-Numerical Ranges Approx. by Sums of Orbits Generalised task: approximate A 0 by elements on the sum of orbits unitary similarity: unitary equivalence: min N U j A j U j A 0 2 U j SU(n) 2 j=1 min U j,v j SU(n) N U j A j V j A 0 2 j=1 2 unitary t-congruence: min N U j A j Uj t A 0 2 U j SU(n) 2 j=1 -congruence: min N S j A j S j A 0 2 S j SL(n) 2 j=1

18 Systems Theory Simulability Spin Systems

19 Systems Theory: Controllability/Simulability Simulability Spin Systems Consider 1 linear control system: ẋ(t) = Ax(t) + Bv 2 bilinear control system: Ẋ(t) = (A + j u jb j )X(t) Conditions for Full Controllability Universality 1 in linear systems: rank [B, AB, A 2 B,..., A N 1 B] = N 2 in bilinear systems: A, B j j = 1, 2,..., m Lie = su(n) key: system algebra k := A, B j j = 1, 2,..., m Lie reachable set Reach(ρ 0 ) = {K ρ 0 K K exp k}

20 Systems Theory: Controllability/Simulability Simulability Spin Systems Consider 1 linear control system: ẋ(t) = Ax(t) + Bv 2 bilinear control system: Ẋ(t) = (A + j u jb j )X(t) Conditions for Full Controllability Universality 1 in linear systems: rank [B, AB, A 2 B,..., A N 1 B] = N 2 in bilinear systems: A, B j j = 1, 2,..., m Lie = su(n) key: system algebra k := A, B j j = 1, 2,..., m Lie reachable set Reach(ρ 0 ) = {K ρ 0 K K exp k}

21 Systems Theory: Controllability/Simulability Simulability Spin Systems Consider 1 linear control system: ẋ(t) = Ax(t) + Bv 2 bilinear control system: Ẋ(t) = (A + j u jb j )X(t) Conditions for Full Controllability Universality 1 in linear systems: rank [B, AB, A 2 B,..., A N 1 B] = N 2 in bilinear systems: A, B j j = 1, 2,..., m Lie = su(n) key: system algebra k := A, B j j = 1, 2,..., m Lie reachable set Reach(ρ 0 ) = {K ρ 0 K K exp k}

22 Systems Theory: Controllability/Simulability Simulability Spin Systems Consider 1 linear control system: ẋ(t) = Ax(t) + Bv 2 bilinear control system: Ẋ(t) = (A + j u jb j )X(t) Conditions for Full Controllability Universality 1 in linear systems: rank [B, AB, A 2 B,..., A N 1 B] = N 2 in bilinear systems: A, B j j = 1, 2,..., m Lie = su(n) key: system algebra k := A, B j j = 1, 2,..., m Lie reachable set Reach(ρ 0 ) = {K ρ 0 K K exp k} symmetries ad k := {S gl(n 2 ) [S, ad A ] = [S, ad Bj ] = 0, j}

23 Bilinear Control Systems Markovian Settings PRA 84, (2011) Simulability Spin Systems Ẋ(t) = ( A + Σ j u j (t)b j ) X(t) as operator lift of ẋ(t) = ( A + Σ j u j (t)b j ) x(t) X(t) or x(t): state ; A: drift; B j : control Hamiltonians; u j : control amplitudes Setting and Task State Drift Controls X(t) A B j closed systems: pure-state transfer X(t) = ψ(t) ih 0 ih j gate synthesis (fixed global phase) X(t) = U(t) ih 0 ih j state transfer X(t) = ρ(t) iĥ0 iĥj gate synthesis (free global phase) X(t) = Û(t) iĥ0 iĥj open systems: state transfer I X(t) = ρ(t) iĥ0 + Γ iĥj quantum-map synthesis I X(t) = F(t) iĥ0 + Γ iĥj state transfer II X(t) = ρ(t) iĥ0 iĥj, Γ j map synthesis II X(t) = F(t) iĥ0 iĥj, Γ j Ĥ is Hamiltonian commutator superoperator generating Û := U( )U.

24 Symmetry vs. Controllability I Necessary Conditions J. Math. Phys. 52, (2011) Control system Σ with algebra k = ih ν ν = d; 1, 2,..., m Lie. Simulability Spin Systems Theorem (Simplicity) The above system algebra k is an irreducible simple subalgebra of su(n), if both 1 the commutant is trivial, i.e. k = span{1l}, 2 the coupling graph to H d is connected. XY XY XY

25 Irreducible Simple Subalgebras to su(n) up to N = 2 15 J. Math. Phys. 52, (2011) Simulability Spin Systems so(11) su(2) su(2) sp(2) su(4) su(2) sp(4) su(8) su(3) so(8) so(7) su(2) sp(8) sp(2) su(16) so(9) so(16) so(10) su(2) sp(16) so(12) su(32) so(32) su(2) so(13) sp(32) sp(2) sp(3) su(64) su(3) so(6) so(64) su(2) sp(64) so(9) su(128) so(16) so(128) so(15) su(2) sp(128) sp(2) so(17) so(256) su(256) su(4) so(18) su(2) sp(256) so(19) so(20) su(512) su(3) so(7) so(512) sp(3) su(2) so(21) sp(512) sp(2) su(1024) su(5) so(1024) so(22) so(23) so(27) su(2) so(24) su(2) sp(2) su(3) so(7) so(17) so(25) sp(4) so(6) so(8) f4 g2 su(2) so(28) sp(1024) so(2048) sp(2048) so(4096) so(26) sp(4096) so(8192) su(2048) su(4096) su(8192) so(31) su(2) so(29) sp(8192) sp(2) so(16384) su(4) so(30) su(2) sp(16384) su(6) su(3) sp(3) so(32768) so(8) so(32) su(16384) su(32768) g2 so(14)

26 Irreducible Simple Subalgebras to su(n) up to N = 2 15 J. Math. Phys. 52, (2011) Simulability Spin Systems

27 Irreducible Simple Subalgebras to su(n) up to N = 2 15 J. Math. Phys. 52, (2011) Simulability Spin Systems

28 Symmetry vs. Controllability II Single Symmetry Condition JMP (2011), OSID (2017) Theorem (Trivial Second-Order Symmetries) Simulability Spin Systems Let {H ν ν = d; 1, 2,..., m} be drift and control Hamiltonians of control system Σ with system algebra k. Define Φ Θ := { ( 1l ih ν + Θ(iH ν 1l) ) ν = d, 1,..., m}. Then Σ is fully controllable, i.e. k = su(2 n ), iff joint commutant to Φ Θ is two-dimensional, i.e. Φ Θ = span{1l 2, Θ(SWAP)}.

29 Symmetry vs. Controllability II Single Symmetry Condition J. Math. Phys. 52, (2011) Simulability Spin Systems Theorem Let {H ν ν = d; 1, 2,..., m} be drift and control Hamiltonians of control system Σ with system algebra k. Define Φ AB := {(1l B ih ν + ih ν 1l A ) ν = d, 1,..., m}. Then Σ is fully controllable, i.e. k = su(2 n ), iff joint commutant to Φ AB is two-dimensional i.e. Φ AB = span{1l, SWAP AB}. [Φ AB ] = [symmetric] bosonic [anti-symmetric] fermionic

30 Symmetry vs. Controllability II Single Symmetry Condition J. Math. Phys. 52, (2011) Simulability Spin Systems Theorem Let {H ν ν = d; 1, 2,..., m} be drift and control Hamiltonians of control system Σ with system algebra k. Define Φ AB := {(1l B ih ν + ih ν 1l A ) ν = d, 1,..., m}. Then Σ is fully controllable, i.e. k = su(2 n ), iff joint commutant to Φ AB is two-dimensional i.e. Φ AB = span{1l, SWAP AB}. [Φ AB ] = [symmetric] bosonic [anti-symmetric] fermionic

31 Symmetry vs. Controllability II Single Symmetry Condition J. Math. Phys. 52, (2011) Simulability Spin Systems Theorem Let {H ν ν = d; 1, 2,..., m} be drift and control Hamiltonians of control system Σ with system algebra k. Define Φ AB := {(1l B ih ν + ih ν 1l A ) ν = d, 1,..., m}. Then Σ is fully controllable, i.e. k = su(2 n ), iff joint commutant to Φ AB is two-dimensional i.e. Φ AB = span{1l, SWAP AB}. [Φ AB ] = [symmetric] bosonic [anti-symmetric] fermionic

32 Quantum Simulability Algebraic Decision Simulability Spin Systems Corollary (J. Math. Phys. 52, (2011)) Let Σ A, Σ B be control systems with system algebras k A, k B over a given Hilbert space H. Then Σ A can simulate Σ B k B is a subalgebra of k A.

33 Quantum Simulation by Spin Systems Overview J. Math. Phys. 52, (2011) Simulability Spin Systems system type fermionic bosonic system alg. n-spins- 1 no. of levels order of coupling 2 A XX XX n quadratic (i.e. 2) so(2n + 1) NB: no. of spins maps into no. of levels (as in Jordan-Wigner transformation).

34 Quantum Simulation by Spin Systems Overview J. Math. Phys. 52, (2011) Simulability Spin Systems system type fermionic bosonic system alg. n-spins- 1 2 no. of levels order of coupling A XX XX n quadratic (i.e. 2) so(2n + 1) A XX XX B n + 1 quadratic (i.e. 2) so(2n + 2) NB: no. of spins maps into no. of levels (as in Jordan-Wigner transformation).

35 Quantum Simulation by Spin Systems Overview J. Math. Phys. 52, (2011) Simulability Spin Systems system type fermionic bosonic system alg. n-spins- 1 2 no. of levels order of coupling A XX XX n quadratic (i.e. 2) so(2n + 1) A XX XX B n + 1 quadratic (i.e. 2) so(2n + 2) XX XX A for n mod 4 {0, 1} n up to n so(2 n ) NB: no. of spins maps into no. of levels (as in Jordan-Wigner transformation).

36 Quantum Simulation by Spin Systems Overview J. Math. Phys. 52, (2011) Simulability Spin Systems system type fermionic bosonic system alg. n-spins- 1 2 no. of levels order of coupling A XX XX n quadratic (i.e. 2) so(2n + 1) A XX XX B n + 1 quadratic (i.e. 2) so(2n + 2) XX XX A for n mod 4 {0, 1} n up to n so(2 n ) for n mod 4 {2, 3} n up to n sp(2 n 1 ) NB: no. of spins maps into no. of levels (as in Jordan-Wigner transformation).

37 Quantum Simulation by Spin Systems Overview J. Math. Phys. 52, (2011) Simulability Spin Systems system type fermionic bosonic system alg. n-spins- 1 2 no. of levels order of coupling A XX XX n quadratic (i.e. 2) so(2n + 1) A XX XX B n + 1 quadratic (i.e. 2) so(2n + 2) XX XX A for n mod 4 {0, 1} n up to n so(2 n ) for n mod 4 {2, 3} n up to n sp(2 n 1 ) A XX B XX n up to n up to n su(2 n ) NB: no. of spins maps into no. of levels (as in Jordan-Wigner transformation).

38 Quantum Simulation by Spin Systems Overview J. Math. Phys. 52, (2011) Simulability Spin Systems system type fermionic bosonic system alg. n-spins- 1 2 no. of levels order of coupling A XX XX n quadratic (i.e. 2) so(2n + 1) A XX XX B n + 1 quadratic (i.e. 2) so(2n + 2) XX XX A for n mod 4 {0, 1} n up to n so(2 n ) for n mod 4 {2, 3} n up to n sp(2 n 1 ) A XX B XX n up to n up to n su(2 n ) NB: no. of spins maps into no. of levels (as in Jordan-Wigner transformation).

39 Numerical Range and C-Numerical Range Classical features of W (A) and W (C, A): W (A) and W (C, A) are compact and connected. GOLDBERG & STRAUSS 1977 W (A) is convex. HAUSDORFF 1919, TOEPLITZ 1918 W (C, A) is star-shaped. CHEUNG & TSING 96 W (C, A) is convex if C or A Hermitian. WESTWICK 75 W (C, A) is a circular disk centered at the origin if C or A are unitarily similar to block-shift form LI & TSING 91

40 The Relative C-Numerical Range Restricted Quantum Control with G. Dirr & U. Helmke Definition (Lin. Multin. Alg. 56 (2008) 3 26 and ) The relative C-numerical range is the set W K (C, A) := {tr (C KAK ) K K SU(N)} W C (A), where the unitary orbit is restricted to a subgroup K. Ex.:local operations K SU(2) SU(2) SU(2)

41 The Relative C-Numerical Range Restricted Quantum Control with G. Dirr & U. Helmke Definition (Lin. Multin. Alg. 56 (2008) 3 26 and ) The relative C-numerical range is the set W K (C, A) := {tr (C KAK ) K K SU(N)} W C (A), where the unitary orbit is restricted to a subgroup K. Ex.:local operations K SU(2) SU(2) SU(2) Example (I non convex) 1 A := ( ) ( ) i i C := diag (1, 0, 0, 0)

42 The Relative C-Numerical Range Restricted Quantum Control with G. Dirr & U. Helmke Definition (Lin. Multin. Alg. 56 (2008) 3 26 and ) The relative C-numerical range is the set W K (C, A) := {tr (C KAK ) K K SU(N)} W C (A), where the unitary orbit is restricted to a subgroup K. Ex.:local operations K SU(2) SU(2) SU(2) Example (II neither star-shaped nor simply connected) A := C := ( 1 0 ) 3 0 e 2iπ/3 ( )

43 The Relative C-Numerical Range Restricted Quantum Control with G. Dirr & U. Helmke Definition (Lin. Multin. Alg. 56 (2008) 3 26 and ) The relative C-numerical range is the set W K (C, A) := {tr (C KAK ) K K SU(N)} W C (A), where the unitary orbit is restricted to a subgroup K. Ex.:local operations K SU(2) SU(2) SU(2) 1 Example (III distinct circ. symmetry) ( ) A := , C := diag ( ) 0.5 Im 0 (a) (b)

44 The Relative C-Numerical Range Restricted Quantum Control Example (IV K = USp(4/2) vs SO(4) vs SU(2) 2 ) A= 1 3 diag(0, e i 2π/3, e i 4 π/3, 1) C=diag(1, 0, 0, 0) C = 1 2 ( 0 ) 1l 1l 0

45 The Relative C-Numerical Range Restricted Quantum Control Example (V K = USp(4/2) vs SO(4) vs SU(2) 2 ) A= 1 8 diag(1 + i, 1 i, 1 i, 1 + i) C=diag(1, 0, 0, 0) C = C= ( ) 0 1l =: 1l J

46 The Relative C-Numerical Range Restricted Quantum Control Example (VI K = USp(4/2) vs SO(4) vs SU(2) 2 ) A = C = =: 1 2 K

47 The Relative C-Numerical Range Restricted Quantum Control Example (VII K = USp(4/2) vs SO(4) vs SU(2) 2 ) A = C = = 1 2 KJ

48 The Relative C-Numerical Range Restricted Quantum Control Example (VIII K = USp(4/2) vs SO(4) vs SU(2) 2 ) A {( ) , ( ) , ( ) , ( )} C = 1 2 K Im Re

49 The Relative C-Numerical Range Restricted Quantum Control Example (IX K = USp(4/2) vs SO(4) vs SU(2) 2 ) A= 1 3 diag(0, e i 2π/3, e i 4π/3, 1) C= = 1 2 KJ

50 Relative C-Numerical Range Condition for Rotational Symmetry Lin.Multilin.Alg. 56 (2008), 27 Theorem Let K be a compact connected subgroup of U(N) with Lie algebra k, and let t be a torus algebra of k. Then the relative C-numerical range W K (C, A + ) of a matrix A + Mat N (C ) is a circular disc centered at the origin of the complex plane for all C Mat N (C ) if and only if there exists a K K and a t such that KA + K is an eigenoperator to ad with a non-zero eigenvalue ad (KA + K ) [, KA + K ] = ip (KA + K ) and p 0.

51 Relative C-Numerical Range Condition for Rotational Symmetry Lin.Multilin.Alg. 56 (2008), 27 Theorem Let K be a compact connected subgroup of U(N) with Lie algebra k, and let t be a torus algebra of k. Then the relative C-numerical range W K (C, A + ) of a matrix A + Mat N (C ) is a circular disc centered at the origin of the complex plane for all C Mat N (C ) if and only if there exists a K K and a t such that KA + K is an eigenoperator to ad with a non-zero eigenvalue ad (KA + K ) [, KA + K ] = ip (KA + K ) and p 0. If KA + K is an eigenoperator of ad to eigenvalue +ip and A := A +, then KA K has eigenvalue ip. A + and A share the same relative C-numerical range of circular symmetry, W K (C, A + ) = W K (C, A ).

52 Gradient Flows on Riemannian Manifolds Abstract Optimisation Task Rep. Math. Phys. 64, 93 (2009) Abstract Optimisation Flows on Lie Groups Similarity Orbits f

53 Gradient Flows on Riemannian Manifolds Abstract Optimisation Task Rep. Math. Phys. 64, 93 (2009) Abstract Optimisation Flows on Lie Groups Similarity Orbits f quality function f : M R, X f (X) drives into (local) maximum by gradient flow to Ẋ = grad f (X) on M

54 Gradient-Flow on Compact Lie Groups Construction in a Nutshell Abstract Optimisation Flows on Lie Groups Similarity Orbits Example: f (U) := Re tr{c UAU } calculate Lie derivative Df (U)(ΩU) = [UAU, C ] S U ΩU obtain gradient vector field grad f (U) = [UAU, C ] S U write [, ] S as skew-herm. part identifying Df (U)(ΩU) = grad f (U) ΩU, where ξ T U SU(N) reads ξ = ΩU and Ω su(n); integrate gradient system U = grad f (U) by discretisation scheme U k+1 = e α k [U k AU k,c ] S U k

55 Gradient-Flow on Compact Lie Groups Construction in a Nutshell Abstract Optimisation Flows on Lie Groups Similarity Orbits Example: f (U) := Re tr{c UAU } calculate Lie derivative Df (U)(ΩU) = [UAU, C ] S U ΩU obtain gradient vector field grad f (U) = [UAU, C ] S U write [, ] S as skew-herm. part identifying Df (U)(ΩU) = grad f (U) ΩU, where ξ T U SU(N) reads ξ = ΩU and Ω su(n); integrate gradient system U = grad f (U) by discretisation scheme U k+1 = e α k [U k AU k,c ] S U k

56 Gradient-Flow on Compact Lie Groups Construction in a Nutshell Abstract Optimisation Flows on Lie Groups Similarity Orbits Example: f (U) := Re tr{c UAU } calculate Lie derivative Df (U)(ΩU) = [UAU, C ] S U ΩU obtain gradient vector field grad f (U) = [UAU, C ] S U write [, ] S as skew-herm. part identifying Df (U)(ΩU) = grad f (U) ΩU, where ξ T U SU(N) reads ξ = ΩU and Ω su(n); integrate gradient system U = grad f (U) by discretisation scheme U k+1 = e α k [U k AU k,c ] S U k

57 Gradient-Flow on Compact Lie Groups Construction in a Nutshell Abstract Optimisation Flows on Lie Groups Similarity Orbits Example: f (U) := Re tr{c UAU } calculate Lie derivative Df (U)(ΩU) = [UAU, C ] S U ΩU obtain gradient vector field grad f (U) = [UAU, C ] S U write [, ] S as skew-herm. part identifying Df (U)(ΩU) = grad f (U) ΩU, where ξ T U SU(N) reads ξ = ΩU and Ω su(n); integrate gradient system U = grad f (U) by discretisation scheme U k+1 = e α k [U k AU k,c ] S U k

58 Generalisation: Flows on Sums of Orbits Approx. by Sums of Similarity Orbits Math. Comp. 80, 1601 (2011) Generalised task: approximate A 0 by elements on the sum of orbits Abstract Optimisation Flows on Lie Groups Similarity Orbits unitary similarity: min U j N U j A j U j A 0 2 j=1 2 N coupled gradient flows: notations: A (j) k := U (j) k U (j) k+1 = exp { α (j) k [A(j) k, A 0jk ] } (j) U S k A j U (j) k ; A0jk := A 0 N ν=1 ν j A (ν) k ; [, ] S skew-herm. part

59 Generalisation: Flows on Sums of Orbits Approx. by Sums of Similarity Orbits Math. Comp. 80, 1601 (2011) Generalised task: approximate A 0 by elements on the sum of orbits Abstract Optimisation Flows on Lie Groups Similarity Orbits unitary similarity: min U j N U j A j U j A 0 2 j=1 2 N coupled gradient flows: notations: A (j) k := U (j) k U (j) k+1 = exp { α (j) k [A(j) k, A 0jk ] } (j) U S k A j U (j) k ; A0jk := A 0 N ν=1 ν j A (ν) k ; [, ] S skew-herm. part

60 Generalisation: Flows on Sums of Orbits Approx. by Sums of Similarity Orbits Math. Comp. 80, 1601 (2011) Generalised task: approximate A 0 by elements on the sum of orbits Abstract Optimisation Flows on Lie Groups Similarity Orbits unitary similarity: min U j N U j A j U j A 0 2 j=1 2 N coupled gradient flows: notations: A (j) k := U (j) k U (j) k+1 = exp { α (j) k [A(j) k, A 0jk ] } (j) U S k A j U (j) k ; A0jk := A 0 N ν=1 ν j A (ν) k ; [, ] S skew-herm. part

61 Generalisation: Sums of Orbits Approx. by Sums of Similarity Orbits Math. Comp. 80, 1601 (2011) Example (unitary similarity) Abstract Optimisation Flows on Lie Groups Similarity Orbits Define in C A j := diag (1, 3, 5,..., 19) + j l 10 A (N) 0 := diag (a 1,..., a 10 ), where a 1,..., a 10 are eigenvalues of N j=1 U(r) j A j U (r) j with random unitaries (distributed by Haar measure). For N = 2 and N = 10 the above flows give: N = 2 A (N) 0 N * Σ Uj j=1 A j U j N = iteration

62 Conclusion for Reachable Sets by Group Orbits J. Math. Phys. 52, (2011) no vs constant vs switchable noise Γ L closed coherently controllable (cc) systems: Reach ρ 0 = O K (ρ 0 ) := {K ρ 0 K K K SU(N)}, where K = exp k generated by system algebra k open systems, cc with constant Markovian noise Γ: Reach ρ 0 = S vec ρ 0, where S e A le A l 1 e A 1 with A 1, A 2,..., A l w open systems, cc with switchable Markovian noise: unital: Reach ρ 0 = {ρ pos 1 ρ ρ 0 } non-unital: Reach ρ 0 = pos 1

63 Conclusion for Reachable Sets by Lie-Semigroup Orbits IEEE TAC 57, 2050 (2012) no vs constant vs switchable noise Γ L closed coherently controllable (cc) systems: Reach ρ 0 = O K (ρ 0 ) := {K ρ 0 K K K SU(N)}, where K = exp k generated by system algebra k open systems, cc with constant Markovian noise Γ: Reach ρ 0 = S vec ρ 0, where S e A le A l 1 e A 1 with A 1, A 2,..., A l w open systems, cc with switchable Markovian noise: unital: Reach ρ 0 = {ρ pos 1 ρ ρ 0 } non-unital: Reach ρ 0 = pos 1

64 Conclusion for Reachable Sets by Unprecedented Noise Control arxiv: & no vs constant vs switchable noise Γ L closed coherently controllable (cc) systems: Reach ρ 0 = O K (ρ 0 ) := {K ρ 0 K K K SU(N)}, where K = exp k generated by system algebra k open systems, cc with constant Markovian noise Γ: Reach ρ 0 = S vec ρ 0, where S e A le A l 1 e A 1 with A 1, A 2,..., A l w open systems, cc with switchable Markovian noise: unital: Reach ρ 0 = {ρ pos 1 ρ ρ 0 } non-unital: Reach ρ 0 = pos 1

65 The Semigroup C-Numerical Range Full Open Systems Control Example (Semigroup Orbit S LK (diag (1, 0, 0, 0)) ) C = 1 diag(1, i, i, 1) 2 A = diag (1, 0, 0, 0) diag (0.9, 0.1, 0, 0) diag (0.8, 0.2, 0, 0) diag (0.7, 0.3, 0, 0) diag (0.6, 0.4, 0, 0) diag (0.5, 0.5, 0, 0) 1 diag (1, 1, 1, 0) 3

66 The Semigroup C-Numerical Range Full Open Systems Control Example (Semigroup Orbit S LK (diag (1, 0, 0, 0)) ) C = 1 diag(1, i, i, 1) 2 A = diag (1, 0, 0, 0) diag (0.9, 0.1, 0, 0) diag (0.8, 0.2, 0, 0) diag (0.7, 0.3, 0, 0) diag (0.6, 0.4, 0, 0) diag (0.5, 0.5, 0, 0) 1 diag (1, 1, 1, 0) 3

67 The Semigroup C-Numerical Range Full Open Systems Control Example (Semigroup Orbit S LK (diag (1, 0, 0, 0)) ) C = 1 diag(1, i, i, 1) 2 A = diag (1, 0, 0, 0) diag (0.9, 0.1, 0, 0) diag (0.8, 0.2, 0, 0) diag (0.7, 0.3, 0, 0) diag (0.6, 0.4, 0, 0) diag (0.5, 0.5, 0, 0) 1 diag (1, 1, 1, 0) 3

68 The Semigroup C-Numerical Range Full Open Systems Control Example (Semigroup Orbit S LK (diag (1, 0, 0, 0)) ) C = 1 diag(1, i, i, 1) 2 A = diag (1, 0, 0, 0) diag (0.9, 0.1, 0, 0) diag (0.8, 0.2, 0, 0) diag (0.7, 0.3, 0, 0) diag (0.6, 0.4, 0, 0) diag (0.5, 0.5, 0, 0) 1 diag (1, 1, 1, 0) 3

69 1 Relative C-Num. Ranges W K (C, A) := tr{c KAK } reflect specific quantum dynamics (via K = exp(k)) come with a geometry not fully explored yet can naturally be generalised to projections of semigroup orbits in the sense W S (C, A) := C SA 2 Gradient Flows on Riem. Manifolds & Lie Groups powerful tool for geometric optimisation and control determine Kraus-rank, decide error correctability

70 1 Relative C-Num. Ranges W K (C, A) := tr{c KAK } reflect specific quantum dynamics (via K = exp(k)) come with a geometry not fully explored yet can naturally be generalised to projections of semigroup orbits in the sense W S (C, A) := C SA 2 Gradient Flows on Riem. Manifolds & Lie Groups powerful tool for geometric optimisation and control determine Kraus-rank, decide error correctability

71 Acknowledgements Thanks go to: II. Relative WC (A) Gunther Dirr, Uwe Helmke, Robert Zeier Chi-Kwong Li, Yiu-Tung Poon, Frederik vom Ende integrated EU programme; excellence network; DFG research group NV centres References: Lin. Multilin. Alg. 56, 3 (2008), Lin. Multilin. Alg. 56, 27 (2008), Rev. Math. Phys. 22, 1 71 (2010), Math. Comput. 80, 1601 (2011) J. Math. Phys. 52, (2011) and 56, (2015) Phys. Rev. A 92, (2015), arxiv: Open Syst. Info. Dyn (2017)