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Mutually Unbiased Bases in Low Dimensions

STEPHEN D. BRIERLEY

Published Nov 2, 2009• 9 citations
OpenAlex

Abstract

The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. The required observables only exist if it is possible to identify d mutually unbiased (MU) complex (d×d) Hadamard matrices, defining a complete set of d+1 MU bases. This thesis is an exploration of sets of r≤d+1 MU bases in low dimensions. We derive all inequivalent sets of MU bases in dimensions two to five confirming that in these dimensions, the complete sets of (d+1) MU bases are unique. In dimension six, we prescribe a first Hadamard matrix and construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and never find more than two that are mutually unbiased. We also study subsets of a complete set of MU bases by considering sets of pure states which satisfy the desired properties. We use this concept to provide the strongest numerical evidence so far that no seven MU bases exist in dimension six. In the final part of the thesis, we introduce a new quantum key distribution protocol that uses d-level quantum systems to encode an alphabet with c letters. It has the property that the error rate introduced by an intercept-and-resend attack is higher than the BB84 or six-state protocols when the legitimate parties use a complete set of MU bases.

Related papers

Constructing mutually unbiased bases in dimension six

STEPHEN D. BRIERLEY, Stefan Weigert

The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. In dimension six, the required observables only exist if it is possible to identify six mutually unbiased complex $(6\ifmmode\times\else\texttimes\fi{}6)$ Hadamard matrices. Prescribing a first Hadamard matrix, we construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and we never find more than two that are mutually unbiased. This result adds considerable support to the conjecture that no seven mutually unbiased bases exist in dimension six.

arXiv:0901.4051•Physical Review A•May 14, 2009•116 citations

All Mutually Unbiased Bases in Dimensions Two to Five

STEPHEN D. BRIERLEY, Stefan Weigert, Ingemar Bengtsson

All complex Hadamard matrices in dimensions two to five are known. We use this fact to derive all inequivalent sets of mutually unbiased (MU) bases in low dimensions. We find a three-parameter family of triples of MU bases in dimension four and two inequivalent classes of MU triples in dimension five. We confirm that the complete sets of (d+1) MU bases are unique (up to equivalence) in dimensions below six, using only elementary arguments for d less than five.

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All mutually unbiased bases in dimensions two to five

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All complex Hadamard matrices in dimensions two to five are known. We use this fact to derive all inequivalent sets of mutually unbiased (MU) bases in low dimensions. We find a three-parameter family of triples of MU bases in dimension four and two inequivalent classes of MU triples in dimension five. We confirm that the complete sets of (d+1) MU bases are unique (up to equivalence) in dimensions below six, using only elementary arguments for d less than five.

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in Dimensions Two to Five

STEPHEN D. BRIERLEY, Stefan Weigert, Ingemar Bengtsson

All complex Hadamard matrices in dimensions two to five are known. We use this fact to derive all inequivalent sets of mutually unbiased (MU) bases in low dimensions. We find a three-parameter family of triples of MU bases in dimension four and two inequivalent classes of MU triples in dimension five. We confirm that the complete sets of (d+1) MU bases are unique (up to equivalence) in dimensions below six, using only elementary arguments for d less than five.

•Jan 1, 2010•0 citations

More mutually unbiased bases

Mateo Cárdenes Wuttig, Joseph Tindall

Mutually unbiased bases (MUBs) describe quantum measurements for which certainty in one basis gives uniform outcomes in the others. Their maximum number remains unknown in dimensions that are not powers of a prime. We introduce an ansatz that, in many dimensions, allows us to construct more MUBs than have previously been found. Each basis in the ansatz consists of a diagonal phase matrix applied to a fixed unitary, which is a tensor product of Fourier matrices and, optionally, any real Hadamard matrix. This construction yields five MUBs in dimension 12, six in dimensions 48, 96, and 192, seven in dimensions 36 and 108, and ten in dimension 648. These bases can then be extended to exceed the tensor product bound, asymptotically, in every eighteenth dimension. Moreover, we show that Paley's real Hadamard matrix can be used to construct $q+1$ bases in dimension $d = q(q+1)$ for every prime power $q\equiv3\pmod4$. This count grows as $\sqrt d$ and cannot be exceeded by tensor products of smaller sets.

quant-pharXiv:2609.40311•Oct 1, 2026•0 citations

Mutually Unbiased Product Bases

Daniel McNulty

A pair of orthonormal bases are mutually unbiased (MU) if the inner products across all their elements have equal magnitude. In quantum mechanics, these bases represent observables that are complementary, i.e. a measurement of one observable implies maximal uncertainty about the possible outcome of a subsequent measurement of a second observable. MU bases have attracted interest in recent years because their properties seem to depend dramatically on the dimension d of the quantum system in hand. If the dimension is given by a prime or prime-power, the state space is known to accommodate a complete set of d+1 MU bases. However, for composite dimensions, such as d = 6, 10, 12, ..., complete sets seem to be absent and it is not understood why. In this thesis we carry out a comprehensive study of MU product bases in dimension six. In particular, we construct all MU bases in dimension six consisting of product states only. The exhaustive classification leads to several non-existence results. We also present a new construction of complex Hadamard matrices of composite order, which is a consequence of our work on MU product bases. Based on this construction we obtain several new isolated Hadamard matrices of Butson-type. \n

•Apr 1, 2013•0 citations
The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. The required observables only exist if it is possible to identify d mutually unbiased (MU) complex (d×d) Hadamard matrices, defining a complete set of d+1 MU bases. This thesis is an exploration of sets of r≤d+1 MU bases in low dimensions. We derive all inequivalent sets of MU bases in dimensions two to five confirming that in these dimensions, the complete sets of (d+1) MU bases are unique. In dimension six, we prescribe a first Hadamard matrix and construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and never find more than two that are mutually unbiased. We also study subsets of a complete set of MU bases by considering sets of pure states which satisfy the desired properties. We use this concept to provide the strongest numerical evidence so far that no seven MU bases exist in dimension six. In the final part of the thesis, we introduce a new quantum key distribution protocol that uses d-level quantum systems to encode an alphabet with c letters. It has the property that the error rate introduced by an intercept-and-resend attack is higher than the BB84 or six-state protocols when the legitimate parties use a complete set of MU bases.
The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. In dimension six, the required observables only exist if it is possible to identify six mutually unbiased complex $(6\ifmmode\times\else\texttimes\fi{}6)$ Hadamard matrices. Prescribing a first Hadamard matrix, we construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and we never find more than two that are mutually unbiased. This result adds considerable support to the conjecture that no seven mutually unbiased bases exist in dimension six.
Mutually unbiased bases (MUBs) describe quantum measurements for which certainty in one basis gives uniform outcomes in the others. Their maximum number remains unknown in dimensions that are not powers of a prime. We introduce an ansatz that, in many dimensions, allows us to construct more MUBs than have previously been found. Each basis in the ansatz consists of a diagonal phase matrix applied to a fixed unitary, which is a tensor product of Fourier matrices and, optionally, any real Hadamard matrix. This construction yields five MUBs in dimension 12, six in dimensions 48, 96, and 192, seven in dimensions 36 and 108, and ten in dimension 648. These bases can then be extended to exceed the tensor product bound, asymptotically, in every eighteenth dimension. Moreover, we show that Paley's real Hadamard matrix can be used to construct q+1q+1q+1 bases in dimension d=q(q+1)d = q(q+1)d=q(q+1) for every prime power q≡3(mod4)q\equiv3\pmod4q≡3(mod4). This count grows as d\sqrt dd​ and cannot be exceeded by tensor products of smaller sets.
A pair of orthonormal bases are mutually unbiased (MU) if the inner products across all their elements have equal magnitude. In quantum mechanics, these bases represent observables that are complementary, i.e. a measurement of one observable implies maximal uncertainty about the possible outcome of a subsequent measurement of a second observable. MU bases have attracted interest in recent years because their properties seem to depend dramatically on the dimension d of the quantum system in hand. If the dimension is given by a prime or prime-power, the state space is known to accommodate a complete set of d+1 MU bases. However, for composite dimensions, such as d = 6, 10, 12, ..., complete sets seem to be absent and it is not understood why. In this thesis we carry out a comprehensive study of MU product bases in dimension six. In particular, we construct all MU bases in dimension six consisting of product states only. The exhaustive classification leads to several non-existence results. We also present a new construction of complex Hadamard matrices of composite order, which is a consequence of our work on MU product bases. Based on this construction we obtain several new isolated Hadamard matrices of Butson-type. \n