Masakazu Jimbo Nagoya University

Transcription

1 2012 Shanghai Conference on Algebraic Combinatorics August 17 22, 2012, Shanghai Jiao Tong University Mutually orthogonal t-designs over C related to quantum jump codes Masakazu Jimbo Nagoya University jimbo@is.nagoya-u.ac.jp This is partially joint work with K. Shiromoto and Y. Lin.

2 Extension of designs from binary to C A quantum jump code was introduced by Alber et al. (2001). It has a close connection with combinatorial designs. ) A combinatorial design (V, B) is considered as a map f : ( V k {0, 1} according as a block B ( ) V k being a member of block set B or not. In this talk, we consider a family of mutually orthogonal partial t-designs ( partial t-mod ) {(V, f i ) i = 1, 2,..., m} with membership functions f i : ( ) V k C, which is equivalent to a quantum jump code introduced by Alber et al. (2001).

3 References of quantum jump codes [1 ] G. Alber, T. Beth, C. Charnes, A. Delgado, M. Grassl and M. Mussinger, Stabilizing distinguishable qubits against spontaneous decay by detected-jump correcting quantum codes, Physical Review Letters, 86 (2001) [2 ] G. Alber, T. Beth, C. Charnes, A. Delgado, M. Grassl and M. Mussinger, Detected-jump-error-correcting quantum codes, quantum error designs, and quantum computation. Physical Review A, 68 (2003), [3 ] T. Beth, C. Charnes, M. Grassl, G. Alber, A. Delgado, M. Mussinger, A new class of designs which protect against quantum jumps. Designs, Codes and Cryptography, 29 (2003), [4 ] O. Kern and G. Alber, Suppressing decoherence of quantum algorithms by jump codes, European Physical J. D, 36 (2005) [5 ] M. Jimbo and K. Shiromoto, Quantum jump codes and related combinatorial designs. In D. Crnković and V. Tonchev eds. Nato Series - D: ICS Vol. 29, Information Security, Coding Theory and Related Combinatorics, (2011)

4 A design over C Let V be a v-set. For a positive integer k( v), let f : ( ) V k C be a map, where C is the set of complex numbers. A pair (V, f) is called a design or a configuration over C and f is called the membership function of a design (V, f). If we restrict the image of f to {0, 1} ( C), (V, f) is called a binary design (configuration). Classical design!!

5 An inner product of two designs (V, f 1 ) and (V, f 2 ) is defined by f 1 f 2 = B ( V k ) f 1 (B)f 2 (B), where x is the complex conjugate of x. The (L 2 -)norm of a design (V, f) is defined by f 2 = f f. For an s-subset S V and B ( ) V \S k s, we define f S (B) = f(b S). (V \ S, f S ) is called a derived design by S of (V, f). Note that f = f.

6 Mutually orthogonal designs over C For a positive integer t < k, fix an index function µ : ( ) V t R 0, where R 0 is the set of nonnegative real numbers. For a family of m designs (V, f i ) (i = 1, 2,..., m), let f = t (f 1,..., f m ). (V ; f) = (V ; f 1,..., f m ) is called a mutually orthogonal partial t-designs over C, denoted by partial t-[v, k, µ; m] MOD, if for any T ( ) V t and for any i and j, holds. f i T f j T = µ(t ) f i 2 for i = j, 0 for i j

7 If we restrict the image of f i to {0, 1}, a partial t-mod is called a binary partial t-mod, or a t-seed (t-spontaneous emmission error design). In this case, we can identify f i with B (i) = {B f i (B) = 1}. Hence, a partial t-mod (V ; f 1,..., f m ) is also written by (V ; B (1),..., B (m) ). For a binary partial t-mod, the following hold: 1. The orthogonality f i T f j T = 0 implies disjointness B (i) B (j) =. 2. f i 2 = B (i) is the number of blocks in B (i). 3. λ(t ) = f i T f i T = f i T 2 = µ(t ) f i 2 = {B B (i) B T } is the number of blocks in B (i) containing T.

8 A t-mod and a t-design over C A partial t-mod is called a t-[v, k, µ t ; m] MOD if µ(t ) is constant (= µ t ). For a t-mod, each design is called t-[v, k, µ t ] design. A binary t-[v, k, µ t ] design is a (classical) binary t-(v, k, λ) design with λ = µ t f 2.

9 Examples of t-mods Example 1 A binary 1-[4, 2, 1 2 ; 3] MOD Let V = {0, 1, 2, 3}, and k = 2. f f f K 4 1 factoriza on This is a 1-factorization of K 4. f i 2 = 2 and f i T 2 = 1 for T = 1. Hence µ 1 = 1 2. There is a binary 1-[v, 2, 2 v ; v 1] MOD for v even, which is a 1-factorization of K v.

10 Example 2 A 1-[7, 2, 7 2 ; 6] MOD over C: f ε ε 2 ε 2 ε 1 f ε 2 ε ε ε 2 1 f ε ε ε 2 ε f ε 2 ε 1 0 ε ε f ε ε ε 2 ε f ε 2 ε 1 0 ε ε ε is a primitive cubic root of 1. f i 2 = 14 and f i T f i T = f i T 2 = 4 for T = 1, hence µ 1 = 2 7. And f i T f j T = 0.

11 Example 3 A 1-[7, 2, 7 2 ; 6] MOD over C: f ε ε 2 ε 2 ε 1 f ε 2 ε ε ε 2 1 f ε ε ε 2 ε f ε 2 ε 1 0 ε ε f ε ε ε 2 ε f ε 2 ε 1 0 ε ε ε is a primitive cubic root of 1. f i 2 = 14 and f i T f i T = f i T 2 = 4 for T = 1, hence µ 1 = 2 7. And f i T f j T = 0. As an example, let T = {1}, then, f i T 2 = 4, f 3 T f 4 T = ε ε 2 + ε 2 ε = 0.

12 Example 4 A binary 2-[7, 3, 1 7 ; 3] MOD over {0, 1}: Let V = {0, 1,..., 6} and define f i s as follows: f f f Then, f 1 2 = f 2 2 = 7, f 3 2 = 21, f i T f j T = 0 for any i j, and µ(t ) = µ t = f i T 2 f i 2 = 1 7 for T = 2.

13 Fundamental properties of a partial t-mod 1. A partial t-[v, k, µ; m] MOD is a partial (t 1)-[v, k, µ ; m] MOD for some µ. 2. If (V ; f) is a partial t-[v, k, µ; m] MOD, then (V ; Df) and (V ; Uf) are also partial t-[v, k, µ; m] MODs for any diagonal matrix D and unitary matrix U. 3. Complement design: For a partial t-[v, k, µ; m] MOD (V ; f), let fi c(b) = f i(b c ) for any B ( ) V v k. Then, (V ; f c 1,, fm c ) is a partial t-[v, v k, µ c ; m] MOD for some µ c. 4. Derived design: If (V ; f 1,..., f m ) is a partial t-[v, k, µ; m] MOD and let S ( ) V s for s < t, then (V \S; f1 S,..., f m S ) is a partial (t s)-[v s, k s, µ S ; m] MOD, where µ S (S ) = µ(s S).

14 An upper bound for m Theorem 1 (Beth et al. (2003)) The number of designs m of a t-[v, k, µ; m] MOD is bounded by { ( v t) m min ( v t) }, ( v t ). (1) k t k 2 v t A partial t-[v, k, µ; m] MOD attainning the bound (1) at the left hand side inequality is called optimal. Whereas, a partial t-[v, v 2, µ; m] MOD attaining the bound (1) at the right hand side inequality is called totally optimal.

15 Any optimal partial t-mod is t-mod! Theorem 2 An optimal partial t-[v, k, µ; m] MOD is a t- [v, k, µ t ; m] MOD. That is, µ(t ) is constant (= µ t ) for any T ( ) V t. Moreover, m i=1 f i (B) 2 is also constant (= µ t ) for any B ( ) V k.

16 Example 5 An optimal 1-[7, 2, 2 7 ; 6] MOD over C: f ε ε 2 ε 2 ε 1 f ε 2 ε ε ε 2 1 f ε ε ε 2 ε f ε 2 ε 1 0 ε ε f ε ε ε 2 ε f ε 2 ε 1 0 ε ε ε is a primitive cubic root of 1. f i 2 = 14 and f i T 2 = 4 for any T ( ) V 1, hence µ1 = 2 7.

17 An optimal binary t-mod is a large set of t-designs! A large set LS 1 (t, k, v) of binary t-designs is a family of t-(v, k, 1) designs (V, B (i) ) satisfying B (i) B (j) = and i B (i) = ( ) V k. It is obvious that if a LS 1 (t, k, v) exists, then it is an optimal binary t-[v, k, µ t ; m] MOD. Theorem 2 derives the following: Corollary 1 If an optimal binary partial t-[v, k, µ; m] MOD exists, then it is an LS 1 (t, k, v).

18 Problem 1 Does there exist an optimal non-binary t-mod when no large set of t-designs exists? Example 5 is the only one known example to Problem 1 at present.

19 Does there exist a totally optimal t-mod? The case of t = 1: Lemma 1 (Beth et al. (2003)) There exists a totally optimal binary 1-[2k, k, 2 1 ; ( ) 2k 1 k 1 ] MOD for any positive integer k. Example 6 A binary 1-[4, 2, 1 2 ; 3] MOD Let k = 2 and V = {0, 1, 2, 3}. f f f

20 Non-existence of a totally optimal t-[2(t+1), t+1, µ; 2] MOD for even t The case of k t = 1: k v = 2k Problem 2 Does a totally optimal t-[2(t+1), t+1, µ; t+2] MOD exist? Totally optimal 1-MOD exists. 3? k - t = 1 2 Existence of totally optimal t-mods t

21 Theorem 3 There does not exist a t-[2(t + 1), t + 1, µ; 3] MOD over C for any even integer t. Theorem 4 There exists a t-[2(t + 1), t + 1, µ; 2] MOD over C for any integer t. Take a (t+1) 2 t linear orthogonal array O 1 with strength t and 2-symbols. And take the coset O 2 of O 1 which are disjoint to O 1.

22 A t-[2(t + 1), t + 1, µ; m] MOD for odd t t = 1: It is known that a 1-[4, 2, 1 2 ; 3] MOD exists. t = 2: A 3-[8, 4, µ; 5] MOD does not exist since a (derived) 2-[7, 3, µ; 5] MOD does not exist. t 3:??? Problem 3 Find the maximum value of m such that there exists a t-[2(t + 1), t + 1, µ; m] MOD over C for given odd integer t?

23 Thank you!