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Journal of Physics A Mathematical and Theoretical

Mutually orthogonal Latin squares from the inner products of vectors in mutually unbiased bases

Joanne L. Hall, Asha Rao

Published Mar 10, 2010• 9 citations
Publisher

Abstract

Mutually unbiased bases (MUBs) are important in quantum information theory. While constructions of complete sets of d + 1 MUBs in are known when d is a prime power, it is unknown if such complete sets exist in non-prime power dimensions. It has been conjectured that complete sets of MUBs only exist in if a maximal set of mutually orthogonal Latin squares (MOLS) of side length d also exists. There are several constructions (Roy and Scott 2007 J. Math. Phys. 48 072110; Paterek, Dakić and Brukner 2009 Phys. Rev. A 79 012109) of complete sets of MUBs from specific types of MOLS, which use Galois fields to construct the vectors of the MUBs. In this paper, two known constructions of MUBs (Alltop 1980 IEEE Trans. Inf. Theory 26 350–354; Wootters and Fields 1989 Ann. Phys. 191 363–381), both of which use polynomials over a Galois field, are used to construct complete sets of MOLS in the odd prime case. The MOLS come from the inner products of pairs of vectors in the MUBs.

DOI
10.1088/1751-8113/43/13/135302

Related papers

Mutually Unbiased Bases and Orthogonal Latin Squares

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On first sight, the number of mutually unbiased bases (MUBs) in d dimensions and the number of mutually orthogonal Latin squares (MOLS) of order d seem to be closely related. A careful look however reveals the connections cannot be really strong. In this talk a few similarities and differences between MUBs and MOLS are presented. The first similarity involves their maximal numbers. There are at most d + 1 MUBs and also d + 1 squares in an augmented set of MOLS. This number is attained if d is a power of a prime and we give an algorithm which relates MUBs and MOLS in these dimensions [1]. Another similarity follows from considerations of composite d. One can think about a Hilbert space of a composite dimension as describing several subsystems of prime-power dimensions. Since we can always build MUBs in the global space as tensor products of MUBs for the subspaces, there are at least M + 1 of them for the whole system, where M is the smallest prime-power factor of d. A similar construction exists for MOLS: squares of composite orders can be built as direct products of squares of prime-power orders. The corresponding lower bound on the number of MOLS is called the MacNeish’s bound. In general, both the MacNeish’s bound and the number of MUBs which follows from the tensor product construction are known not to be tight. However, the simplest example today involves dimension d = 676 for MUBs [2], whereas MacNeish’s bound is not tight already for d = 10. We therefore checked whether the connection [1] also works for d = 10, and found it no longer links all MOLS to MUBs [3]. The general results of Ref. [4] reveal in this context that the connection fails whenever MacNeish’s bound is not tight. The last (apparent) similarity to be presented is related to incomplete sets of MOLS and MUBs. There is a Latin square of order 4 which does not have an orthogonal mate (so called bachelor Latin square) and connection [1] links it with a set of MUBs which also cannot be extended. It might be tempting to conjecture that every unextendible augmented set of MOLS has a corresponding unextendible set of MUBs. We already disproved such conjecture for the connection [1] and now we briefly describe that there is no other connection for which the conjecture would hold true. One can see this in at least two ways [5]. The first involves bachelor Latin square of order 5. It was recently shown that bachelor Latin squares exist for all orders d > 3 [6]. If there is a strong relation between MOLS and MUBs, there should correspondingly exist a set of 3 MUBs in all dimensions d > 3. To the contrary, it is known that for d = 5 every set of 3 MUBs can be extended to a complete set of 6 MUBs [7]. The second way is more dramatic. It turns out that for d = 6 there exists a set of two MUBs which cannot be extended any further (and six is the smallest dimension in which this happens). Since an augmented set of MOLS contains at least 3 elements, the strong relation is excluded. For more arguments against a close connection between MOLS and MUBs see Ref. [8].

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Mutually unbiased bases (MUBs) are widely used in quantum information processing and play an important role in quantum cryptography, quantum state tomography and communications. It's difficult to construct MUBs and remains unknown whether complete MUBs exist for any non prime power. Therefore, researchers have proposed the solution to construct approximately mutually unbiased bases (AMUBs) by weakening the inner product conditions. This paper constructs q AMUBs of ℂq, (q+1) AMUBs of ℂq-1 and q AMUBs of ℂq-1 by using character sums over Galois rings and finite fields, where q is a power of a prime. The first construction of q AMUBs of ℂq is new which illustrates K AMUBs of ℂK can be achieved. The second and third constructions in this paper include the partial results about AMUBs constructed by W. Wang et al. in [9].

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences•Feb 6, 2024•0 citations
Mutually unbiased bases (MUBs) are important in quantum information theory. While constructions of complete sets of d + 1 MUBs in are known when d is a prime power, it is unknown if such complete sets exist in non-prime power dimensions. It has been conjectured that complete sets of MUBs only exist in if a maximal set of mutually orthogonal Latin squares (MOLS) of side length d also exists. There are several constructions (Roy and Scott 2007 J. Math. Phys. 48 072110; Paterek, Dakić and Brukner 2009 Phys. Rev. A 79 012109) of complete sets of MUBs from specific types of MOLS, which use Galois fields to construct the vectors of the MUBs. In this paper, two known constructions of MUBs (Alltop 1980 IEEE Trans. Inf. Theory 26 350–354; Wootters and Fields 1989 Ann. Phys. 191 363–381), both of which use polynomials over a Galois field, are used to construct complete sets of MOLS in the odd prime case. The MOLS come from the inner products of pairs of vectors in the MUBs.
On first sight, the number of mutually unbiased bases (MUBs) in d dimensions and the number of mutually orthogonal Latin squares (MOLS) of order d seem to be closely related. A careful look however reveals the connections cannot be really strong. In this talk a few similarities and differences between MUBs and MOLS are presented. The first similarity involves their maximal numbers. There are at most d + 1 MUBs and also d + 1 squares in an augmented set of MOLS. This number is attained if d is a power of a prime and we give an algorithm which relates MUBs and MOLS in these dimensions [1]. Another similarity follows from considerations of composite d. One can think about a Hilbert space of a composite dimension as describing several subsystems of prime-power dimensions. Since we can always build MUBs in the global space as tensor products of MUBs for the subspaces, there are at least M + 1 of them for the whole system, where M is the smallest prime-power factor of d. A similar construction exists for MOLS: squares of composite orders can be built as direct products of squares of prime-power orders. The corresponding lower bound on the number of MOLS is called the MacNeish’s bound. In general, both the MacNeish’s bound and the number of MUBs which follows from the tensor product construction are known not to be tight. However, the simplest example today involves dimension d = 676 for MUBs [2], whereas MacNeish’s bound is not tight already for d = 10. We therefore checked whether the connection [1] also works for d = 10, and found it no longer links all MOLS to MUBs [3]. The general results of Ref. [4] reveal in this context that the connection fails whenever MacNeish’s bound is not tight. The last (apparent) similarity to be presented is related to incomplete sets of MOLS and MUBs. There is a Latin square of order 4 which does not have an orthogonal mate (so called bachelor Latin square) and connection [1] links it with a set of MUBs which also cannot be extended. It might be tempting to conjecture that every unextendible augmented set of MOLS has a corresponding unextendible set of MUBs. We already disproved such conjecture for the connection [1] and now we briefly describe that there is no other connection for which the conjecture would hold true. One can see this in at least two ways [5]. The first involves bachelor Latin square of order 5. It was recently shown that bachelor Latin squares exist for all orders d > 3 [6]. If there is a strong relation between MOLS and MUBs, there should correspondingly exist a set of 3 MUBs in all dimensions d > 3. To the contrary, it is known that for d = 5 every set of 3 MUBs can be extended to a complete set of 6 MUBs [7]. The second way is more dramatic. It turns out that for d = 6 there exists a set of two MUBs which cannot be extended any further (and six is the smallest dimension in which this happens). Since an augmented set of MOLS contains at least 3 elements, the strong relation is excluded. For more arguments against a close connection between MOLS and MUBs see Ref. [8].
Two orthonormal bases in the complex space of dimension d, are said to be mutually unbiased if the square of the magnitude of the inner product of any vector from one basis with any vector in other basis is equal to the reciprocal of the dimension d. Mutually unbiased bases are used for optimal state determination of mixed quantum states. It is known that in any dimension d, the number of mutually unbiased bases is at most d+1. Ivanovic found a complete set of mutually unbiased bases for prime dimensions. His construction was generalized by Wootters and Fields for prime power dimensions. There is a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. Based on this connection, there exits a constructive proof of the existence of a complete set of mutually unbiased bases for prime power dimensions. This thesis is an exploration on construction of mutually unbiased bases.
Mutually unbiased bases (MUBs) are widely used in quantum information processing and play an important role in quantum cryptography, quantum state tomography and communications. It's difficult to construct MUBs and remains unknown whether complete MUBs exist for any non prime power. Therefore, researchers have proposed the solution to construct approximately mutually unbiased bases (AMUBs) by weakening the inner product conditions. This paper constructs q AMUBs of ℂq, (q+1) AMUBs of ℂq-1 and q AMUBs of ℂq-1 by using character sums over Galois rings and finite fields, where q is a power of a prime. The first construction of q AMUBs of ℂq is new which illustrates K AMUBs of ℂK can be achieved. The second and third constructions in this paper include the partial results about AMUBs constructed by W. Wang et al. in [9].