Mutually unbiased bases are complex projective 2-designs

Abstract

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

Graph-state formalism for mutually unbiased bases

“…Here, the resulting states may no longer be graph-states, but belong to the class of LME states [25]. This gives a new perspective on other constructions such as the one by Alltop [12], which for p ≥ 5 was shown to be equivalent to the construction by Wootters and Fields up to a permutation of the vector components [52]. Note that for LME states, such a permutation can always be rephrased in terms of general phase gates [25].…”

“…. , t}, and any |φ ∈ H, where dψ is a unitarily invariant and normalized measure [12,13,53]. In other words, t-designs enable to compute uniformly weighted integrals over the Hilbert space H = C d , where the integrand is a polynomial in | φ |ψ | 2 of degree at most t, by averaging over a finite set of vectors D t = {|ψ i }.…”

“…In Ref. [12], it was shown that the union of the basis vectors of complete sets of MUBs constitute such a design. Namely, mutually unbiased bases are complex projective 2-designs.…”

“…Moreover, they have recently been shown to be useful for entanglement detection [10]. Furthermore, it was discovered that they have interesting connections to symmetric informationally complete positive-operator-valued measures [11], and complex t-designs [12,13].…”

A Criterion for Attaining the Welch Bounds with Applications for Mutually Unbiased Bases

“…The current research originated in the problem of constructing a complete set of mutually unbiased bases and is inspired mostly by [22].…”