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arXiv:quant-ph/0403095 · Physical Review A

Mutually unbiased bases and trinary operator sets for N qutrits

Jay Lawrence

Published Jul 2, 2004• 47 citations
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Abstract

A compete orthonormal basis of $N$-qutrit unitary operators drawn from the Pauli group consists of the identity and ${9}^{N}\ensuremath{-}1$ traceless operators. The traceless ones partition into ${3}^{N}+1$ maximally commuting subsets (MCS's) of ${3}^{N}\ensuremath{-}1$ operators each, whose joint eigenbases are mutually unbiased. We prove that Pauli factor groups of order ${3}^{N}$ are isomorphic to all MCS's and show how this result applies in specific cases. For two qutrits, the 80 traceless operators partition into 10 MCS's. We prove that 4 of the corresponding basis sets must be separable, while 6 must be totally entangled (and Bell-like). For three qutrits, 728 operators partition into 28 MCS's with less rigid structure, allowing for the coexistence of separable, partially entangled, and totally entangled (GHZ-like) bases. However a minimum of $16\phantom{\rule{0.3em}{0ex}}\mathrm{GHZ}$-like bases must occur. Every basis state is described by an $N$-digit trinary number consisting of the eigenvalues of $N$ observables constructed from the corresponding MCS.

DOI
10.1103/physreva.70.012302

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A compete orthonormal basis of NNN-qutrit unitary operators drawn from the Pauli group consists of the identity and 9N\ensuremath−1{9}^{N}\ensuremath{-}19N\ensuremath−1 traceless operators. The traceless ones partition into 3N+1{3}^{N}+13N+1 maximally commuting subsets (MCS's) of 3N\ensuremath−1{3}^{N}\ensuremath{-}13N\ensuremath−1 operators each, whose joint eigenbases are mutually unbiased. We prove that Pauli factor groups of order 3N{3}^{N}3N are isomorphic to all MCS's and show how this result applies in specific cases. For two qutrits, the 80 traceless operators partition into 10 MCS's. We prove that 4 of the corresponding basis sets must be separable, while 6 must be totally entangled (and Bell-like). For three qutrits, 728 operators partition into 28 MCS's with less rigid structure, allowing for the coexistence of separable, partially entangled, and totally entangled (GHZ-like) bases. However a minimum of 16GHZ16\phantom{\rule{0.3em}{0ex}}\mathrm{GHZ}16GHZ-like bases must occur. Every basis state is described by an NNN-digit trinary number consisting of the eigenvalues of NNN observables constructed from the corresponding MCS.
The Pauli operators (tensor products of Pauli matrices) provide a complete basis of operators on the Hilbert space of N qubits. We prove that the set of ${4}^{N}\ensuremath{-}1$ Pauli operators may be partitioned into 2N+1{2}^{N}+12N+1 distinct subsets, each consisting of ${2}^{N}\ensuremath{-}1$ internally commuting observables. Furthermore, each such partitioning defines a unique choice of 2N+1{2}^{N}+12N+1 mutually unbiased basis sets in the N-qubit Hilbert space. Examples for 2 and 3 qubit systems are discussed with emphasis on the nature and amount of entanglement that occurs within these basis sets.
We define mutually complementary observable sets for N qubits via the operational requirement that a state with a definite outcome for one set of (commuting) binary observables must give completely random results in all other sets. The bases formed by the eigenvectors of such complementary sets are mutually unbiased. We prove that the full set of 4^N-1 Pauli operator products may be partitioned into 2^N+1 distinct sets, each set consisting of 2^N-1 internally commuting observables. Furthermore we prove that each such partitioning defines a unique choice of 2^N+1 mutually unbiased bases. Examples for 2 and 3 qubit systems are discussed with emphasis on the nature and amount of entanglement that occurs within these basis sets.
A few simply stated rules govern the entanglement patterns that can occur in mutually unbiased basis sets (MUBs) and constrain the combinations of such patterns that can coexist in full complements of MUBs. We consider Hilbert spaces of prime power dimensions (D=pND={p}^{N}D=pN), as realized by systems of NNN prime-state particles, where full complements of D+1D+1D+1 MUBs are known to exist, and we assume only that MUBs are eigenbases of generalized Pauli operators, without using any particular construction. The general rules include the following: (1) In any MUB, a given particle appears either in a pure state or totally entangled and (2) in any full MUB complement, each particle is pure in (p+1p+1p+1) bases (not necessarily the same ones) and totally entangled in the remaining (${p}^{N}\ensuremath{-}p$). It follows that the maximum number of product bases is p+1p+1p+1 and, when this number is realized, all remaining (${p}^{N}\ensuremath{-}p$) bases in the complement are characterized by the total entanglement of every particle. This ``standard distribution'' is inescapable for two-particle systems (of any ppp), where only product and generalized Bell bases are admissible MUB types. This and the following results generalize previous results for qubits [Phys. Rev. A 65. 032320 (2002); Phys. Rev. A 72, 062310 (2005)] and qutrits [Phys. Rev. A 70, 012302 (2004)], drawing particularly upon [Phys. Rev. A 72, 062310 (2005)]. With three particles there are three MUB types, and these may be combined in (p+2p+2p+2) different ways to form full complements. With N=4N=4N=4, there are 6 MUB types for p=2p=2p=2, but new MUB types become possible with larger ppp, and these are essential to realizing full complements. With this example, we argue that new MUB types that show new entanglement patterns should enter with every step in NNN and, also, when NNN is a prime plus 1, at a critical ppp value, $p=N\ensuremath{-}1$. Such MUBs should play critical roles in filling complements.
We investigate the interplay between mutual unbiasedness and product bases for multiple qudits of possibly different dimensions. A product state of such a system is shown to be mutually unbiased to a product basis only if each of its factors is mutually unbiased to all the states which occur in the corresponding factors of the product basis. This result implies both a tight limit on the number of mutually unbiased product bases which the system can support and a complete classification of mutually unbiased product bases for multiple qubits or qutrits. In addition, only maximally entangled states can be mutually unbiased to a maximal set of mutually unbiased product bases.
We propose a systematic procedure to construct all the possible bases with a definite factorization structure (eigenstates of n commuting monomials constructed as products of Pauli operators) for an n -qubit system, as well as the possibility of collecting them into mutually unbiased sets. We also discuss an algorithm for the determination of basis separability and propose a criteria for complementarity between such bases. The results are applied to generate non-isomorphic complete sets of mutually unbiased bases.
We provide a construction of sets of (d/2+1) mutually unbiased bases (MUBs) in dimensions d=4,8 using maximal commuting classes of Pauli operators. We show that these incomplete sets cannot be extended further using the operators of the Pauli group. However, specific examples of sets of MUBs obtained using our construction are shown to be strongly unextendible; that is, there does not exist another vector that is unbiased with respect to the elements in the set. We conjecture the existence of such unextendible sets in higher dimensions (d=2^{n}, n>3) as well. Furthermore, we note an interesting connection between these unextendible sets and state-independent proofs of the Kochen-Specker Theorem for two-qubit systems. Our construction also leads to a proof of the tightness of a H_{2} entropic uncertainty relation for any set of three MUBs constructed from Pauli classes in d=4.