Symplectic Structures in Quantum Information Vlad Gheorghiu epartment of Physics Carnegie Mellon University Pittsburgh, PA 15213, U.S.A. June 3, 2010 Vlad Gheorghiu (CMU) Symplectic struct. in Quantum Information June 3, 2010 1 / 14
Outline 1 Symplectic forms Skew-symmetric forms Symplectic vector spaces 2 Applications: Weyl-Heisenberg operators General properties Commutations and the symplectic structure of Z 2n Symplectomorphisms of Z 2n and Clifford operations A summary of this talk is available online at http://quantum.phys.cmu.edu/qip Vlad Gheorghiu (CMU) Symplectic struct. in Quantum Information June 3, 2010 2 / 14
Symplectic forms Symplectic forms Skew-symmetric forms Consider an m-dimensional vector space V over a finite field F. Let Ω : V V F be a bilinear map, i.e. linear in both arguments. The map Ω is called skew-symmetric if Ω(u, v) = Ω(v, u), u, v V. Theorem (Standard form for Skew-symmetric Bilinear Maps) Let Ω be a skew-symmetric bilinear map on V. Then there exist a basis u 1,... u k, e 1,..., e n, f 1,..., f n of V such that Ω(u i, v) = 0, for all i and all v V ; Ω(e i, e j ) = 0 = Ω(f i, f j ), for all i, j; Ω(e i, f j ) = δ ij, for all i, j. Vlad Gheorghiu (CMU) Symplectic struct. in Quantum Information June 3, 2010 3 / 14
Sketch of the proof Symplectic forms Skew-symmetric forms Induction proof, skew-symmetric version of Gram-Schmidt process. Let U := {u V Ω(u, v) = 0 for all v V }. Choose a basis u 1,..., u k of U, and choose a complementary space W to U in V, V = U W. Take any nonzero e 1 W. Then there is f 1 W such that Ω(e 1, f 1 ) 0. W.l.o.g. assume that Ω(e 1, f 1 ) = 1. Let W 1 = span of e 1, f 1 W Ω 1 = {w W Ω(w, v) = 0 for all v W 1 }. Vlad Gheorghiu (CMU) Symplectic struct. in Quantum Information June 3, 2010 4 / 14
Symplectic forms Sketch of the proof, cont. Skew-symmetric forms Claim 1. W 1 W Ω 1 = {0}. Proof: Assume that v = ae 1 + bf 1 W 1 W Ω 1. Then hence v = 0. 0 = Ω(v, e 1 ) = b 0 = Ω(v, f 1 ) = a, Claim 2. W = W 1 W Ω 1. Proof: Suppose v W has Ω(v, e 1 ) = c and Ω(v, f 1 ) = d. Then v = ( cf 1 + de 1 ) + (v + cf }{{} 1 de 1 ). }{{} W 1 W Ω 1 Vlad Gheorghiu (CMU) Symplectic struct. in Quantum Information June 3, 2010 5 / 14
Symplectic forms Sketch of the proof, cont. Skew-symmetric forms Go on: let e 2 W Ω 1, e 2 0. There is f 2 W Ω 1 s.t. Ω(e 2, f 2 ) 0. W.l.o.g. assume that Ω(e 2, f 2 ) = 1. Let W 2 = span of e 2, f 2. Etc. The process eventually stops because dim V <. We hence obtain V = U W 1 W 2... W n where all summands are orthogonal with respect to Ω, and where W i has basis e i, f i with Ω(e i, f i ) = 1, q.e.d.. The dimension of the subspace U = {u V Ω(u, v) = 0, for all v V } does not depend on the choice of basis. Hence k := dim U is an invariant of (V, Ω). Since k + 2n = m = dim V, then n is an invariant of (V, Ω);2n is called the rank of Ω. Vlad Gheorghiu (CMU) Symplectic struct. in Quantum Information June 3, 2010 6 / 14
Observations Symplectic forms Skew-symmetric forms 1 The basis of the Theorem is not unique, though it is traditionally also called a canonical basis or a arboux basis. 2 In matrix notation with respect to such basis, we have Ω(u, v) = ( u ) 0 0 0 0 0 I v 0 I 0 Vlad Gheorghiu (CMU) Symplectic struct. in Quantum Information June 3, 2010 7 / 14
Symplectic maps Symplectic forms Symplectic vector spaces efine a map Ω : V V by v Ω v, Ω v (u) := Ω(v, u). The kernel of Ω is the subspace U above. efinition A skew-symmetric bilinear map Ω is symplectic (or nondegenerate) if Ω is bijective, i.e. U = {0}. The map Ω is then called a linear symplectic structure on V, and (V, Ω) is called a symplectic vector space. Vlad Gheorghiu (CMU) Symplectic struct. in Quantum Information June 3, 2010 8 / 14
Symplectic forms Properties of a symplectic map Ω Symplectic vector spaces 1 k = dim U = 0, so dim V = 2n is even. 2 By the theorem above, a symplectic vector space (V, Ω) has a basis e 1,..., e n, f 1,..., f n satisfying Ω(e i, f j ) = δ ij and Ω(e i, e j ) = 0 = Ω(f i, f j ). 3 Such a basis is called a symplectic basis of (V, Ω). With respect to a symplectic basis, we have Ω(u, v) = ( u ) ( ) 0 I v. I 0 4 A subspace W of a symplectic vector space (V, Ω) is called symplectic if Ω W 0. For example, the span of e 1, f 1 is symplectic. 5 A subspace W of a symplectic vector space (V, Ω) is called isotropic if Ω W 0. For instance, the span of e 1, e 2 is isotropic. Vlad Gheorghiu (CMU) Symplectic struct. in Quantum Information June 3, 2010 9 / 14
Applications: Weyl-Heisenberg operators General properties General properties Let pe a prime number. efine the one-qudit operators X = 1 j=0 j j 1 and Z = 1 j=0 ω j j j. The operators X x Z z, with x, z Z are called Weyl-Heisenberg operators (or generalized Pauli operators). Extend to n qudits. efine X x := X x 1... X xn and similarly Z z, with x, z Z n. A generalized Weyl-Heisenberg operator has the form X x Z z, so can be uniquely specified by a 2n dimensional row vector (x z). Operators correspond to vectors in Z 2n. Vlad Gheorghiu (CMU) Symplectic struct. in Quantum Information June 3, 2010 10 / 14
Applications: Weyl-Heisenberg operators Commutations and the symplectic structure of Z 2n Commutations and symplectic maps on Z 2n Consider X x Z z and X x Z z. One has X x Z z X x Z z = ω x z x z X x Z z X x Z z where inner products are taken mod. The map Z 2n Z2n [(x z), (x z )] x z x z Z is a symplectic map, sometimes called a symplectic innder product on Z 2n (Z 2n, <, > symp) is a symplectic vector space. A subspace S of Z 2n is not necessary symplectic. But, by the standard form theorem, there always exist a arboux basis and S = S isotropic S symplectic. Intuitively, given a set of W. H. operators that correspond to a subspace of Z 2n, one can always find a closed subset that will commute with everything else (the center), and then the rest of the basis operator anti-commute in pairs.. Vlad Gheorghiu (CMU) Symplectic struct. in Quantum Information June 3, 2010 11 / 14
Applications: Weyl-Heisenberg operators Commutations and the symplectic structure of Z 2n This result has important consequences. For example, consider a (noisy) quantum channel for which there exists a subgroup S of the W. H. group such that the eigenvectors of any operator in the subgroup is perfectly transmitted to the channel. Using the standard form theorem, one can easily get a lower bound on the quantum capacity of such a channel: count the number of anti-commuting pairs, then each pair contributes with 1 qudit to the capacity. Equivalently, Q 1 2 (dims symp). For additive graph codes, there is always a W. H. subgroup of the input that is perfectly preserved down to a subset of the carriers, and the remainder of the W. H. group is totally absent. The channel from the input to the subset is additive, so the above formula is precisely the quantum capacity of the channel. See Phys. Rev. A 81, 032326 (2010), Gheorghiu, Looi and Griffiths, for more details. Vlad Gheorghiu (CMU) Symplectic struct. in Quantum Information June 3, 2010 12 / 14
Applications: Weyl-Heisenberg operators Symplectomorphisms of Z 2n Symplectomorphisms of Z 2n and Clifford operations and Clifford operations The isotropic subspace correspond to classical information (and does not contribute to the quantum capacity), whereas the symplectic one to quantum information. efinition A symplectomorhism Υ : Z 2n Z2n is a linear isomorphism which preserves the symplectic inner product. Any symplectomorhism can be expressed in the form of a symplectic matrix M over Z 2n, ( ) M T 0 I M = I 0 ( ) 0 I. I 0 M is invertible (from the definition; why?) Vlad Gheorghiu (CMU) Symplectic struct. in Quantum Information June 3, 2010 13 / 14
Applications: Weyl-Heisenberg operators Symplectomorphisms of Z 2n and Clifford operations Theorem For any symplectomorphism Υ over Z 2n, there is a unitary map U Υ on H n that maps the corresponding W. H. operators. Formally XZ[Υ(u)] = U Υ XZ[u]U Υ u Z2n. Actually, U Υ is a Clifford operator. Most of the concepts and results of this talk can be generalized to composite dimensions. For example, we used the above theorem and constructed an explicit unitary encoding circuit for additive graph codes of arbitrary dimensionality. All these results contribute to a better understanding of arbitrary-dimensionality stabilizer states/codes, as well as to discrete models of quantum mechanics. Vlad Gheorghiu (CMU) Symplectic struct. in Quantum Information June 3, 2010 14 / 14