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Refs. [1, 8] outline a symplectic Gram-Schmidt or- given above. The number c of ebits is also equal to thogonalization procedure (SGSOP) that uniquely deter- rank (ΩH ) /2 because the matrix G is full rank. The code mines the optimal (i.e., minimal) number of ebits that is an [[n, k + c; c]] entanglement-assisted quantum block the code requires, and Ref. [9] proves that the SGSOP code by the construction in Refs. [1, 8]. gives the optimal number of ebits. The code construc- Our formula (3) is equivalent to the formula at the top tion in Refs. [1, 8] shows that the resulting entanglement- of page 14 in Ref. [8], but it provides the quantum code assisted quantum code requires at most c ebits. The designer with a quick method to determine how many essence of the argument in Ref. [9] is that the resulting ebits an entanglement-assisted code requires, by simply entanglement-assisted quantum code requires at least c “plugging in” the generators of the code. ebits because any fewer ebits would not be able to resolve The formula (3), like the CSS formula in (1), is a mea- the anticommutativity of the generators on Alice’s side sure of how far a set of generators is from being a com- of the code. muting set, or equivalently, how far it is from giving a The SGSOP performs row operations that do not standard stabilizer code. change the error-correcting properties of the quantum Corollary 1 below gives a formula for the optimal num- code (because the code is additive), but these row opera- ber of ebits required by a CSS entanglement-assisted tions do change the symplectic product relations. These quantum code. It is generally a bit less difficult to com- row operations are either a row swap S (i, j), where pute than the above formula in (3). This reduction in S (i, j) is a full-rank (n k) (n k) matrix that swaps complexity occurs because of the special form of a CSS row i with j, or a row addition− × A−(i, j), where A (i, j) is quantum code and because the size of the matrices in- a full-rank (n k) (n k) matrix that adds row i to volved are generally smaller for a CSS code than for a row j. These row− operations× − multiply the matrix H from general code with the same number of generators and the left. The SGSOP then is equivalent to a full-rank physical qubits. (n k) (n k) matrix G that contains all of the row operations− × and− produces a new quantum check matrix Corollary 1 Suppose we import two classical [n, k1, d1] H′ = GH with corresponding symplectic product matrix and [n, k2, d2] binary codes with respective parity ′ T T T check matrices H1 and H2 to build an entanglement- ΩH = G HX HZ + HZ HX G . In particular, the re- sulting symplectic product matrix ΩH′ is in a standard assisted quantum code. The resulting code is an form so that [[n, k1 + k2 n + c, min (d1, d2); c]] entanglement- assisted code,− and requires c ebits where c n−k−2c ΩH′ = J [0] , (8) T ⊕ c = rank H1H2 . (13) i=1 j=1 M M Proof. The quantum check matrix has the following where the small and large correspond to the direct sum form: operation, J is the matrix⊕ H1 0 0 1 H = . (14) J = , (9) 0 H2 1 0 The symplectic product matrix Ω is then and [0] is the one-element zero matrix. Each matrix J H in the direct sum corresponds to a symplectic pair and H1 T 0 T ΩH = 0 H2 + H1 0 (15) has rank two. Each symplectic pair corresponds to ex- 0 H2 actly one ebit that the code requires [1, 8]. Each matrix T 0 H1H2 [0] has rank zero and corresponds to an ancilla qubit. = T . (16) H2H1 0 The optimal number of ebits required for the code is rank (Ω ′ ) /2: H The above matrix is equivalent by a full rank permutation T T c n−k−2c matrix to the matrix H1H2 H2H1 , so the rank of ΩH is ⊕ rank (ΩH′ ) = rank J [0] (10) =1 ⊕ =1 i j T T M M rank (ΩH ) = rank H1H2 H2H1 (17) c n−k−2c ⊕ T T = rank (J)+ rank ([0]) (11) = rank H1H2 + rank H2H1 (18) i=1 j=1 T X X = 2 rank H1H2 (19) =2c. (12) The second line follows because the rank of a direct The second line follows because the rank of a direct sum is equivalent to the sum of the individual ranks, sum is the sum of the individual matrix ranks, and and the third line follows because the rank is invari- the third line follows from the individual matrix ranks ant under matrix transposition. The number of ebits 3 required for the resulting entanglement-assisted quan- matrices over the field GF (4). Consider the following T tum code is rank H1H2 , using the result of the pre- full-rank GF (4) matrices: vious theorem. The construction in Refs. [1, 8] produces 1ω ¯ 1 0 an [[n, k1 + k2 n + c, min (d1, d2); c]] entanglement- A1 = I, A2 = I. (27) − 0 1 ⊗ 1 1 ⊗ assisted quantum block code.
We premultiply and postmultiply the matrix ΩHQ as fol- Corollary 2 Suppose we import an [n,k,d]4 classical lows and obtain a matrix with the same rank as ΩH : code over GF (4) with parity check matrix H for use as Q an entanglement-assisted quantum code according to the † † 0 0 † 1 0 T A2A1ΩH A1A2 = HH + HH¯ (28) construction in Ref. [1, 8]. Then the resulting quantum Q 0 1 ⊗ 0 0 ⊗ code is an [[n, 2k n + c; c]] entanglement-assisted quan- HH¯ T 0 − † = (29) tum code where c = rank HH and denotes the con- 0 HH† jugate transpose operation over matrices† in GF (4). = HH¯ T HH† (30) ⊕ Proof. Ref. [1] shows how to produce an entanglement- Therefore, the rank of Ω is assisted quantum code from the parity check matrix H HQ T † of a classical code over GF (4). The resulting quantum rank ΩH = rank HH¯ HH (31) Q ⊕ parity check matrix HQ in symplectic binary form is T † = rank HH¯ + rank HH (32) ωH † H = γ , (20) = 2 rank HH . (33) Q ωH¯ The second line holds because the rank of a direct sum where γ denotes the isomorphism between elements of is the sum of the individual ranks and the third holds GF (4) and symplectic binary vectors that represent because the rank is invariant under the matrix trans- −1 Pauli matrices. Specifically, γ (h)= ωhx +¯ωhz where pose operation. Therefore, the resulting entanglement- † h is a symplectic binary vector and hx and hz denote its assisted quantum code requires c = rank HH ebits “X” and “Z” parts respectively. The symplectic product by applying the result of the original theorem. The con- between binary vectors is equivalent to the trace product struction in Refs. [1, 8] produces an [[n, 2k n + c; c]] − of their GF (4) representations (see, e.g., Ref. [1]): entanglement-assisted quantum code.
−1 −1 Corollary 3 We can construct a continuous-variable hi hj = tr γ (hi) γ (hj ) , (21) ⊙ · entanglement-assisted quantum code from generators cor- n o responding to the rows in quantum check matrix H = where hi and hj are any two rows of HQ, denotes the · HZ HX where H is (n k) 2n-dimensional, H inner product, the overbar denotes the conjugate opera- is a real matrix representing− the× quantum code [6], tion, and tr x = x +¯x denotes the trace operation over { } and both HZ and HX are (n k) n-dimensional. elements of GF (4). We exploit these correspondences to The resulting code is an [[n, k + c;−c]] continuous-variable× write the symplectic product matrix Ω for the quan- HQ entanglement-assisted code and requires c entangled tum check matrix HQ as follows: modes where
† T T ωH ωH c = rank HX H HZ H /2. (34) Ω = tr (22) Z − X HQ ωH¯ ωH¯ ( ) Proof. The proof is similar to the proof of the first theo- ωH rem but requires manipulations of real vectors instead of = tr ωH¯ † ωH† (23) ωH¯ binary vectors. See Ref. [6] for details of the symplectic HH† ωHH¯ † geometry required for continuous-variable entanglement- = tr (24) ωHH† HH† assisted codes. 1ω ¯ Remark 1 A similar formula holds for entanglement- = tr HH† (25) ω 1 ⊗ assisted qudit codes by replacing the subtraction operation above with subtraction modulo d. Specifically, we can con- 1ω ¯ 1 ω = HH† + HH¯ T (26) struct a qudit entanglement-assisted quantum code from ω 1 ⊗ ω¯ 1 ⊗ generators corresponding to the rows in check matrix where the “tr” operation above is an element-wise trace H = HZ HX whose matrix entries are elements of the finite field Zd. The code requires c edits (a d- operation over GF (4) (it is not the matrix trace oper- d−1 √ ation.) The matrix ΩHQ is over the field GF (2), but dimensional state i=0 i i / d) where we can consider it as being over the field GF (4) with- | i | i P T T out changing its rank. Therefore, we can multiply it by c = rank HX H d HZ H /2 Z ⊖ X 4 and d is subtraction modulo d. We use subtraction where h (D)=1+ D. This code requires two ebits and modulo⊖ d because the symplectic form over d-dimensional one ancilla qubit for quantum redundancy and encodes variables includes subtraction modulo d. two information qubits. The shifted symplectic product T −1 T −1 matrix HX (D)HZ (D )+ HZ (D)HX (D ) [7, 11] for We can also consider the case when the binary, d- this code is as follows: variable, or real matrix H specifies one party’s genera- tors of a respective bipartite qubit, qudit, or continuous- variable stabilizer state. In that case, our formula is equivalent to the entanglement measure found in Ref. [5]. 01000 We finally conjecture two formulas for the number 10000 of ebits required in a general (non-CSS) entanglement- 00000 (37) assisted quantum convolutional code. 00001 00010 Conjecture 1 The optimal number c of ebits necessary per frame for an entanglement-assisted quantum convo- The rank of the above matrix is four and the code re- lutional code is quires two ebits. Therefore, the conjecture holds for this example. T −1 T −1 c = rank HX (D)HZ (D )+ HZ (D)HX (D ) /2 (35) where H(D)= HZ (D) HX (D) represents the parity check matrix for a set of quantum convolutional genera- tors that do not necessarily form a commuting set.
Conjecture 2 The optimal number c of ebits required MMW and TAB acknowledge support from NSF per frame for an entanglement-assisted quantum convo- Grants CCF-0545845 and CCF-0448658. lutional code imported from a classical quaternary convo- lutional code with parity check matrix H (D) is
† −1 c = rank H(D)H (D ) . (36) ∗ Electronic address: [email protected] We know the number of ebits required for a [1] T. A. Brun, I. Devetak, and M.-H. Hsieh, Science 314, CSS entanglement-assisted quantum convolutional code pp. 436 (2006). T −1 [2] D. Gottesman, Ph.D. thesis, California Institue of Tech- is rank H1(D)H (D ) [7] where H1(D) and H2(D) 2 nology (1997). correspond to the classical binary convolutional codes 76 [3] M.-H. Hsieh, I. Devetak, and T. Brun, Phys. Rev. A , that we import to correct respective bit and phase flips. 062313 (2007). Comparing the CSS convolutional formula, the CSS block [4] M.-H. Hsieh, T. A. Brun, and I. Devetak, in Asian Con- formula in Corollary 1, and the general block formula in ference on Quantum Information Science (2007), pp. Theorem 1, the above conjecture seems natural. 101–102. We finally provide an example of the above conjec- [5] D. Fattal, T. S. Cubitt, Y. Yamamoto, S. Bravyi, and I. L. Chuang, arXiv:quant-ph/0406168 (2004). ture. It is a slight modification of the code presented in [6] M. M. Wilde, H. Krovi, and T. A. Brun, Phys. Rev. A Ref. [10]. 76, 052308 (2007). [7] M. M. Wilde and T. A. Brun, arXiv:0712.2223 (2007). Example 1 Consider the quantum convolutional code [8] T. A. Brun, I. Devetak, and M.-H. Hsieh, arXiv:quant- with quantum check matrix as follows: ph/0608027 (2006). [9] D. Gottesman, URL http://www.perimeterinstitute.ca/personal/dgottesman/QECC2007/Sols9.pdf. 0 000 0 h (D) 0 D 1 h (D) [10] M. M. Wilde and T. A. Brun, arXiv:0801.0821 (2008). h (D) D 0 1 h (D) 0 000 0 [11] M. M. Wilde, H. Krovi, and T. A. Brun, arXiv:0708.3699 0 0 DDD 0 101 1 , (2007). 1 1 0 1 0 0 000 0 D1 D 0 00 0 0 010 0 D