One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. In past years, many good quantum error-correcting codes had been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over Fq in terms of classical codes over F q 2 is provided that generalizes the well-known notion of additive codes over F4 of the binary case. This paper derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum BCH codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper.

Two orthonormal bases B and B ′ of a d-dimensional complex inner-product space are called mutually unbiased if and only if | b|b ′ | 2 = 1/d holds for all b ∈ B and b ′ ∈ B ′ . The size of any set containing pairwise mutually unbiased bases of C d cannot exceed d + 1. If d is a power of a prime, then extremal sets containing d + 1 mutually unbiased bases are known to exist. We give a simplified proof of this fact based on the estimation of exponential sums. We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.

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Abstract-Mutually unbiased bases (MUBs) are a primitive used in quantum information processing to capture the principle of complementarity. While constructions of maximal sets of d + 1 such bases are known for system of prime power dimension d, it is unknown whether this bound can be achieved for any non-prime power dimension. In this paper we demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0, 1/d}. We also give a new and simple proof that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}. I. INTRODUCTIONTwo quantum mechanical observables are called complementary if and only if precise knowledge of one of them implies that all possible outcomes are equally probable when measuring the other, see for example [19, p. 561]. The principle of complementarity was introduced by Bohr [6] in 1928, and it had a profound impact on the further development of quantum mechanics. A recent application is the quantum key exchange protocol by Bennett and Brassard [3] that exploits complementarity to secure the key exchange against eavesdropping.We mention a simple mathematical consequence of this complementarity principle, which motivates some key notion. Suppose that O and O ′ are two hermitian d × d matrices representing a pair of complementary observables. We assume that the eigenvalues of both matrices are multiplicity free. It follows that the observables O and O ′ respectively have orthonormal eigenbases B and B ′ with basis vectors uniquely determined up to a scalar factor.The complementarity of O and O ′ implies that if a quantum system is prepared in an eigenstate b ′ of the observable O ′ , and O is subsequently measured, then the probability to find the system after the measurement in the state b ∈ B is given by | b|b ′ | 2 = 1/d. Recall that two orthonormal bases B and B ′ of C d are said to be mutually unbiased precisely when | b|b ′ | 2 = 1/d holds for all b ∈ B and b ′ ∈ B ′ . Thus the eigenbases of non-degenerate complementary observables are mutually unbiased. Conversely, we can associate to a pair of mutually unbiased bases a pair of non-degenerate complementary observables.There is a fundamental property of mutually unbiased bases that is invaluable in quantum information processing. Suppose

Recently, quantum error-correcting codes have been proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit-and phase-flip errors. An example for a channel that exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit and phase flips can be related to relaxation and dephasing time, respectively. We study asymmetric quantum codes that are obtained from the Calderbank-Shor-Steane (CSS) construction. For such codes, we derive upper bounds on the code parameters using linear programming. A central result of this paper is the explicit construction of some new families of asymmetric quantum stabilizer codes from pairs of nested classical codes. For instance, we derive asymmetric codes using a combination of Bose-ChaudhuriHocquenghem (BCH) and finite geometry low-density parity-check (LDPC) codes. We show that the asymmetric quantum codes offer two advantages, namely to allow a higher rate without sacrificing performance when compared with symmetric codes and vice versa to allow a higher performance when compared with symmetric codes of comparable rates. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.