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Mutually complementary and compatible binary measurements on N qubits

W. E. Lawrence, Časlav Brukner, Anton Zeilinger

Published Apr 2, 2001• 2 citations
OpenAlex

Abstract

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.

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Mutually unbiased binary observable sets onNqubits

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One of the essential features of quantum mechanics is that most pairs of observables cannot be measured simultaneously. This phenomenon manifests itself most strongly when observables are related to mutually unbiased bases. In this paper, we shed some light on the connection between mutually unbiased bases and another essential feature of quantum mechanics, quantum entanglement. It is shown that a complete set of mutually unbiased bases of a bipartite system contains a fixed amount of entanglement, independent of the choice of the set. This has implications for entanglement distribution among the states of a complete set. In prime-squared dimensions we present an explicit experiment-friendly construction of a complete set with a particularly simple entanglement distribution. Finally, we describe the basic properties of mutually unbiased bases composed of product states only. The constructions are illustrated with explicit examples in low dimensions. We believe that the properties of entanglement in mutually unbiased bases may be one of the ingredients to be taken into account to settle the question of the existence of complete sets. We also expect that they will be relevant to applications of bases in the experimental realization of quantum protocols in higher-dimensional Hilbert spaces.

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Maximal complementarity in the n-qubit Pauli group

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Observables in quantum mechanics are generally complementary, that is, they reveal mutually incompatible pieces of information about a given system. This property is not only a fundamental tenet of the quantum formalism, but also a key component in quantum cryptographic protocols. As such it has fuelled much research into finding sets of highly complementary observables. In its strongest form -- the one we consider in this work -- the information between complementary observables is not merely incompatible but mutually exclusive: maximal information about one observable implies no information about the other, and vice versa. Maximal sets of non-degenerate complementary observables are known to exist in systems of prime power dimension such as n-qubit systems. Here, we study complementarity of degenerate observables, specifically we prove that observables associated with the n-qubit Pauli group also exhibit complementarity under this restriction: first, we obtain a criterion for two such observables to be complementary and, second, we relate maximal sets of complementary observables with informational pure state complementarity equalities, further studied in two companion papers. Equivalently, these results can be formulated in terms of maximal complementary sets of (not necessarily maximal) Abelian subgroups of the Pauli group linking with (possibly coarse-grained) mutually unbiased bases. Finally, we prove that these complementarity sets entail strong entropic uncertainty relations.

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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.

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Entanglement patterns in mutually unbiased basis sets

Jay Lawrence

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 ($D={p}^{N}$), as realized by systems of $N$ prime-state particles, where full complements of $D+1$ 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 ($p+1$) 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 $p+1$ 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 $p$), 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 ($p+2$) different ways to form full complements. With $N=4$, there are 6 MUB types for $p=2$, but new MUB types become possible with larger $p$, 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 $N$ and, also, when $N$ is a prime plus 1, at a critical $p$ value, $p=N\ensuremath{-}1$. Such MUBs should play critical roles in filling complements.

arXiv:1103.3818•Physical Review A•Aug 25, 2011•22 citations
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.
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.
One of the essential features of quantum mechanics is that most pairs of observables cannot be measured simultaneously. This phenomenon manifests itself most strongly when observables are related to mutually unbiased bases. In this paper, we shed some light on the connection between mutually unbiased bases and another essential feature of quantum mechanics, quantum entanglement. It is shown that a complete set of mutually unbiased bases of a bipartite system contains a fixed amount of entanglement, independent of the choice of the set. This has implications for entanglement distribution among the states of a complete set. In prime-squared dimensions we present an explicit experiment-friendly construction of a complete set with a particularly simple entanglement distribution. Finally, we describe the basic properties of mutually unbiased bases composed of product states only. The constructions are illustrated with explicit examples in low dimensions. We believe that the properties of entanglement in mutually unbiased bases may be one of the ingredients to be taken into account to settle the question of the existence of complete sets. We also expect that they will be relevant to applications of bases in the experimental realization of quantum protocols in higher-dimensional Hilbert spaces.
A compete orthonormal basis of NNN-qutrit unitary operators drawn from the Pauli group consists of the identity and ${9}^{N}\ensuremath{-}1$ traceless operators. The traceless ones partition into 3N+1{3}^{N}+13N+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 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.
Observables in quantum mechanics are generally complementary, that is, they reveal mutually incompatible pieces of information about a given system. This property is not only a fundamental tenet of the quantum formalism, but also a key component in quantum cryptographic protocols. As such it has fuelled much research into finding sets of highly complementary observables. In its strongest form -- the one we consider in this work -- the information between complementary observables is not merely incompatible but mutually exclusive: maximal information about one observable implies no information about the other, and vice versa. Maximal sets of non-degenerate complementary observables are known to exist in systems of prime power dimension such as n-qubit systems. Here, we study complementarity of degenerate observables, specifically we prove that observables associated with the n-qubit Pauli group also exhibit complementarity under this restriction: first, we obtain a criterion for two such observables to be complementary and, second, we relate maximal sets of complementary observables with informational pure state complementarity equalities, further studied in two companion papers. Equivalently, these results can be formulated in terms of maximal complementary sets of (not necessarily maximal) Abelian subgroups of the Pauli group linking with (possibly coarse-grained) mutually unbiased bases. Finally, we prove that these complementarity sets entail strong entropic uncertainty relations.
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.
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.