Unitary t-designs. Artem Kaznatcheev. February 13, McGill University

Transcription

1 Unitary t-designs Artem Kaznatcheev McGill University February 13, 2010 Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

2 Preliminaries Basics of quantum mechanics A particle s state is represented by a vector ψ C d Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

3 Preliminaries Basics of quantum mechanics A particle s state is represented by a vector ψ C d Given an orthonormal basis e 1,..., e d, the probability of finding the particle in state e i is e i ψ 2 Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

4 Preliminaries Basics of quantum mechanics A particle s state is represented by a vector ψ C d Given an orthonormal basis e 1,..., e d, the probability of finding the particle in state e i is e i ψ 2 Since the particle has to be in one of the d possible states, we want e 1 ψ e d ψ 2 = 1. This is called a normalized state. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

5 Preliminaries Basics of quantum mechanics A particle s state is represented by a vector ψ C d Given an orthonormal basis e 1,..., e d, the probability of finding the particle in state e i is e i ψ 2 Since the particle has to be in one of the d possible states, we want e 1 ψ e d ψ 2 = 1. This is called a normalized state. States are evolved by acting on them by matrices; i.e. ψ t+1 = M ψ t Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

6 Preliminaries Basics of quantum mechanics A particle s state is represented by a vector ψ C d Given an orthonormal basis e 1,..., e d, the probability of finding the particle in state e i is e i ψ 2 Since the particle has to be in one of the d possible states, we want e 1 ψ e d ψ 2 = 1. This is called a normalized state. States are evolved by acting on them by matrices; i.e. ψ t+1 = M ψ t However, we want the state to remain normalized. Thus, any M must be norm-preserving. In C d this is the group of d-by-d unitary matrices. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

7 Preliminaries Preliminaries: is the topologically compact and connected group of norm preserving (unitary) operators on C d. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

8 Preliminaries Preliminaries: is the topologically compact and connected group of norm preserving (unitary) operators on C d. We can introduce the Haar measure and use it to integrate functions f of U to find their averages: f = f (U) du. For convenience we normalize integration by assuming that du = 1. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

9 Preliminaries Preliminaries: is the topologically compact and connected group of norm preserving (unitary) operators on C d. We can introduce the Haar measure and use it to integrate functions f of U to find their averages: f = f (U) du. For convenience we normalize integration by assuming that du = 1. The goal of unitary t-designs is to evaluate averages of polynomials via a finite sum. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

10 Preliminaries Preliminaries: Hom(r, s) Definition Hom(r, s) is the set of polynomials homogeneous of degree r in entries of U and homogeneous of degree s in U. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

11 Preliminaries Preliminaries: Hom(r, s) Definition Hom(r, s) is the set of polynomials homogeneous of degree r in entries of U and homogeneous of degree s in U. Examples U, V U V UV Hom(2, 2) U U V UV Hom(1, 1) Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

12 Preliminaries Preliminaries: Hom(r, s) Definition Hom(r, s) is the set of polynomials homogeneous of degree r in entries of U and homogeneous of degree s in U. Examples U, V U V UV Hom(2, 2) U U V UV Hom(1, 1) U tr(u U) Hom(1, 1) d Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

13 Preliminaries Preliminaries: Hom(r, s) Definition Hom(r, s) is the set of polynomials homogeneous of degree r in entries of U and homogeneous of degree s in U. Examples U, V U V UV Hom(2, 2) U U V UV Hom(1, 1) U tr(u U) Hom(1, 1) d U, V tr(u V )U 2 + VU VU Hom(3, 1) U tr(u V )U 2 + VU }{{}}{{ VU} Hom(2,1) Hom(1,1) / Hom(2, 1) Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

14 Functional definition Functional definition of unitary t-designs Definition A function w : X (0, 1] is a weight function on X if for all U X we have w(u) > 0 and U X w(u) = 1 Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

15 Functional definition Functional definition of unitary t-designs Definition A function w : X (0, 1] is a weight function on X if for all U X we have w(u) > 0 and U X w(u) = 1 Definition A tuple (X,w) with finite X and weight function w on X is a unitary t-design if w(u)f (U) = f (U) du for all f Hom(t, t). U X Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

16 Functional definition Functional definition is general enough Proposition Every t-design is a (t 1)-design. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

17 Functional definition Functional definition is general enough Proposition Every t-design is a (t 1)-design. Proposition For any f Hom(r, s) with r s f (U) du = 0 Lemma For any f Hom(r, s), U, and c C we have f (cu) = c r c s f (U) Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

18 Functional definition Strengths and shortcomings of the functional definition Strengths: Average of any polynomial with degrees in U and U less than t can be evaluated one summand at a time. Multi-variable polynomials can be evaluated: f (U 1,..., U n )du 1...dU n = U 1 X... U n X w(u 1 )...w(u n )f (U 1,..., U n ). Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

19 Functional definition Strengths and shortcomings of the functional definition Strengths: Average of any polynomial with degrees in U and U less than t can be evaluated one summand at a time. Multi-variable polynomials can be evaluated: f (U 1,..., U n )du 1...dU n Shortcomings: = U 1 X... U n X w(u 1 )...w(u n )f (U 1,..., U n ). Not clear how to test if a given (X, w) is a t-design. If (X, w) is not a design, then how far away is it? Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

20 Tensor product definition Tensor product definition of unitary t-designs Definition A tuple (X,w) with finite X and weight function w on X is a unitary t-design if w(u)u t (U ) t = U t (U ) t du U X Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

21 Tensor product definition Tensor product definition of unitary t-designs Definition A tuple (X,w) with finite X and weight function w on X is a unitary t-design if w(u)u t (U ) t = U t (U ) t du U X More tractable for checking if an arbitrary (X, w) is a t-design. Literature has explicit formula for the RHS for many choices of d and t [Col03, CS06]. Still not metric. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

22 Metric definition Metric definition of unitary t-designs Definition A weight function w is a proper weight function on X if for all other choices of weight function w on X, we have: w(u)w(v ) tr(u V ) 2t w (U)w (V ) tr(u V ) 2t. U,V X U,V X The trace double sum is a function Σ defined for finite X as: Σ(X ) = w(u)w(v ) tr(u V ) 2t, U,V X Definition A finite X is a unitary t-design if Σ(X ) = tr(u) 2t Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

23 Metric definition Strengths and shortcomings of the metric definition Strengths: Σ(X ) > tr(u) 2t if X is not a t-design. This gives us a useful metric to say how far a set with proper weight function is from being a design. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

24 Metric definition Strengths and shortcomings of the metric definition Strengths: Σ(X ) > tr(u) 2t if X is not a t-design. This gives us a useful metric to say how far a set with proper weight function is from being a design. tr(u) 2t has a nice combinatorial interpertation: the number of permutations of {1,..., t} with no increasing subsequences of order greater than d [DS94, Rai98]. If d t then RHS is t!. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

25 Metric definition Strengths and shortcomings of the metric definition Strengths: Σ(X ) > tr(u) 2t if X is not a t-design. This gives us a useful metric to say how far a set with proper weight function is from being a design. tr(u) 2t has a nice combinatorial interpertation: the number of permutations of {1,..., t} with no increasing subsequences of order greater than d [DS94, Rai98]. If d t then RHS is t!. One of the easiest way to test if X is a t-design Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

26 Metric definition Strengths and shortcomings of the metric definition Strengths: Σ(X ) > tr(u) 2t if X is not a t-design. This gives us a useful metric to say how far a set with proper weight function is from being a design. tr(u) 2t has a nice combinatorial interpertation: the number of permutations of {1,..., t} with no increasing subsequences of order greater than d [DS94, Rai98]. If d t then RHS is t!. One of the easiest way to test if X is a t-design Shortcomings: Does not give any insight into what t-designs are useful for. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

27 Small designs Characterization of minimal t-designs Definition A minimal t-design X is a t-design such that all Y X are not t-designs. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

28 Small designs Characterization of minimal t-designs Definition A minimal t-design X is a t-design such that all Y X are not t-designs. Theorem A t-design X is minimal if and only if it has a unique proper weight function w. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

29 Small designs Characterization of minimal t-designs Definition A minimal t-design X is a t-design such that all Y X are not t-designs. Theorem A t-design X is minimal if and only if it has a unique proper weight function w. Useful tool for proving minimality. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

30 Small designs Characterization of minimal t-designs Definition A minimal t-design X is a t-design such that all Y X are not t-designs. Theorem A t-design X is minimal if and only if it has a unique proper weight function w. Useful tool for proving minimality. Sadly, minimal designs are not necessarily minimum. Currently working on finding correspondences between minimal and minimum designs. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

31 Small designs A lower bound on the size of t-designs Proposition If X is a t-design then X d 2t tr(u) 2t. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

32 Small designs A lower bound on the size of t-designs Proposition If X is a t-design then X d 2t tr(u) 2t. Best known bounds are by Roy and Scott [RS08]: X ( ) d 2 +t 1 t Asymptotically, for large d and fixed t, both bounds are Θ(d 2t ) Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

33 Using designs 1-design 1-design construction Let e 1... e d be an orthonormal basis of C d that is mutually unbiased with the standard basis. Define I i = ddiag( e i ) for 1 i d. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

34 Using designs 1-design 1-design construction Let e 1... e d be an orthonormal basis of C d that is mutually unbiased with the standard basis. Define I i = ddiag( e i ) for 1 i d. Consider the cyclic permutation group of order d, represented as d-by-d matrices: C 1...C d where C d = C 0 = I. Define C m i = C m I i Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

35 Using designs 1-design 1-design construction Let e 1... e d be an orthonormal basis of C d that is mutually unbiased with the standard basis. Define I i = ddiag( e i ) for 1 i d. Consider the cyclic permutation group of order d, represented as d-by-d matrices: C 1...C d where C d = C 0 = I. Define C m i = C m I i For any tuple 1 i, j, m, n d we have: tr((ci m ) Cj n ) = tr(ii C d m+n I j ) = { d if i = j and m = n 0 otherwise Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

36 Using designs Evaluating [, V ] Evaluating the average commutator over Theorem For any V and [U, V ] = U V UV we have: [, V ] = tr(v ) V d Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

37 Using designs Evaluating [, V ] Proof of EAC Consider the diagonalization of V, i.e. V = P DP, with D = diag(λ 1,..., λ d ). [ U V UV du = ] [ ] U V U du V = U P DPU du V Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

38 Using designs Evaluating [, V ] Proof of EAC Consider the diagonalization of V, i.e. V = P DP, with D = diag(λ 1,..., λ d ). [ U V UV du = ] [ ] U V U du V = U P DPU du V But we know a symmetry that allows substituting PU U without changing the average. U P DPU du = U DU du Let f (U) = U DU. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

39 Using designs Evaluating [, V ] Proof of EAC Consider the diagonalization of V, i.e. V = P DP, with D = diag(λ 1,..., λ d ). [ U V UV du = ] [ ] U V U du V = U P DPU du V But we know a symmetry that allows substituting PU U without changing the average. U P DPU du = U DU du Let f (U) = U DU. Look at the elements of the design: f (Ci m ) = Ii (C m ) DC m I i. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

40 Using designs Evaluating [, V ] Proof of EAC Consider the diagonalization of V, i.e. V = P DP, with D = diag(λ 1,..., λ d ). [ U V UV du = ] [ ] U V U du V = U P DPU du V But we know a symmetry that allows substituting PU U without changing the average. U P DPU du = U DU du Let f (U) = U DU. Look at the elements of the design: f (Ci m ) = Ii (C m ) DC m I i. (C m ) DC m = diag(λ c m (1),..., λ c m (d)) Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

41 Using designs Evaluating [, V ] Proof of EAC Consider the diagonalization of V, i.e. V = P DP, with D = diag(λ 1,..., λ d ). [ U V UV du = ] [ ] U V U du V = U P DPU du V But we know a symmetry that allows substituting PU U without changing the average. U P DPU du = U DU du Let f (U) = U DU. Look at the elements of the design: f (Ci m ) = Ii (C m ) DC m I i. (C m ) DC m = diag(λ c m (1),..., λ c m (d)) Thus, f = λ λ d d I Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

42 Using designs t-designs are non-commuting t-designs are non-commuting Theorem For all d 2 if X is a minimal t-design then X has a trivial center. Supports our intuition that designs must be well spread out. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

43 Conclusion Concluding remarks Introduced 3 definitions of unitary t-designs Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

44 Conclusion Concluding remarks Introduced 3 definitions of unitary t-designs Classified minimal designs: a t-design is minimal if and only if it has a unique proper weight function. Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

45 Conclusion Concluding remarks Introduced 3 definitions of unitary t-designs Classified minimal designs: a t-design is minimal if and only if it has a unique proper weight function. Used an orthonormal basis of C d d as a 1-design. Evaluated the average commutator on : [, V ] = tr(v ) d V Showed that t-designs are non-commuting Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

46 Conclusion Concluding remarks Introduced 3 definitions of unitary t-designs Classified minimal designs: a t-design is minimal if and only if it has a unique proper weight function. Used an orthonormal basis of C d d as a 1-design. Evaluated the average commutator on : [, V ] = tr(v ) d V Showed that t-designs are non-commuting Thank you for listening! Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

47 Conclusion References I B. Collins. Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability. International Mathematics Research Notices, pages , B. Collins and P. Śniady. Integration with respect to the haar measure on unitary, orthogonal and symplectic group. Communications in Mathematical Physics, 264: , P. Diaconis and M. Shahshahani. On the eigenvalues of random matrices. Journal of Applied Probability, 31A:49 62, E. M. Rains. Increasing subsequences and the classical groups. Electronic Journal of Combinatorics, 5:Research Paper 12, 9 pp., Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16

48 Conclusion References II A. Roy and A. J. Scott. Unitary designs and codes Artem Kaznatcheev (McGill University) Unitary t-designs February 13, / 16