Numerical radius, entropy and preservers on quantum states

Transcription

1 2th WONRA, TSIMF, Sanya, July 204 Numerical radius, entropy and preservers on quantum states Kan He (å Ä) Joint with Pro. Jinchuan Hou, Chi-Kwong Li and Zejun Huang Taiyuan University of Technology, Taiyuan, Shanxi,

2 The talk is based on the following papers. [] Kan He, Jinchuan Hou, Chi-Kwong Li, A geometric characterization of invertible quantum measurement maps, Journal of Functional Analysis, 264 (203), [2] Kan He, Jinchuan Hou, Quantum measurement maps, quantum gates and convex combinations preserving maps on Hilbert spaces, preprint [3] Kan He, Jinchuan Hou, Entropy-preserving maps on quantum sates, preprint, many thanks for many suggestions from Professor Chi-Kwong Li. [4] Kan He, Jinchuan Hou, Transformations on quantum states leaving numerical radius invariant, preprint, putted forward by Professor Chi- Kwong Li. [5] Kan He, Zejun Huang, Jinchuan Hou, Chi-Kwong Li, Linear maps preserving quantum entropy of tensor product of matrices, preprint

3 Outline

4 Outline (I) Numerical radius and quantum states (II) Convex combinations preserving transformations on quantum states (III) Transformations on quantum states leaving numerical radius (or quantum entropy) invariant (IV) Preservers of tensor product operators (V) Further discussion

5 (I) Numerical radius and quantum states

6 (I) Numerical radius and quantum states Recall that the numerical range of an operator A is the set W (A) = { Ax, x : x H, x = }, and the numerical radius of A is w(a) = sup{ λ : λ W (A)}.

7 The C-numerical range of A is the set W C (A) = {tr(cuau ) : U is unitary}, and the C-numerical radius of A is w C (A) = sup{ λ : λ W C (A)}. Here, C can be chosen as a trace-class operator in infinite dimensional case.

8 The C-numerical range of A is the set W C (A) = {tr(cuau ) : U is unitary}, and the C-numerical radius of A is w C (A) = sup{ λ : λ W C (A)}. Here, C can be chosen as a trace-class operator in infinite dimensional case. If C is a rank one projection, then W C (A) = W (A).

9 The topic of numerical range and numerical radius plays an important role in Mathematics and Physics. Thomas Schulte-Herbruggen, Gunther Dirr, Uwe Helmke, Steffen J. Glaser, The significance of the C-numerical range and the local C- numerical range in quantum control and quantum information, Linear and Multilinear Algebra, Volume 56, Issue,2(2008), pages F. F. Bonsall. Duncan, Numerical Ranges, Vol. I and II, Cambridge University Press, 97 and 973.

10 The topic of numerical range and numerical radius plays an important role in Mathematics and Physics. Thomas Schulte-Herbruggen, Gunther Dirr, Uwe Helmke, Steffen J. Glaser, The significance of the C-numerical range and the local C- numerical range in quantum control and quantum information, Linear and Multilinear Algebra, Volume 56, Issue,2(2008), pages F. F. Bonsall. Duncan, Numerical Ranges, Vol. I and II, Cambridge University Press, 97 and 973. In the Mathematical framework of quantum physics, a quantum system is a Hilbert space H, quantum states are trace-one positive operators on H.

11 The topic of numerical range and numerical radius plays an important role in Mathematics and Physics. Thomas Schulte-Herbruggen, Gunther Dirr, Uwe Helmke, Steffen J. Glaser, The significance of the C-numerical range and the local C- numerical range in quantum control and quantum information, Linear and Multilinear Algebra, Volume 56, Issue,2(2008), pages F. F. Bonsall. Duncan, Numerical Ranges, Vol. I and II, Cambridge University Press, 97 and 973. In the Mathematical framework of quantum physics, a quantum system is a Hilbert space H, quantum states are trace-one positive operators on H. The numerical radius and C-numerical radius of the quantum state are applied into the theory of quantum control of quantum systems.

12 How is the C-numerical range (radius) applied in quantum control?

13 How is the C-numerical range (radius) applied in quantum control? In a quantum system, let A be the initial state and C the target state,

14 How is the C-numerical range (radius) applied in quantum control? In a quantum system, let A be the initial state and C the target state, as we know, to arrive to the target state C from A, the approach is unitary similarity transformation: UAU C.

15 How is the C-numerical range (radius) applied in quantum control? In a quantum system, let A be the initial state and C the target state, as we know, to arrive to the target state C from A, the approach is unitary similarity transformation: UAU C. Generally speaking, there is no unitary U such that UAU = C. So one task in quantum control is to find points closed to C in the unitary orbit of A {UAU U is unitary}.

16 The minimal Euclidean distance C UAU 2 is a measure of difference between UAU and C.

17 The minimal Euclidean distance C UAU 2 is a measure of difference between UAU and C. Since C UAU 2 2 = A C 2 2 2Retr(CUAU ),

18 The minimal Euclidean distance C UAU 2 is a measure of difference between UAU and C. Since C UAU 2 2 = A C 2 2 2Retr(CUAU ), maximality of the real part of the C-numerical range of A means the minimal Euclidean distance.

19 The minimal Euclidean distance C UAU 2 is a measure of difference between UAU and C. Since C UAU 2 2 = A C 2 2 2Retr(CUAU ), maximality of the real part of the C-numerical range of A means the minimal Euclidean distance. So C-numerical ranges can be a measure to help us finds points in unitary orbit of A with minimal distance to C.

20 Another measure of difference between UAU and C can be developed by the C-numerical radius of A: cos(a, C) U = C, UAU A 2 C 2 = tr(cuau ) A 2 C 2.

21 Another measure of difference between UAU and C can be developed by the C-numerical radius of A: cos(a, C) U = C, UAU A 2 C 2 = tr(cuau ) A 2 C 2. The C-numerical radius of A is the maximal value of tr(cuau ), and means the minimal angle of UAU, C.

22 (II) Convex combinations preserving transformations on quantum states

23 (II) Convex combinations preserving transformations on quantum states The topic of transformations on quantum states is one of important aspects in quantum information.

24 (II) Convex combinations preserving transformations on quantum states The topic of transformations on quantum states is one of important aspects in quantum information. For example, quantum channels and quantum operations are completely positive linear maps;

25 (II) Convex combinations preserving transformations on quantum states The topic of transformations on quantum states is one of important aspects in quantum information. For example, quantum channels and quantum operations are completely positive linear maps; quantum gates are unitary operators.

26 (II) Convex combinations preserving transformations on quantum states The topic of transformations on quantum states is one of important aspects in quantum information. For example, quantum channels and quantum operations are completely positive linear maps; quantum gates are unitary operators. Therefore, it is helpful to know the characterizations of maps on quantum states leaving invariant some important subsets or quantum properties. Such questions have attracted the attention of many researchers.

27 Let S(H) be the convex set of all quantum states on Hilbert space H.

28 Let S(H) be the convex set of all quantum states on Hilbert space H. Kadison s Theorem φ is a convex isomorphism from S(H) to S(H), i.e. φ is bijective and for any states ρ, ρ 2 and any non-negative scalar p, φ satisfies φ(pρ + ( p)ρ 2 ) = pφ(ρ ) + ( p)φ(ρ 2 ) if and only if there is a unitary or anti-unitary operator U on H such that φ(ρ) = UρU for all ρ S(H). I. Bengtsson, K. Życzkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, Cambridge University Press, Cambridge, 2006.

29 We generalize the Kadison s theorem. Theorem [Kan He, Jinchuan Hou, Chi-Kwong Li, J Func. Anal., 264, ] A bijective map φ : S(H) S(H) satisfies φ([ρ, ρ 2 ]) [φ(ρ ), φ(ρ 2 )] for any ρ, ρ 2 S(H) if and only if there is an invertible bounded linear operator M B(H) such that φ has the form ρ MρM tr(mρm ) or ρ MρT M tr(mρ T M ), where ρ T is the transpose of ρ with respect to an orthonormal basis.

30 Observe that φ([ρ, ρ 2 ]) [φ(ρ ), φ(ρ 2 )] for any t [0, ], there exists s [0, ] such that φ(tρ + ( t)ρ 2 ) = sφ(ρ ) + ( s)φ(ρ 2 ), such a map φ is called convex combinations preserving map.

31 Observe that φ([ρ, ρ 2 ]) [φ(ρ ), φ(ρ 2 )] for any t [0, ], there exists s [0, ] such that φ(tρ + ( t)ρ 2 ) = sφ(ρ ) + ( s)φ(ρ 2 ), such a map φ is called convex combinations preserving map. What is the physical meaning of Theorem?

32 Observe that φ([ρ, ρ 2 ]) [φ(ρ ), φ(ρ 2 )] for any t [0, ], there exists s [0, ] such that φ(tρ + ( t)ρ 2 ) = sφ(ρ ) + ( s)φ(ρ 2 ), such a map φ is called convex combinations preserving map. What is the physical meaning of Theorem? Recall that in quantum mechanics, a quantum measurement is described by a collection {M m } of measurement operators acting on the state space H satisfying Σ m M mm m = I.

33 Observe that φ([ρ, ρ 2 ]) [φ(ρ ), φ(ρ 2 )] for any t [0, ], there exists s [0, ] such that φ(tρ + ( t)ρ 2 ) = sφ(ρ ) + ( s)φ(ρ 2 ), such a map φ is called convex combinations preserving map. What is the physical meaning of Theorem? Recall that in quantum mechanics, a quantum measurement is described by a collection {M m } of measurement operators acting on the state space H satisfying Σ m M mm m = I. Let M j be a quantum measurement operator. If the state of the quantum system is ρ S(H) before the measurement, then the state after the M j ρmj measurement is tr(m j ρmj ) whenever M jρmj 0.

34 If M j is fixed, we get a quantum measurement map φ j defined by φ j (ρ) = M jρmj tr(m j ρmj ) convex subset S M (H) = {ρ : M j ρmj from the 0} of the (convex) set S(H) of states into S(H). If M j is invertible, then φ j : S(H) S(H) is bijective and will be called an invertible measurement map.

35 If M j is fixed, we get a quantum measurement map φ j defined by φ j (ρ) = M jρmj tr(m j ρmj ) convex subset S M (H) = {ρ : M j ρmj from the 0} of the (convex) set S(H) of states into S(H). If M j is invertible, then φ j : S(H) S(H) is bijective and will be called an invertible measurement map. Physical meaning of Theorem : It follows that an invertible quantum measurement map φ is represented as a convex combinations preserving bijective map or the composition of such a preserver and the transpose.

36 Next we turn to convex combinations preserving transformations on unit balls of Hilbert spaces.

37 Next we turn to convex combinations preserving transformations on unit balls of Hilbert spaces. A quantum system can be simulated well by wave functions, which are described by many different ways. Being regarded as square integrable functions with respect to an appropriate measure, wave functions are unit vectors in the Hilbert space H = L 2 (µ). Denote by B (H) the closed unit ball of H, W(H) the sphere of B (H).

38 Next we turn to convex combinations preserving transformations on unit balls of Hilbert spaces. A quantum system can be simulated well by wave functions, which are described by many different ways. Being regarded as square integrable functions with respect to an appropriate measure, wave functions are unit vectors in the Hilbert space H = L 2 (µ). Denote by B (H) the closed unit ball of H, W(H) the sphere of B (H). Mx For a pure state x, a measurement map φ satisfies φ(x) = Mx, where M is a bounded linear operator on H and Mx = 0, and if M is invertible, we also call φ the invertible measurement map of pure states.

39 Theorem 2 [Kan He, Jinchuan Hou, preprint] Assume φ : B (H) B (H) is a bijective map, then the following statements are equivalent: (i) for arbitrary t [0, ] and x, y B (H), there exists s [0, ] such that φ(tx+( t)y) = sφ(x)+ ( s)φ(y); (ii) there exists an invertible bounded real linear operator M on H, a nonzero constant real scalar r and a bounded real linear functional h : H R such that φ has the form x Mx h(x) + r, for all x B (H), and h(x) + r = Mx for all unit vectors x.

40 Quantum gates are unitary operators. Regarding a quantum gate as a map φ on state spaces, we have where U is unitary. φ(x) = Ux,

41 Quantum gates are unitary operators. Regarding a quantum gate as a map φ on state spaces, we have where U is unitary. φ(x) = Ux, The following corollary uncovers partially connection between quantum gates and convex isomorphisms on B (H).

42 Theorem 3 [Kan He, Jinchuan Hou, preprint] Assume φ : B (H) B (H) is a bijective map, then the following statements are equivalent: (i) φ is a convex isomorphism, i.e., φ(tx + ( t)y) = tφ(x) + ( t)φ(y) for arbitrary t [0, ] and x, y B (H). (ii) there exists an invertible isometric real linear operator U on H such that φ has the form x Ux, for all x B (H).

43 (III) Transformations on quantum states leaving numerical radius (or quantum entropy) invariant

44 (III) Transformations on quantum states leaving numerical radius (or quantum entropy) invariant There have been a great deal of different and active researches on numerical ranges and numerical radii.

45 (III) Transformations on quantum states leaving numerical radius (or quantum entropy) invariant There have been a great deal of different and active researches on numerical ranges and numerical radii. One of such subjects is to characterize linear or general maps on operator algebras or operator spaces which preserve the numerical range or numerical radius. Z.-F. Bai, J.-C. Hou, Z. B. Xu, Maps preserving numerical radius on C*-algebras, Studia Math., 62 (2004), J.-T. Chan, Numerical radius preserving operators on B(H), Proc. Amer. Math. Soc., 23 (995),

46 J.-T. Chan, Numerical radius preserving operators on C -algebras, Arch. Marh. (Basel), 70 (998), J.-C. Hou, J.-L. Cui, Introdution to linear maps on operator algebra, Science Press, Beijing, C.-K. Li and N.-K. Tsing, Linear preserver problem: A brief introduction and some special techniques, Linear Algebra Appl. S. Pierce et al., A survey of linear preserver problems, Linear and Multilinear Algebra 33 (992), -2. Z.-F. Bai, J.-C. Hou, Numerical radius distance preserving maps on B(H), Proc. Amer. Math. Soc., 32 (2004), J.-L. Cui, J.-C. Hou, Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras, J. Funct. Anal., 206 (2004), 44-

47 448. J.-C. Hou, Q.-H. Di, Maps preserving numerical range of operator products, Proc. Amer. Math. Soc., 34 (2006), C.-K. Li, N.-S. Sze, Product of Operators and Numerical Range Preserving Maps, Studia Math., 74 (2006), Kan He, Jinchuan Hou, Xiuling Zhang, Maps preserving numerical radius or cross norms of products of self-adjoint operators, Acta Mathematica Sinica, English Series, 26 (200), Jinchuan Hou, Kan He, Xiuling Zhang, Nonlinear maps preserving numerical radius of indefinite skew products of operators, Linear Algebra and its Applications, 430 (2009),

48 A map Φ : S(H) S(H) preserves numerical radius of convex combinations if Φ satisfies that w(λρ + ( λ)σ) = w(λφ(ρ) + ( λ)φ(σ)) for ρ, σ S(H) and all λ [0, ].

49 A map Φ : S(H) S(H) preserves numerical radius of convex combinations if Φ satisfies that w(λρ + ( λ)σ) = w(λφ(ρ) + ( λ)φ(σ)) for ρ, σ S(H) and all λ [0, ]. Here, we devote to characterizing surjective maps on quantum states preserving numerical radius of convex combinations. Such a problem is from discussion with Professor Chi-Kwong Li.

50 A map Φ : S(H) S(H) preserves numerical radius of convex combinations if Φ satisfies that w(λρ + ( λ)σ) = w(λφ(ρ) + ( λ)φ(σ)) for ρ, σ S(H) and all λ [0, ]. Here, we devote to characterizing surjective maps on quantum states preserving numerical radius of convex combinations. Such a problem is from discussion with Professor Chi-Kwong Li. However, we obtain the following more refined result.

51 Theorem 4 [Kan He, Jinchuan Hou, preprint] If Φ : S(H) S(H) is a surjective map, then the following statements are equivalent: (i) Φ preserves numerical radius of convex combinations of quantum states; (ii) there exists some λ 0 (0, ) such that w(λ 0 ρ+ ( λ 0 )σ) = w(λ 0 Φ(ρ) + ( λ 0 )Φ(σ)) for ρ, σ S(H); (iii) there exists a unitary or anti-unitary operator U on H such that Φ(ρ) = UρU for all ρ S(H).

52 Remark that since the numerical radius of a quantum state equals to its operator norm and spectral radius, one can obtain the same characterization of maps on quantum states preserving the operator norm (spectral radius) of convex combinations.

53 We also are interested in characterizing maps on quantum states (positive operators) preserving other kinds of functions. David Li-Wei Kuo, Ming-Cheng Tsai, Ngai-Ching Wong, and Jun Zhang, Maps Preserving Schatten p-norms of Convex Combinations, Abstract and Applied Analysis Volume 204 (204), Article ID G. Nagy, Isometries on positive operators of unit norm, Publicationes Mathematicae Debrecen, vol. 82, pp , 203. Yuan Li, Paul Busch, Von Neumann entropy and majorization, arxiv: v2 L. Molnár, G. Nagy, Transformations on density operators that leave the Holevo bound invariant, Int. J. Theor. Phys., to appear.

54 For a quantum state ρ, the quantum entropy of ρ is defined as follows: S(ρ) = tr(ρlog 2 ρ).

55 For a quantum state ρ, the quantum entropy of ρ is defined as follows: S(ρ) = tr(ρlog 2 ρ). Also quantum entropy is called von Neumman entropy, and plays an important role in the theory of quantum information. M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer-Verlag, Heidelberg, 993. Kan He, Jinchuan Hou, Ming Li, A von Neumann entropy condition of unitary equivalence of quantum states, Applied Mathematics Letters, 25(202),

56 M. Nielsen, I. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge, 2000.

57 M. Nielsen, I. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge, In finite dimensional case, assume that {p i } n i= are eigenvalues of ρ S(H) with dimh = n < and the Shannon entropy H({p i } n i=) = Σ n i=p i log 2 p i then quantum entropy of ρ equals to the Shannon entropy H({p i } n i= ), where 0log 20 = 0 and log 2 = 0.

58 From the definition of quantum entropy, an elementary discussion show the following interesting conclusions.

59 From the definition of quantum entropy, an elementary discussion show the following interesting conclusions. Quantum entropy is non-negative, i.e., S(ρ) 0 for arbitrary ρ S(H). S(ρ) = 0 if and only if ρ is a rank one projection. S(ρ) = log 2 n if and only if ρ = n I, where I is the identity on H.

60 From the definition of quantum entropy, an elementary discussion show the following interesting conclusions. Quantum entropy is non-negative, i.e., S(ρ) 0 for arbitrary ρ S(H). S(ρ) = 0 if and only if ρ is a rank one projection. S(ρ) = log 2 n if and only if ρ = n I, where I is the identity on H. Let ρ i s be quantum states, i =, 2,..., k and p i s are non-negative scalars with Σ k i= p i =, then Σ k i= p is(ρ i ) S(Σ k i= p iρ i ) H(p i ) + Σ k i= p is(ρ i ).

61 Here, we also characterizing surjective maps on quantum states preserving quantum entropy of convex combinations.

62 Theorem 5 [Kan He, Jinchuan Hou, preprint] Let S(H) be the set of all quantum states on a complex Hilbert space H with dimh = n and < n <. For a surjective map φ : S(H) S(H), the following statements are equivalent: (I) φ satisfies for arbitrary ρ, σ S(H) and t [0, ], S(tρ + ( t)σ) = S(tφ(ρ) + ( t)φ(σ)); (II) there exists a unitary or anti-unitary operator U such that φ(ρ) = UρU for all ρ S(H).

63 (IV) Preservers of tensor product operators

64 (IV) Preservers of tensor product operators A multipartite quantum system is described by the tensor product of Hilbert spaces. For the finite dimensional bipartite system H A H B, a state ρ AB S(H A H B ) is separable if ρ AB = Σ i p i ρ A i σi B, where ρa i and σi B are states on H A, H B respectively. Otherwise, ρ AB is called an entangled state. Entanglement is one of the most important aspects of quantum information.

65 Many authors pay their attentions to determining the structure of linear preservers of tensor product operators. S. Friedland, C.K. Li, Y.T. Poon and N.S. Sze, The automorphism group of separable states in quantum information theory, Journal of Mathematical Physics 52 (20), E. Alfsen and F. Shultz, Unique decompositions, faces, and automorphisms of separable states, Journal of Mathematical Physics 5 (200),

66 In the finite dimensional bipartite system H A H B with dimh A = m and dimh B = n, let H mn be the space of all mn mn Hermitian matrices, D mn the set of all mn mn quantum states and S mn the set of all mn mn separable states. Denote by S(ρ) the quantum entropy of ρ D mn and σ(a) the spectrum of A.

67 In the finite dimensional bipartite system H A H B with dimh A = m and dimh B = n, let H mn be the space of all mn mn Hermitian matrices, D mn the set of all mn mn quantum states and S mn the set of all mn mn separable states. Denote by S(ρ) the quantum entropy of ρ D mn and σ(a) the spectrum of A. Ajda Fošner, Zejun Huang, Chi-Kwong Li, Nung- Sing Sze show that for a linear map φ : H mn H mn, σ(φ(a B)) = σ(a B) for A H m and B H n if and only if there exists a unitary matrix U such that φ(a B) = Uφ (A) φ 2 (B)U, where φ j is the identity or transposition, j =, 2. Ajda Fošner, Zejun Huang, Chi-Kwong Li, Nung-Sing Sze, Linear preservers and quantum information science, Linear and Multilinear algebra, 203, 6,

68 The structure of linear maps preserving numerical radius of tensor products of matrices is determined in the following paper. Ajda Fošner, Zejun Huang, Chi-Kwong Li, Nung-Sing Sze, Linear maps preserving numerical radius of tensor products of matrices, J Math. Anal. Appl., 203, 407, 83-89

69 Theorem 6 [Kan He, Zejun Huang, Jinchuan Hou, Chi-Kwong Li, preprint] Let φ : H mn H mn be a linear map, then the following statements are equivalent: (I) φ(d mn ) D mn, S(φ(ρ 2 )) = S(ρ 2 ) for ρ 2 S mn ; (II) σ(φ(a B)) = σ(a B) for A H m and B H n ; (III) there exists a unitary matrix U such that φ(a B) = Uφ (A) φ 2 (B)U, where φ j is the identity or transposition, j =, 2.

70 (V) Further discussion

71 (V) Further discussion Determining the structure of C-numerical radius preserving maps on quantum states.

72 (V) Further discussion Determining the structure of C-numerical radius preserving maps on quantum states. Determining the structure of more generalized functional values preserving maps on quantum states, for instance, Shur-convex functions (putted forward by and many suggestions from Professor Chi- Kwong Li).

73 . Thanks! 2