On mutually unbiased bases

Acronyms

Introduction

Two orthonormal bases of a Hilbert space are said to be mutually unbiased (MU) if the transition probabilities from each state in one basis to all states of the other basis are the same irrespective of which pair of states is chosen. Put differently, if the physical system is prepared in a state of the first basis, then all outcomes are equally probable when we conduct a measurement that probes for the states of the second basis. This situation is symmetrical, it does not matter from which of the two bases we choose the prepared state and which is the other basis that is measured: Unbiasedness of bases is a mutual property, possessed jointly by both bases. Familiar examples are the bases of position and momentum eigenstates for a particle moving along a line, and the spin states of a spin-$\frac{1}{2}$ particle for two perpendicular directions.

When the Hilbert space dimension $N$ is a prime power, ${N=p^{\mathnormal{\textsc{m}}}}$, there exist sets of ${N+1}$ mutually unbiased bases (MUB). These sets are maximal in the sense that it is not possible to find more than ${N+1}$ MUB in any $N$-dimensional Hilbert space, there is simply no room for the ${(N+2)}$th basis. Such a maximal set of MUB is also complete because when we know all the probabilities of transition of a given quantum state towards the states of the bases of this set — exceptional situations aside, there are $(N+1)(N-1)=N^{2}-1$ independent probabilities — we can reconstruct the statistical operator that characterizes this quantum state; in other words we can perform full tomography or complete quantum state determination.

The existence of a maximal set of MUB for ${N=p^{\mathnormal{\textsc{m}}}}$ is demonstrated by an explicit construction, not by an abstract existence proof. Various methods have been used for the construction of maximal sets of MUB, including the Galois–Fourier approach of this review. Other constructions are based on generalized Pauli matrices, discrete Wigner functions, abelian subgroups, mutually orthogonal Latin squares, and finite-geometry methods.

All known constructions rely on the fact that $N$ is the power of a prime and, therefore, they say nothing about other dimensions, of which ${N=6}$ is the smallest one and also the one that has been studied most intensely. At present, there is a widely shared conviction that one cannot have a maximal set of seven MUB for ${N=6}$ and that the largest sets of MUB have no more than three bases. This conviction is strongly founded in a solid body of evidence but, strictly speaking, it is an unproven conjecture.

This situation is reminiscent of seemingly similar existence questions about finite affine planes, Graeco-Latin squares, and related geometrical structures where prime-power dimensions also play a privileged role. As suggestive as these similarities may be, there is, however, no known connection as yet between the two kinds of existence problems.

There is a plethora of applications whenever maximal sets of MUB are available, in particular when the physical system is composed of many q-bits (${N=2^{\mathnormal{\textsc{m}}}}$), the building blocks of devices for quantum information processing. Not surprisingly, then, the rise of quantum information science has triggered fresh interest in MUB and, as a consequence, our knowledge about MUB and their applications is much richer now. But the various facts are scattered over a large number of publications, and the many pieces of the puzzle do not readily fit together and do not compose a uniform picture.

We are here reviewing the state of affairs in an attempt to offer a unified view, with emphasis on both the structural properties of MUB and their use in quantum-information applications. As in all constructions of MUB in prime power dimension, a crucial element is a finite commutative division ring — a Galois field of $N$ elements.A ring is a set that is closed under two operations: addition and multiplication. They obey the usual rules, associativity and commutativity of both operations, the distributive law, existence of a unique neutral element for the addition and a neutral element $1$ for the multiplication. A field, or division ring, is a ring with multiplicative inverses for every nonzero element. Finite fields with $N$ elements exist if and only if $N$ is a power of a prime, and the mathematical properties of Galois fields are exploited in all constructions of maximal sets of MUB. Modifications of these constructions in the absence of a finite field do not yield maximal sets of MUB for other dimensions.

The paper is structured as follows. We begin with a brief survey of elements of quantum kinematics in Sec. 1. The legacy of Weyl and Schwinger: the notion of complementary observables and their algebraic completeness, the MUB associated with them, and the ${N\to\infty}$ limit of continuous degrees of freedom — all these are central to the story told in Sec. 1.1. It is supplemented by remarks on the Heisenberg–Weyl group of unitary operators and the related Clifford group as well as, in Sec. 1.2, a geometrically motivated “measure of unbiasedness” of two bases, a distance in a real euclidean vector space.

Section 2 deals with the construction of a maximal set of MUB in prime power dimension, ${N=p^{\mathnormal{\textsc{m}}}}$, systematically treated as a composite system of m $p$-dimensional subsystems. For the purpose of introducing some notational conventions, but also for the benefit of the typical working physicist for whom Galois fields are hardly the daily bread, we recall the most important and most relevant properties of Galois fields in Sec. 2.1. We are making extensive use of a formalism in which the numbers $0,1,2,\dots,N-1$ play a dual role — they are elements of a Galois field, but also ordinary integers. This somewhat unconventional approach enables us to give a compact, transparent construction of a maximal set of MUB in Sec. 2.2–2.4. A fitting version of the discrete Heisenberg–Weyl group, also known as the generalized Pauli group, is an important tool for the construction; its abelian subgroups define the MUB. In passing, we establish the contact between these MUB and the complementary observables of the Weyl–Schwinger methodology (Sec. 2.5).

The survey of applications of the maximal set of MUB in Sec. 3 begins with the construction of a complete set of maximally entangled states, the analogs of the familiar Bell states of two–q-bit systems, in Sec. 3.1. After brief accounts of their use for quantum dense coding (Sec. 3.2) and teleportation (Sec. 3.3), we discuss in Sec. 3.4 how the generalized Bell states facilitate quantum cryptography and eavesdropping with the aid of covariant cloning machines and comment on the role of the Heisenberg–Weyl operators in error correction. Section 3 closes with a brief discussion of entanglement swapping (Sec. 3.5).

The prime-power version of the so-called Mean King’s problem (Sec. 4.1) opens the section on quantum state tomography. The Mean King’s problem is, in fact, very closely related to the discrete analog of Wigner’s continuous phase space function which — jointly with its Fourier partner, the analog of Weyl’s characteristic function — is the subject matter of Sec. 4.2. We comment on the covariance of the Wigner-type operator basis and discuss the $N\to\infty$ limit of continuous degrees of freedom. The relation to finite affine planes in Sec. 4.3 provides further insights into the underlying geometry.

Section 5 is devoted to the matrices that transform pairs of MUB into each other: the complex Hadamard matrices. Pairs of bases may be equivalent or not, in the sense that one can map the basis states of one pair on those of the other pair by a unitary transformation in conjunction with permutations of the basis states (Sec. 5.1). The equivalence of triplets of MUB is more difficult to check (Sec. 5.2). Mutually unbiased Hadamard matrices (MUHM) are encountered when there are more than two MUB. Accordingly, one can investigate sets of MUB by studying the corresponding sets of MUHM, and vice versa. All Hadamard matrices of size ${N\leq 5}$ have been classified (Sec. 5.3), and all sets of MUB are known for ${N<6}$ (Sec. 5.7). The situation is not so clear, and thus more interesting, for ${N=6}$; we report what is known about the families of ${6\times 6}$ Hadamard matrices in Sec. 5.5, after a general discussion of affine families and tensor products in Sec. 5.4, and we deal with MUB for ${N=6}$ in Secs. 5.8–5.10. Hadamard matrices for ${N>6}$ get their share of attention in Sec. 5.6.

We close with a brief summary and concluding remarks (Sec. 6) and provide some additional technical details in four appendixes. The standard set of MUHM for prime dimension is given in B, and a prime-distinguishing function related to this standard set is introduced in C. Finally, D deals with MUB for the two–q-bit case of ${N=4}$.

Elements of quantum kinematics

As emphasized by Bohr in his 1927 Como lecture, quantum systems have properties that are complementary: equally real but mutually exclusive. If one such property is known accurately, then the complementary property is completely unknown. Here, “known accurately” means that the outcome of a measurement can be predicted with certainty, whereas “completely unknown” means that all outcomes are equally likely — the two properties are maximally incompatible. Familiar examples are the position and momentum of a particle moving along a line, and the $x$ and $z$ spin components of a spin-$\frac{1}{2}$ object. These are, in fact, the extreme cases of a continuous degree of freedom and a binary degree of freedom — the latter being the “q-bit” of recent quantum information terminology.

Intermediate are “q-nits,” $N$-dimensional quantum degrees of freedom ($N>1$), for which the measurement of a physical property can have at most $N$ exclusive outcomes. Following Weyl and Schwinger, we call a pair of observables, $A$ and $B$, complementary if their eigenvalues are not degenerate (there is the full count of $N$ different possible measurement results) and the sets of normalized kets $|a_{j}\rangle$ and $|b_{k}\rangle$ that describe states with predictable measurement outcomes for $A$ and $B$, respectively, are MU,

The important detail is not the value on the right, which is implied by the normalization to unit total probability, but that the transition probabilities on the left do not depend on the quantum numbers $a_{j}$ and $b_{k}$.In fact, there can be different right-hand sides for infinite degrees of freedom, when normalization is more subtle; see Secs. 1.1.8–1.1.11. We will mostly deal with finite degrees of freedom.

Technically speaking, $A$ and $B$ are normal operatorsA normal operator $A$ commutes with its adjoint $A^{\dagger}$: $AA^{\dagger}=A^{\dagger}A$, and can be regarded either as a function of a more fundamental hermitian operator or as a function of a unitary operator. and $|a_{j}\rangle$, $|b_{k}\rangle$ are their eigenkets, which make up two bases that are orthonormal and complete,

where $\mathbf{1}$ is the identity operator. We recognize that the complementarity of $A$ and $B$ is in fact a property of their respective eigenket bases. The particular eigenvalues are irrelevant, we just need to know that they are not degenerate. It follows in particular that, if $A$ and $B$ are complementary, then $\alpha A$ and $\beta B$ with ${\alpha\beta\neq 0}$ are complementary as well. And if a unitary transformation turns $A$ into $A^{\prime}$ and $B$ into $B^{\prime}$, then the pair $A^{\prime},B^{\prime}$ is complementary if the pair $A,B$ is. Therefore, we can shift the focus from the pair $A,B$ of complementary observables to the pair $\{|a_{j}\rangle\},\{|b_{k}\rangle\}$ of MUB.

Whenever is it expedient to be specific about the observables associated with a basis, we will follow the guidance of Weyl and SchwingerA brief account of the history of the subject can be found in Ref. 7. and choose unitary operators to represent physical quantities. In the present context, these will be nondegenerate cyclic operators with period $N$,

with products of fewer than $N$ factors not equaling the identity. The eigenvalues of $A$ and $B$ are then the $N$ different $N$th roots of unity,

That these cyclic operators are a pair of complementary operators can be stated as

which is the operator version of (1.1). Indeed, (1.1) and (1.5) imply each other.

1.2 Existence of a basic pair of complementary observables

The first question we address is whether there always is a pair of complementary observables for each quantum degree of freedom. The affirmative answer begins with selecting an orthonormal reference basis $|0\rangle$, $|1\rangle$, …, $|N-1\rangle$ — we will refer to it as the computational basis from Sec. 2.2 onwards. Then we define a second orthonormal basis $|\widehat{0}\rangle,|\widehat{1}\rangle,\dots,|\widehat{N-1}\rangle$ by means of the discrete quantum Fourier transformation,

by construction — the two bases are MU, indeed.

In analogy with the Pauli operators $\sigma_{x}$ and $\sigma_{z}$, we introduce the cyclic operators $X$ and $Z$ in accordance with

As an immediate consequence of (1.7), we note that $X$ and $Z$ are unitary shift operators that permute the kets or bras of the respective other basis cyclically,

and (1.5) holds for $(A,B)=(X,Z)$, as it must. The fundamental Weyl commutation rule ${ZX=\gamma_{N}^{\ }XZ}$ follows. It is the analog of the familiar $N=2$ identity ${\sigma_{z}\sigma_{x}=-\sigma_{x}\sigma_{z}}$ and is more generally, and more usefully, stated as

valid for all integer values of $m$ and $n$, both positive and negative.

1.3 Algebraic completeness of the basic pair of operators

The second question, which also has an affirmative answer, is whether the pair $X,Z$ of complementary observables parameterizes the degree of freedom completely. Put differently: Are all other operators functions of $X$ and $Z$?

As a first step, we observe that the projectors onto the respective eigenstates are polynomials of $X$ or $Z$,

where the Kronecker delta symbols are to be understood in the usual sense of an operator function, as exemplified by

The second step in writing an arbitrary operator $F$ as a function of $X$ and $Z$ is to exploit the completeness of the two bases,

where the denominator is assuredly nonvanishing.Numbers of the form of $f_{j,k}$ are known as “weak values” of $F$ in the context of “weak measurements.” This answers the second question by giving an explicit expression for $F$ as a polynomial of $X$ and $Z$, here written in a unique way as an $XZ$-ordered function: In products, all $X$ operators stand to the left of all $Z$ operators. Of course, quite analogously, we can also write $F$ in a unique $ZX$-ordered form — as an example recall the equivalence of the $XZ$-ordered operator on the left of (1.11) with the $ZX$-ordered product on the right. In summary, there is not just one function of $X$ and $Z$ that equals the given operator $F$, there are many such functions.

The lesson of these considerations is that the pair $X,Z$ is algebraically complete, there are no operators that are not linear combinations of products of powers of $X$ and $Z$. Accordingly, we can phrase Bohr’s Principle of Complementarity, the fundamental principle of quantum kinematics, in the following technical terms: For each degree of freedom the dynamical variables are a pair of complementary observables. For a textbook discussion, see Ref. 17.

1.4 The Heisenberg–Weyl group; the Clifford group

Supplemented with powers of $\gamma_{N}^{\ }$, the $XZ$-ordered products that are implicit in (1.14),

make up the Heisenberg–Weyl group of unitary operators, also called the generalized Pauli group, with operator multiplication as the composition,

In addition to this notion of the Heisenberg–Weyl group as a group of unitary operators that are composed by multiplication, there is also the notion of the Heisenberg–Weyl group as a group of unitary transformations

that are composed by sequential execution. There is no difference in (1.18) between $Y=X^{n}Z^{m}$ and $Y=Z^{m}X^{n}$,

More generally, the powers of $\gamma_{N}^{\ }$ in (1.15) are irrelevant here, and therefore the group of unitary transformations has $N^{2}$ elements and is abelian. By contrast, the group of unitary operators is nonabelian; its abelian subgroups play a crucial role in Sec. 1.1.6 below. Weyl’s view of “quantum kinematics as an abelian group of rotations” with its utter disregard of phase factors in the “ray fields” should be understood in this context; see Ch. IV, Sec. 14 in Ref. 3.

We shall pay due attention to the phase factors in (1.15) where they are relevant, but otherwise remember that the physically more essential factors in (1.15) are the powers of $X$ and $Z$, and thus we will not be overly pedantic when referring to the Heisenberg–Weyl group. In the given context, it will be clear whether we mean the group of unitary operators with its $\gamma_{N}^{l}$ phase factors or the group of unitary transformations. An example is the observation that the $N$th power of $Y_{l,m,n}^{\ }$ can differ from the identity operator,

For the group of unitary operators, the appearance of $(-1)^{mn}$ is crucial, telling us that one quarter of the $Y_{l,m,n}^{\ }$s have period $2N$ for even $N$, whereas this is of no concern for the group of unitary transformations. As an example, consider once more the ${N=2}$ situation with ${X=\sigma_{x}}$ and ${Z=\sigma_{z}}$, for which

The unitary operators $C$ that map the Heisenberg–Weyl group onto itself under conjugation,Anti-unitary operators could be, and often are, included — see Ref. 18, for example — but we have no use for them here. that is: $Y_{l,m,n}\to CY_{l,m,n}C^{\dagger}$ equals one of the $Y_{l,m,n}$s, constitute the so-called Clifford group. It contains the Heisenberg–Weyl group as a subgroup, but is truly larger. For ${N=2}$, the Clifford group contains $24$ unitary transformations (and is isomorphic to the symmetry group of the cube) whereas the Heisenberg–Weyl group contains only four unitary transformations. An example of a transformation belonging to the former but not the latter is the “q-bit Hadamard gate” $(\sigma_{x}+\sigma_{z})/\sqrt{2}$ that is represented by the familiar Hadamard matrix

if we use the standard $2\times 2$ matrices for $\sigma_{x}$ and $\sigma_{z}$.

1.5 Composite degrees of freedom

If $N$ is a composite number, $N=N_{1}N_{2}$ with $N_{1}>1$ and $N_{2}>1$, then some of the Heisenberg–Weyl operators have a shorter period, as exemplified by $(Y_{0,N_{1},0})^{N_{2}}=(X^{N_{1}})^{N_{2}}=X^{N}=\mathbf{1}$. As a consequence, there are Heisenberg–Weyl operators that have different spectral properties and are not related to each other by a unitary transformation.

It is then methodical to regard the $N$-dimensional degree of freedom as composed of a $N_{1}$-dimensional and a $N_{2}$-dimensional degree of freedom. Accordingly, the labels $k$ of the kets $|k\rangle$ of the reference basis are understood as pairs $k_{1},k_{2}$ with ${k=k_{1}+k_{2}N_{1}}$ whereby ${k_{1}=0,1,\dots,N_{1}-1}$ and ${k_{2}=0,1,\dots,N_{2}-1}$. The action of the corresponding cyclic operators $X_{1}$ and $X_{2}$ is given by

and the respective $k_{1}=N_{1}-1$ and $k_{2}=N_{2}-1$ statements are

By construction, $X_{1}$ and $X_{2}$ have periods $N_{1}$ and $N_{2}$, respectively, and as a consequence of the algebraic completeness of the pair $X,Z$ of complementary observables, we can express $X_{1}$ and $X_{2}$ quite explicitly as functions of $X$ and $Z$, with the outcome

Clearly, $X_{1}$ commutes with $X_{2}$ because $Z^{N_{2}}$ commutes with $X^{N_{1}}$ when $N_{1}N_{2}=N$, as is the case here.

Likewise one constructs the complementary partners $Z_{1}$ and $Z_{2}$ as the operators that cyclically advance the respective quantum numbers of the common eigenbras $\langle\widehat{j_{1}},\widehat{j_{2}}|$ of $X_{1}$ and $X_{2}$, which are related to the kets $|k_{1},k_{2}\rangle$ through the analog of (1.7),

In summary, then, the original $N$-dimensional degree of freedom, parameterized by the pair $X,Z$, is decomposed into the product of two degrees of freedom, a $N_{1}$-dimensional and a $N_{2}$-dimensional one, parameterized by the pairs $X_{1},Z_{1}$ and $X_{2},Z_{2}$, respectively.

In passing, we note that the two bases of product kets $|k_{1},k_{2}\rangle$ and $|\widehat{j_{1}},\widehat{j_{2}}\rangle$ are MU. This illustrates how one can construct MUB of a composite degree of freedom from such bases of its constituents.

If $N_{1}$ or $N_{2}$ are composite numbers themselves, this reasoning can be applied again, if necessary repeatedly, until one has one degree of freedom for each prime factor of $N$. These prime degrees of freedom are fundamental and cannot be decomposed further. As emphasized by Schwinger in his teaching, they are the elementary quantum degrees of freedom.

1.6 Prime degrees of freedom

More generally, we can consider any two components ${A=\vec{a}\cdot\vec{\sigma}}$ and ${B=\vec{b}\cdot\vec{\sigma}}$ of Pauli’s vector operator $\vec{\sigma}$ whose cartesian components are $\sigma_{x}$, $\sigma_{y}$, and $\sigma_{z}$. Operators $A$ and $B$ are complementary if the nonvanishing three-dimensional numerical vectors $\vec{a}$ and $\vec{b}$ are orthogonal to each other, ${\vec{a}\cdot\vec{b}=0}$. Since there are at most three pairwise orthogonal vectors, there are at most three pairwise complementary operators and at most three MUB. The choice $\sigma_{x}$, $\sigma_{y}$, $\sigma_{z}$ for the three operators is, therefore, not particular, but typical.

If $N$ is an odd prime, $N=3,5,7,11,13,\dots$, then all unitary Heisenberg–Weyl operators $Y_{l,m,n}^{\ }$ of (1.15) are cyclic with period $N$, except for the identity $\mathbf{1}=Y_{0,0,0}^{\ }$. Further, we observe that the $N+1$ operators

are pairwise complementary, as one verifies most directly with the aid of (1.5) and (1.16) in conjunction with

It follows that the ${N+1}$ bases of eigenkets, one for each of the operators in (1.27), are MU. In addition to the eigenbases of $X$ and $Z$ that we met in Sec. 1.1.2, there are thus ${N-1}$ more such bases.

And there cannot be a ${(N+2)}$th basis because a counting argument shows that one can at most have ${N+1}$ bases that are MU. One way of seeing this is presented in Sec. 1.2 below.

In this context, we note here that the powers of the operators in (1.27) make up ${N+1}$ abelian cyclic subgroups of the Heisenberg–Weyl group with $N$ unitary operators in each subgroup. Remembering that the identity is contained in each subgroup, this gives a total count of ${(N+1)(N-1)+1=N^{2}}$ operators, one representative for each set of $Y_{l,m,n}^{\ }$s with common $m,n$ values, that is: one count for each $X^{m}Z^{n}$ product.

Explicitly, ket $|i,k\rangle$, the $k$th eigenket of the $i$th basis, $XZ^{i}|i,k\rangle=|i,k\rangle\gamma_{N}^{k}$, is given by

in terms of the reference basis of eigenkets of $Z$. For $i=0$ we have the eigenstates of $X$, $|\widehat{k}\rangle=|0,k\rangle$. While (1.29) correctly states the eigenkets of $XZ^{i}$ for all odd $N$, these bases are pairwise MU only if $N$ is prime. With due attention to the extra phase factors required by (1.20) one can give a similar expression for $|i,k\rangle$ when $N$ is even.

In summary, we can systematically construct ${N+1}$ bases that are MU if $N$ is prime. As noted, the construction based on the cyclic operators in (1.27) does not work if $N$ is composite; try $N=4$ to see what goes wrong. We return to the case of $N=6$ in Sec. 5.10, and a general discussion for arbitrary $N\geq 2$ is given in C.

Yet, this is not the end of the story. If $N=p^{\mathnormal{\textsc{m}}}$ is the power of a prime, for which ${N=8=2^{3}}$ and ${N=9=3^{2}}$ are examples, it is possible to modify the construction such that it does work in a closely analogous way. The clue is to replace the modulo-$N$ shifts of (1.9) and (1.10) by shifts of a Galois field arithmetic that treats the $N$-dimensional degree of freedom systematically as composed of m $p$-dimensional constituents. This is the theme of Sec. 2, followed by applications in Secs. 3 and 4.

This Galois cure is, however, not available for ${N=6}$ and ${N=10}$ or other composite $N$ values that are not powers of a prime, simply because the number of elements in a finite field is always a prime power. Section 5 contains a report on what is known about these cases, in particular about ${N=6}$. The question whether there are seven MUB for ${N=6}$ is currently unanswered, but there is a lot of evidence, and a growing conviction in the community, that there are no more than three such bases. And three such bases are immediately available by pairing each of the three q-bit bases ($N_{1}=2$) with one of the four q-trit bases ($N_{2}=3$) to product bases as in (1.26).

1.7 The continuous limit of N→∞→𝑁N\to\infty

Since composite values of $N$ refer to composite quantum degrees of freedom, we take the limit $N\to\infty$ through prime values of $N$, thereby dealing with a single degree of freedom of increasing complexity. The prime nature of $N$ will not be so crucial, however, but we make use of the fact that large primes are odd numbers and relabel the kets of the reference basis $|k\rangle$ and the bras $\langle\widehat{j}|$ of the Fourier-transformed basis such that now ${j,k=0,\pm 1,\pm 2,\dots,\pm\frac{1}{2}(N-1)}$.

Next, we introduce a small, eventually infinitesimal, parameter $\epsilon$ by

to account for the fact that the basic unit of complex phase $2\pi/N$ gets arbitrarily small when $N\to\infty$. Aiming at a continuous degree of freedom in this limit, we also relabel the states in accordance with

The numbers $a$ and $b$ will cover the real axis, ${-\infty<a,b<\infty}$, when ${N\to\infty}$, ${\epsilon\to 0}$.

The unitary operator $X$ acting on $|k\rangle$ increases $k$ by unity, so that it effects ${b\to b+\epsilon}$. Likewise $Z$ applied to $\langle\widehat{j}|$ results in ${a\to a+\epsilon}$. This suggests the identification of hermitian operators $A$ and $B$ such that

The Weyl commutation relation (1.11) then appears as

The said restriction is that, for large $N$, only $a,b$ values from a finite vicinity of matter, which is to say that we break the cyclic nature of the labels $a,b$,

and take for granted that all relevant values of $a,a^{\prime}$ and $b,b^{\prime}$ are such that we stay inside the range $-(\pi/\epsilon-\epsilon/2)\cdots(\pi/\epsilon-\epsilon/2)$. Put differently, we give up the periodicity that would force us to identify ${a=+\infty}$ with ${a=-\infty}$ in the ${\epsilon\to 0}$ limit.

After performing the ${N\to\infty}$, ${\epsilon\to 0}$ limit with this restriction, the statements of (1.1.7) hold with continuous values for $a$ and $b$. We can, therefore, exhibit the terms that are linear in $a$ or $b$ and arrive at

We recognize, of course, Heisenberg’s commutation relation for a pair of complementary hermitian observables of a continuous degree of freedom, such as position $A$ and momentum $B$ (in natural units) for the motion along a line.

These $N\to\infty$ considerations for $X$ and $Z$ have to be supplemented by counterparts for their respective kets and bras. We need to identify

as the continuum versions of the orthogonality and completeness relations in (1.2).

This discussion of the $N\to\infty$ limit is a variant of Schwinger’s treatment in Sec. 1.16 of Ref. 6; see also Sec. 1.2.5 in Ref. 17. It should be appreciated that ${N\to\infty}$ is not a limit in the precise sense that one has in calculus. Rather, it is a systematic method for inferring the properties of the basic operators for continuous degrees of freedom, but these operators then stand on their own and the consistency of the inferred algebraic properties must be verified.

We note that, in addition to the standard symmetric limit that treats $X$ and $Z$ on equal footing and results in the Heisenberg pair of $A$ and $B$ (position and momentum for motion along a line), there are also asymmetric limits. For instance, if the position variable — the analog of the hermitian $A$ of (1.1.7) — is kept periodic over a finite range in the limit, one obtains the pair of azimuth-angle operator and angular-momentum operator for motion on a circle, with which we deal in Sec. 1.1.9. In a third way of taking the $N\to\infty$ limit, the hermitian position variable is kept positive throughout and one arrives at a continuous quantum degree of freedom of the kind that parameterizes radial motion; see Sec. 1.1.10. Finally, there is a fourth procedure, in which the position values cover a finite range without, however, retaining the cyclic nature by identifying the boundaries with each other; this results in a degree of freedom of the kind associated with the polar angle in spherical coordinates (Sec. 1.1.11).

1.8 Continuous degree of freedom 1: Motion along a line

Knowing that there are ${N+1}$ pairwise complementary observables for prime degrees of freedom, we expect to find an infinite number of them for a continuous degree of freedom. Indeed, there is a continuum of pairwise complementary observables and, therefore, a continuum of MUB, although an interesting complication can be observed too.

Harking back to Sec. 1.1.6, we recall that each basis in the set of MUB consists of the joint eigenstates of the unitary operators that one gets by taking products of one of the unitary operators in the list (1.27) with itself. Translated into the continuum case of Sec. 1.1.7, the corresponding unitary operators are those of (1.34), and since

It is expedient to choose the single-exponent form

for the subgroup elements, so that the subgroup composition rule

involves no additional phase factors, and we denote the common eigenkets and eigenbras of all unitary operators in the $(\alpha,\beta)$ subgroup by $|\alpha,\beta;y\rangle$ and $\langle\alpha,\beta;y|$,

If one wishes, one can regard $|\alpha,\beta;y\rangle$ and $\langle\alpha,\beta;y|$ as eigenstates of the hermitian operator ${\beta A+\alpha B}$ with eigenvalue $y$, but we prefer to work with the sets of bounded unitary operators rather than the unbounded hermitian operators.

As usual, the eigenstates are normalized to the Dirac delta function,

which implies that, up to a phase factor of no consequence,

for $\lambda\neq 0$, consistent with $Y(\lambda\alpha,\lambda\beta;t/\lambda)=Y(\alpha,\beta;t)$. The subgroup for $(\lambda\alpha,\lambda\beta)$ is identical with the subgroup for $(\alpha,\beta)$, with the elements parameterized differently. The respective eigenstates are in one-to-one correspondence, but differ from each other by a normalization factor (except when $\lambda=-1$).

These statements have no analogs for finite $N$, when the normalization of states is unambiguous and the parameterization of the abelian subgroups is essentially unique. In the continuous case, by contrast, there is more than one way of parameterizing the continuous abelian subgroups, and one would have to impose constraints on $\alpha$ and $\beta$ to avoid this innocuous ambiguity, such as insisting on $\alpha=\cos\theta$ and $\beta=\sin\theta$ with ${0\leq\theta<\pi}$ or, equivalently, permitting only $(\alpha,\beta)=(0,1)$ and $\alpha=1$ with arbitrary $\beta$. Clearly, constraints of this sort are a bit awkward, and they are not necessary.

The projector $|\alpha,\beta;y\rangle\langle\alpha,\beta;y|$ is given by

as one verifies by, for instance, checking that

follows and confirms that we have a basis for each of the abelian subgroups.

which is Eq. (11) in Ref. 24. Since the value of \bigl{|}\langle\alpha,\beta;y|\alpha^{\prime},\beta^{\prime};y^{\prime}\rangle\bigr{|}^{2} does not depend on the quantum numbers $y$ and $y^{\prime}$ that label the states of the two bases, the two bases are MU. This is true for the bases to any two different abelian subgroups. Indeed, we have a continuum of MUB for a continuous degree of freedom.

We could have arrived at the same conclusion by the following more direct argument that exploits the observations made after (1.2). There are unitary transformations that turn ${\beta A+\alpha B}$ into $\kappa A$ and ${\beta^{\prime}A+\alpha^{\prime}B}$ into $\kappa^{\prime}B$ with ${\kappa\kappa^{\prime}=\alpha\beta^{\prime}-\beta\alpha^{\prime}}\neq 0$. Now, since the pair $A,B$ is complementary, so is the pair $\kappa A,\kappa^{\prime}B$, which implies that the pair ${\beta A+\alpha B},{\beta^{\prime}A+\alpha^{\prime}B}$ is complementary as well, and their bases of eigenstates are MU.

Whereas the right-hand side of (1.1) has the same value of $N^{-1}$ for any pair of MUB for a $N$-dimensional degree of freedom, this is not the case for the right-hand side of (1.52); recall footnote ‘2’. For a given pair of bases specified by the coefficients $(\alpha,\beta)$ and $(\alpha^{\prime},\beta^{\prime})$, we can either choose $(\alpha^{\prime\prime},\beta^{\prime\prime})=(\alpha+\alpha^{\prime},\beta+\beta^{\prime})$ or $(\alpha^{\prime\prime},\beta^{\prime\prime})=(\alpha-\alpha^{\prime},\beta-\beta^{\prime})$ to supplement them with a third basis such that these three MUB have the same numerical value for the constant transition probability densities between each pair of bases. The basis for any fourth choice $(\alpha^{\prime\prime\prime},\beta^{\prime\prime\prime})$ will have a different value for one or more of its transition probability densities with the earlier three bases. This observation by Weigert and Wilkinson means that the continuous set of MUB, composed of the bases of (1.46), contains three-element subsets that are polytopes of MUB in the sense of Sec. 1.2.

1.9 Continuous degree of freedom 2: Motion along a circle

We parameterize the position around the circle by the $2\pi$-periodic azimuth $\varphi$ — with $|\varphi\rangle=|\varphi+2\pi\rangle$, for instance — and normalize the corresponding bras and kets in accordance with the orthogonality and completeness relations

where the integration range is any $2\pi$ interval and $\delta^{(2\pi)}(\ )$ denotes the $2\pi$-periodic version of Dirac’s delta function,

We regard the azimuthal states $|\varphi\rangle$ as eigenstates of a unitary operator $E$,

This $E$ is the proper $N\to\infty$ limit of $X$ in the present context.

All azimuthal wave functions $\psi(\varphi)=\langle\varphi|\ \rangle=\psi(\varphi+2\pi)$ are periodic, and the Fourier series of $\langle\varphi|$ identifies the eigenstates of the associated angular momentum operator $L$,

Their orthonormality and completeness relations are

the $\varphi$-basis and the $l$-basis are MU. The respective unitary shift operators are powers of $E$ and exponential functions of $L$,

Despite these analogies and the great structural similarities with the situation of Sec. 1.1.8, there is a striking difference: There is no third basis that is MU with respect to both the $\varphi$-basis and the $l$-basis.

To make this point, let us assume that ket $|\ \rangle$ belongs to such a third basis. Then it must be true that

The completeness relations in (1.53) and (1.57) then imply

which contradict each other. It follows that there is not even a single ket with the properties (1.61); indeed, there is no third basis.

This situation of a missing third basis is a unique feature of the $E,L$-type continuous degree of freedom. There is always a third basis for finite $N$ — the three eigenbases to $X$, $Z$, and $XZ$ of (1.27) are pairwise MU for all $N>1$ — and there is a continuum of MUB for the continuous degrees of freedom of the three other types. It appears that the combination of the continuous position variable $E$ with the discrete momentum variable $L$ is at the heart of the matter. For the other continuous degrees of freedom, the respective position and momentum variables are both continuous, as will be discussed below.

The nonexistence of a third basis that supplements the $\varphi$-basis and the $l$-basis does not exclude the possibility that there are other bases that are MU, perhaps with sets of MUB that have more than two elements. Currently, we are not aware of any such set, however, but its bases would have to be rather unusual. For, two different discrete bases (such as the $l$-basis) cannot be MU, nor can two different continuous bases (such as the $\varphi$-basis) be MU. And if one basis is discrete and the other continuous, the dilemma of (1.1.9) cannot be avoided.

1.10 Continuous degree of freedom 3: Radial motion

In spherical coordinates, radial motion is characterized by a positive position operator $R>0$,

whereas the eigenvalues of its complementary partner $S$ are all real numbers,

confirm that the $r$-basis and the $s$-basis are MU and that $R$ and $S$ are a pair of complementary observables.

as follows from (1.65). The resulting Weyl commutation relation

tell us that $S$ is the hermitian generator of scaling transformations,

The unitary operator products in (1.67) make up the Heisenberg–Weyl group here, and the abelian subgroups can be characterized by common values of $\tau$ and $\mu$ in $(t,\lambda)=\kappa(\tau,\mu)$. In full analogy with (1.46)–(1.52), then, the bases defined by

for $(\tau,\mu)\neq(0,0)$ are pairwise MU,

which follows from (1.68) or from (1.69).

1.11 Continuous degree of freedom 4: Motion within a segment

After dealing with the azimuthal and radial degrees of freedom in Secs. 1.1.9 and 1.1.10, we now turn to the degree of freedom associated with the polar angle $\vartheta$ of spherical coordinates, $(x,y,z)=(r\sin\vartheta\,\cos\varphi,r\sin\vartheta\,\sin\varphi,r\cos\vartheta)$. Since the values of $\vartheta$ are restricted to a finite interval $0\leq\vartheta\leq\pi$, where the endpoints are not identified with each other as is the case for the periodic azimuth $\varphi$, we speak of “motion within a segment,” very much like the popular textbook example of the “particle in a box,” about which some non-textbook material is reported in Ref. 25. The relations between the position and momentum operators for cartesian and spherical coordinates are discussed in A.

The eigenstates of the position variable $\Theta$ and its complementary partner $\Omega$ are related to each other by

and the respective orthonormality and completeness relations are

for the $\omega$-basis. Accordingly, the unitary shift operators are specified by

from which we get the Heisenberg commutator

More generally, we have the analog of (1.72),

identifies $\log\tan\frac{\Theta}{2}$ and $\Omega$ as a Heisenberg pair of complementary observables. Remembering the lessons of Secs. 1.1.8 and 1.1.10, we conclude that the abelian subgroups of the Heisenberg–Weyl group composed of the unitary operators of (1.77) define a continuum of MUB, with the set of MUB having three-element subsets that are MUB polytopes in the sense of Ref. 24.

We close this excursion into the realm of continuous degrees of freedom with a comment on the completeness and orthonormality relations (1.63) and (1.74). Why did we not absorb the factors $r$ and $\sin\vartheta$ into the normalization of the respective bras and kets? There are two good reasons: (i) Such a change of normalization would spoil the relations (1.65) and (1.73); (ii) these factors would re-appear in a disturbing way when the orthonormality and completeness relations are rewritten in terms of the eigenstates for the Heisenberg partners $\log R$ and $\log\tan\frac{\Theta}{2}$ of $S$ and $\Omega$, respectively. In other words, it is very natural to have the factors $r$ and $\sin\vartheta$ in (1.63) and (1.74).

In view of the various subtle issues regarding the normalization of eigenkets and eigenbras for continuous degrees of freedom, the definition of what constitutes a pair of complementary observables — given above in the context of (1.1) — should perhaps be modified to state more carefully that two nondegenerate observables $A$ and $B$ are complementary if one can normalize their respective eigenstates consistently such that \bigl{|}\langle a|b\rangle\bigr{|}^{2} has the same value for all eigenbras $\langle a|$ of $A$ and all eigenkets $|b\rangle$ of $B$.

2 A geometrically motivated measure of mutual unbiasedness

The kets $|\ \rangle$ in $N$-dimensional Hilbert space, and their adjoint bras $\langle\ |=|\ \rangle^{\dagger}$, are rather abstract geometrical objects, and so are the linear operators that map kets on kets and bras on bras, among them the statistical operator $\rho$ that summarizes our knowledge about the state of the physical $N$-dimensional degree of freedom under consideration. With reference to a specified basis, the kets are represented by numerical column vectors $\psi$ ($N\times 1$ matrices), the bras by row vectors $\psi^{\dagger}$ ($1\times N$ matrices), and the linear operators by $N\times N$ matrices, among them the density matrix $\varrho$ for the statistical operator $\rho$. We denote these relationships by $\psi\mathrel{\widehat{=}}|\ \rangle$, $\psi^{\dagger}\mathrel{\widehat{=}}\langle\ |$, and $\varrho\mathrel{\widehat{=}}\rho$, respectively.

There are many density matrices, one for each reference basis, to one and the same statistical operator, much like there are many trios of components for the velocity vector of the moon, one trio for each coordinate system. One should not confuse the velocity vector with its components, or the statistical operator with the density matrix used to represent it numerically.

When they exist, maximal sets of MUB form a very distinct geometrical pattern in the set of hermitian matrices of unit trace — the real euclidean space that contains the set of density matrices. This is where we begin our story about maximal sets of MUB, although in most of what follows we will prefer to work directly in Hilbert space. The two pictures ought to be considered as complementary, each of them possessing advantages and drawbacks.

The set $\{\varrho\}$ of density matrices is a convex body in the set of hermitian matrices of unit trace. Its pure states are the one-dimensional projectors. The set of its pure states has real dimension $2(N-1)$, and can be identified with the complex projective Hilbert space. The dimension of $\{\varrho\}$ is $N^{2}-1$, and the space in which it sits can be regarded as a vector space, with its origin at the maximally mixed state

With any hermitian matrix $M$ of unit trace we associate a traceless matrix

The set of these traceless matrices forms a vector space, and we will think of them as vectors. The matrix representation is used to define the inner product

Thus the squared distance between the tips of the two vectors $\mathbf{m}_{1}$ and $\mathbf{m}_{2}$ is

With any unit ket $|e\rangle$ in Hilbert space we associate a vector $\mathbf{e}$ in $\mathbf{R}^{N^{2}-1}$, the space of $(N^{2}-1)$-component real vectors, through

so that the squared length of $\mathbf{e}$ is

All vectors in $\mathbf{R}^{N^{2}-1}$ with this specific length sit on the surface of the outsphere of the body $\{\varrho\}$, the smallest sphere containing the body. But it is important to realize that it is only a small $2(N-1)$-dimensional subset of this outsphere that corresponds to vectors in Hilbert space — most of the outsphere lies outside the body. The case ${N=2}$ is an exception: In this case the outsphere is the familiar Bloch sphere, which is identical to the boundary of the body of density matrices.

imply each other. If $|e_{i}\rangle$ is an orthonormal basis of kets, the corresponding vectors $\mathbf{e}_{i}$ form a regular simplex that spans an $(N-1)$-plane, and clearly

Hence the simplex is centered at the origin. We have normalized its edge lengths to unity.

Next consider two MUB with kets $|e_{i}\rangle$ and $|f_{j}\rangle$, respectively, represented by the vectors $\mathbf{e}_{i}$ and $\mathbf{f}_{j}$. The two equations

are equivalent ways of stating that the bases are MU and, therefore, the two planes spanned by a pair of MUB are totally orthogonal: Each vector in one plane is orthogonal to all vectors in the other plane. Since the dimension of our space is $N^{2}-1=(N+1)(N-1)$, we can fit at most $N+1$ totally orthogonal $(N-1)$-planes into it. This is one way of seeing that the maximal number of MUB is $N+1$.

Let us now momentarily forget that our vectors $\mathbf{e}_{i}$, $\mathbf{f}_{i}$, and so on, are supposed to come from unit vectors in Hilbert space. Whatever the value of $N$, we can always find $N+1$ totally orthogonal $(N-1)$-planes in $\mathbf{R}^{N^{2}-1}$, and if we place a regular simplex in each we will obtain a quite interesting convex polytope with $N(N+1)$ vertices. When $N=2$, it is in fact a regular octahedron, but for other values of $N$ it needs a name of its own. We will call it the MUB polytope, without implying that there exists a maximal set of MUB in the $N$-dimensional Hilbert space. The MUB polytope and the body of density matrices share the same outsphere and, in this manner, the existence problem for MUB can be turned into the problem of rotating the MUB polytope in such a way that all its vertices fit into the small subset of pure quantum states that are present in that outsphere. This is a hard problem (unless $N=2$). Indeed, from this perspective it is not obvious that we can find even one pair of MUB but, as we have seen in Sec. 1.1.2, we can always do this. It is the existence of a maximal set, with $N+1$ bases that are pairwise MU, which is in doubt for general $N$.

Viewing bases as $(N-1)$-planes in $\mathbf{R}^{N^{2}-1}$ gives us the means to quantify how close a given pair of bases is to being MU. The trick is to regard $n$-planes in $\mathbf{R}^{m}$ as rank-$n$ projectors in a vector space of real $m\times m$ matrices, in analogy to the way we go from vectors in Hilbert space to density matrices. This gives us an embedding of the Grassmannian of $n$-planes into a flat vector space equipped with a natural euclidean distance, and hence a natural notion of distance between vectors in Hilbert space. To derive it, consider the $N$ vectors $\mathbf{e}_{i}$. Then form the $(N^{2}-1)\times N$ matrix

It has rank $N-1$ because of (1.88). Next form the projector onto the $(N-1)$-plane spanned by the linearly dependent vectors $\mathbf{e}_{i}$. It is

Finally, the square of the chordal Grassmannian distance between a pair of planes is

where the kets $|e_{a}\rangle$ are related to $\Pi_{e}$ through (1.85), (1.90), and (1.91), and the kets $|f_{b}\rangle$ are analogously related to $\Pi_{f}$. The last expression of (1.92) shows that ${D_{c}=0}$ if the projectors $|f_{b}\rangle\langle f_{b}|$ are a permutation of the projectors $|e_{a}\rangle\langle e_{a}|$, in which case we have the same basis twice, possibly with different labeling.

and that the distance is maximal if and only if the two bases are MU. This notion of distance has been used to study packing problems for $n$-planes, and as a measure of “MUness”. If we pick our bases at random, using the unitarily invariant Fubini–Study measure to define “random,” we find that the average squared distance is given by

If the dimension is large, $N\gg 1$, two bases picked at random are likely to be almost MU. Let us finally mention that entropic uncertainty relations in effect provide an interesting alternative measure of “MUness”.

Construction of mutually unbiased bases in prime power dimensions

In what follows, we work in a Hilbert space of prime power dimension $N=p^{\mathnormal{\textsc{m}}}$ with $p$ a prime number and m a positive integer. These are the dimensions for which maximal sets of MUB are known to exist. Moreover, and not coincidentally, there is a finite Galois field with $N=p^{\mathnormal{\textsc{m}}}$ elements. We shall label these elements by integer numbers $i$, $0\leq i\leq N-1$, or, equivalently, by m-tuples $(i_{0},i_{1},\ldots,i_{\mathnormal{\textsc{m}}-1})$ of integers, each integer running from 0 to $p-1$, that we get from the $p$-ary expansion of $i$:

Each field is characterized by two operations, a multiplication and an addition, that we shall denote by $\odot$ and $\oplus$ respectively. As in footnote ‘1’, we shall use the symbols and $1$ for the neutral elements of addition and multiplication, respectively, throughout the paper, consistent with their meaning as integers.

Further, we adopt the particular convention that the elements of the field are labeled in such a way that the addition is equivalent to the component-wise addition modulo $p$, that is

for $n=0,1,\dots,\mathnormal{\textsc{m}}-1$, where $i_{n},j_{n},k_{n}$ are the respective coefficients of (2.1). As a consequence, the summation in (2.1) is also a field summation,

All fields with the same number of elements are equivalent up to a relabeling, and there is no strict obligation for the convention (2.2), but it is natural and convenient in the present context, because it allows us to regard the elements of the field both as labels of basis states and as integer numbers that we can use for getting powers of complex numbers in accordance with the usual computation rules.

Actually, that there exists a relabeling such that the addition is equivalent to the addition modulo $p$ component-wise is a direct consequence of the fact that for all finite fields the characteristics of the field — the smallest number of times that we must add the element $1$ (neutral for the multiplication) to itself before we obtain the element (neutral for the addition) — is always equal to a prime number ($p$ when $N=p^{\mathnormal{\textsc{m}}}$).

Unfortunately, there is no similarly simple convention for the field multiplication $\odot$, and — the exceptions $N=p$ and $N=4$ aside — one has a choice between several equally good ways of defining the field multiplication $\odot$ such that it is consistent with the component-wise definition of the field addition $\oplus$. In view of the associative and distributive nature of $\odot$, that is: $(a\odot b)\odot c=a\odot(b\odot c)$ and $(a\oplus b)\odot c=(a\odot c)\oplus(b\odot c)$, respectively, we only need to state the values of $p^{j}\odot p^{k}$, the products of powers of $p$, with $j,k=0,1,\ldots,\mathnormal{\textsc{m}}-1$.

For $\mathnormal{\textsc{m}}=1$, $N=p$, the field multiplication is just multiplication modulo $p$. For $\mathnormal{\textsc{m}}>1$, we have the Galois construction

Hereby, the coefficients that define the $j+k=\mathnormal{\textsc{m}}$ products are restricted by the requirement that

is an irreducible polynomial over the Galois field with $p$ elements, which is to say that it cannot be factored into two nonconstant polynomials whose coefficients are modulo-$p$ integers.

In a standard textbook parameterization of the Galois field with ${N=p^{\mathnormal{\textsc{m}}}}$ elements, one identifies the field elements with polynomials that are defined by the coefficients of the $p$-ary expansion of (2.1),

Addition and multiplication of the field elements are then carried out as addition and multiplication of the corresponding polynomials modulo the polynomial of (2.5), with the resulting sums and products stated as polynomials of degree $\mathnormal{\textsc{m}}-1$ with modulo-$p$ integer coefficients. Clearly, this gives the component-wise addition of (2.2) and multiplication in accordance with (2.4). Since the field multiplication is not familiar to readers with a typical theoretical-physics background, we now discuss it in some detail.

For instance, the choice $2\odot 2=3$ is unique for $N=4$, and for $p$ odd and $N=p^{2}$, one can always choose $p\odot p=\mu^{\ }_{0}$ with $\mu^{\ }_{0}$ not a square, such as $3\odot 3=2$, $5\odot 5=2$ or $5\odot 5=3$, $7\odot 7=3$ or $7\odot 7=5$ or $7\odot 7=6$, and so forth. For higher powers of $p=2$, there are several choices too; they include $2\odot 4=5$ for $N=8$, $2\odot 8=3$ for $N=16$, and $2\odot 16=5$ for $N=32$.

As a final example, we mention $3\odot 9=(1,2,2)=25$ for $N=3^{3}$.The choice $3\odot 9=25$ is the largest one of the eight permissible values. The other seven values for $(\mu_{0},\mu_{1},\mu_{2})$ are $(1,1,0)=4$, $(2,1,0)=5$, $(2,0,1)=11$, $(1,1,1)=13$, $(2,2,1)=17$, $(1,0,2)=19$, and $(2,1,2)=23$. Each of them yields a consistent implementation of the field multiplication. This implies first $9\odot 9=(2,2,0)=8$ and then

for the multiplication of two arbitrary field elements. The special cases ${3\odot 13=1}$ and ${9\odot 17=1}$ may serve as illustrations.

and the coefficients for $j+k=\mathnormal{\textsc{m}}+1,\mathnormal{\textsc{m}}+2,\ldots,2\mathnormal{\textsc{m}}-2$ are successively calculated with the aid of the recurrence relation

which is valid for $j+k=1,2,\dots,2\mathnormal{\textsc{m}}-2$. The field product of two arbitrary elements is then given by

where $\mathcal{M}_{m}^{\ }=\mathcal{M}_{m}^{T}$ is the symmetric $\mathnormal{\textsc{m}}\times\mathnormal{\textsc{m}}$ matrix

and in the products $a\mathcal{M}_{m}b^{T}$ we regard $a=(a_{0},a_{1},\dots)$ as a row of $p$-ary coefficients and $b^{T}$ as a column. These row$\,\times\,$matrix$\,\times\,$column products are ordinary matrix products with the outcome evaluated modulo $p$. The matrices $\mathcal{M}_{0}^{\ }$, $\mathcal{M}_{1}^{\ }$, …, $\mathcal{M}_{\mathnormal{\textsc{m}}-1}^{\ }$ are invertible, in the sense of modulo-$p$ arithmetic, because there is a unique multiplicative inverse for each non-zero field element. For instance, we have

Having thus established how the field addition $a\oplus b$ and the field multiplication $a\odot b$ are implemented for any two field elements ${a,b=0,1,\ldots,N-1}$ with $N=p^{\mathnormal{\textsc{m}}}$, we can put the Galois field to use. For notational simplicity, let us denote by $\gamma$ the basic $p$th root of unity,

rather than writing $\gamma_{p}$ as in (1.4). Exponentiating $\gamma$ with elements $g$ of the field — regarding now, as noted above, the field elements as integers — we obtain complex phase factors of the type $\gamma^{g}$ with $0\leq g\leq N-1\,$. As $g$ is an integer, such phase factors can take on only $p$ different values, which are completely determined by the first component $g_{0}$ of the $p$-ary expansion of $g$,

The following identity plays a fundamental role:

Indeed, if $i=0$, then $\displaystyle\sum_{j=0}^{N-1}\gamma^{j\odot i}=\sum_{j=0}^{N-1}1=N$. Otherwise,

because the field multiplication is invertible, and then

where the first step exploits (2.19) and recognizes that there are $p^{\mathnormal{\textsc{m}}-1}$ field elements $j^{\prime}$ with the same value of $j^{\prime}_{0}$.

The fact that the field addition is the component-wise addition modulo $p$, combined with the rule (2.19), implies the following useful identity:

In the final expression on the right, the sum $i\oplus j$ is the Galois sum of $i$ and $j$, which is then regarded as an integer, just as we regard the result of the Galois multiplication $j\odot i$ in (2.20) and (2.21) as an integer, and so get integer powers of $\gamma$. Relation (2.23) expresses, in the language of mathematicians, that the $p$th roots of unity are additive characters of the Galois field.

It is important to note, in order to avoid confusions, that different types of operations are present at this level: The internal field operations ($\oplus$ and $\odot$) must not be confused with the modulo-$N$ operations. As an illustration of the differences between these operations, we consider the case $p=2$, $\mathnormal{\textsc{m}}=2$, $N=p^{\mathnormal{\textsc{m}}}=4$ and give the tables for field addition ($\oplus$) and field multiplication ($\odot$) in Table 2.1(a) as well as the tables for modulo-$N$ addition and multiplication ($\oplus_{4}$ and $\odot_{4}$, respectively) in Table 2.1(b).

One can check that the field and modulo-$4$ multiplications are distributive with respect to the associated addition, but that there are no non-zero dividers of only in the case of the field multiplication, whereas we have $0=2\odot_{4}2$ for the modulo-$4$ multiplication. As a consequence, the field multiplication table exhibits an invertible group structure when the first line and first column are removed. All operations are commutative as can be seen from the invariance of all four tables under transposition.

Let us express q-quarts as products of two q-bits, in accordance with the binary encoding of $i=(i_{0},i_{1})$ for $i=0,1,2,3$ as stated by

With the aid of the $\oplus$ subtable in Table 2.1, it is easy to verify that

This illustrates that the field addition is equivalent to the component-wise modulo-$p$ addition.

It is also worth reminding that the properties

2 The computational basis

Consider now a quantum degree of freedom of prime-power dimension $N=p^{\mathnormal{\textsc{m}}}$ — a q-nit composed of m q-pits. The corresponding Hilbert space of kets has a conveniently chosen orthonormal reference basis consisting of $|0\rangle$, $|1\rangle$, …, ${|N-1\rangle}$, which we regard as the computational basis of kets. The adjoint basis of bras comprises all $\langle n|=|n\rangle^{\dagger}$ with $n=0,1,\dots,N-1$. As usual, the inner products $(\,\cdot\,,\,\cdot\,)$ of two kets or two bras are given by Dirac brackets ($\equiv$ bra-kets), for which the orthonormality relations

3 The dual basis

Let us now consider the unitary transformations $V^{0}_{l}$ that shift each label of the states of the computational basis $\{|0\rangle,|1\rangle,\dots,|i\rangle,\dots,|N-1\rangle\}$ by $l$,

so that each $V^{0}_{l}$ implements a permutation among the kets of the computational basis, but does not change the basis as a whole. The shift in (2.28) is a shift modulo $N$ in prime dimensions only ($N$=$p$) and then $V^{0}_{l}$ is identical with $X^{l}$ of Sec. 1.1.2; in prime power dimensions ($N=p^{\mathnormal{\textsc{m}}}$, $\mathnormal{\textsc{m}}>1$) the shift consists of m shifts modulo $p$, component-wise. The transformations effected by $V^{0}_{l}$ with $l=0,1,\dots,N-1$ make up a commutative group of permutations with $N$ elements that is isomorphic to the Galois addition.

Generalizing the procedure outlined in Ref. 33, we employ a suitable discrete Fourier-type transformation — the inverse Galois–Fourier transformation — to define the dual basis as follows:

where the symbol $\ominus$ represents the inverse of the Galois addition $\oplus$, that is: $x=\ominus y$ if $x\oplus y=0$. It is easy to check that these dual kets are joint eigenkets of the unitary permutation operators $V^{0}_{l}$. Indeed, we have

which identifies the eigenvalues $\displaystyle\gamma^{l\odot j}$. These are $p$ different eigenvalues, each occurring $p^{\mathnormal{\textsc{m}}-1}$ times.

Obviously, the dual basis and the computational basis are MU by construction,

When the dimension is prime ($N=p$), the dual basis is the standard discrete Fourier transform of the computational basis, as in (1.6); when $N$ is a power of $2$, the Galois–Fourier transform is a real Hadamard transform.

These permutation operators are diagonal in the computational basis,

This is the dual counterpart of the analogous expression for the shifts in the computational basis,

which is equivalent to (2.30) and follows from that eigenket statement.

As mentioned above, it is immediately clear that $|k\rangle\to V^{0}_{l}|k\rangle=|k\oplus l\rangle$ is a component-wise addition, where the components of the q-nit ket $|k\rangle$ are the m q-pits that compose it, as is illustrated by (2.1) and (2.1) for $p=2$ and $\mathnormal{\textsc{m}}=2$. More generally,

where each factor $|k_{m}\rangle$ in the tensor product is a q-pit ket, and the sums $k_{m}+l_{m}$ are modulo-$p$ sums. It follows that $V_{l}^{0}$ is a product of factors, each of which referring to one of the q-pits,

where the $m$th factor affects the $m$th q-pit only, with $V^{0}_{p^{m}}$ giving a unit shift of the $m$th modulo-$p$ label.

where $\mathcal{M}_{0}$ is the th multiplication matrix in (2.14) and $\underline{k}_{m}$ is the $m$th component of $k\mathcal{M}_{0}=(\underline{k}_{0},\underline{k}_{1},\ldots)$. Since $\mathcal{M}_{0}$ is invertible, we can parameterize the field element $k$ in terms of the coefficients $\underline{k}_{m}$,

with the field elements $g_{m}$ defined such that their $p$-ary coefficients make up the rows of the $\mathnormal{\textsc{m}}\times\mathnormal{\textsc{m}}$ matrix $\mathcal{M}_{0}^{-1}$. Alternatively, we could define the $g_{m}$s by their basic property

Therefore, a unit increase of $\underline{k}_{m}$ means the addition of $g_{m}$ to $k$, and the shift operator $V^{l}_{0}$ factorizes accordingly into a product of powers of single–q-pit Fourier operators, each of which (the $m$th, say) acting on the single–q-pit bras $\langle\widetilde{j_{m}}|$ only and leaving the other ${\mathnormal{\textsc{m}}-1}$ q-pit bras in the products of (2.37) unaffected,

with the $m$th factor affecting the $m$th q-pit only,

For instance, we have $g_{0}=1$, $g_{1}=12$, $g_{2}=3$, and $\underline{k}_{0}=k_{0}$, $\underline{k}_{1}=k_{2}$, $\underline{k}_{2}=k_{1}-k_{2}$ for the $N=27$ example of (2.10), (2.16), and (2.17).

The respective unitary operator factors for unit shifts in (2.36) and (2.40) commute if they refer to different q-pits,

which essentially states that the Galois shifts with their component-wise addition are consistent with the factorization of the ${N=p^{\mathnormal{\textsc{m}}}}$-dimensional degree of freedom into m $p$-dimensional degrees of freedom, as discussed in Sec. 1.1.5. And for the pair of operators to the same q-pit, one easily verifies the Weyl commutation rule

Equations (2.42) and (2.43) are particular cases of (2.45) below.

4 Construction of the remaining N𝑁N-1 mutually unbiased bases

In the previous section we established a pair of MUB, the computational basis, which can be chosen arbitrarily, and its dual basis, defined by (2.29). In this section, we shall generalize this construction in order to obtain the other $N-1$ bases that complement the computational basis and its dual basis such that the bases of each of the $N(N+1)/2$ pairs are MU.

Let us denote by $V^{j}_{i}$ the compositions of the shifts in the computational and the dual bases, obtained by ordinary operator multiplication of $V^{j}_{0}$ and $V^{0}_{i}$ ,

the building blocks of the Heisenberg–Weyl group. This is consistent with the previous expressions for $i=0$ or $j=0$ because $V_{0}^{0}$ is the identity. In particular, for $i=0$ and $j=l$ we get the second sum of (2.33), and for $i=l$ and $j=0$ we have the first sum of (2.34).

We note that the order of multiplication of $V^{j}_{0}$ and $V^{0}_{j}$ matters in the definition (2.44) because these unitary shift operators do not commute,

We recognize here the Weyl commutation rule for the two unitary operators $V^{j}_{0}$ and $V^{0}_{i}$, which is their basic algebraic relation.

In dimension $N=p=2$, the commutation relation (2.45) is that of the Pauli group (identify $V_{0}^{1}$ with $\sigma_{x}$ and $V_{1}^{0}$ with $\sigma_{z}$ once more). When the dimension is a prime number, the field operations are the addition and multiplication modulo $p$, and the properties of MUB are well-known; recall the discussion in Sec. 1.1.6 with its emphasis on the Heisenberg–Weyl group.

Currently, we consider the Heisenberg–Weyl group associated with the Galois addition and multiplication rather than the Heisenberg–Weyl group associated with the usual modulo-$N$ operations. These groups coincide in prime dimensions but differ for non-prime but prime-power dimensions. Notably, the Galois field is isomorphic to the modulo-$N$ ring in prime dimensions only ($N=p$). Nevertheless, the Heisenberg–Weyl group factorizes in dimension $p^{\mathnormal{\textsc{m}}}$ into products of operators that belong to the local q-pit Heisenberg–Weyl group. In the case of translations of the computational basis, the factorization is straightforward and given above in (2.36). And in the case of translations of the dual basis, where the mapping from global operator labels to local operator labels is more intricate, see (2.37)–(2.39), the factorization is stated in (2.40).

The composition law of the $N^{2}$ unitary operators introduced in (2.44) is

which is reminiscent of (1.20) and once again shows a difference between the single even prime ${p=2}$ and the odd primes ${p>2}$.

Yet another implication is the orthonormality relation for the $V^{j}_{i}$s, with respect to the Hilbert–Schmidt inner product,

because all $V^{j}_{i}$s are traceless, except $V^{0}_{0}=\mathbf{1}$. The other side of this coin is the relation

which one may regard as a manifestation of Schur’s lemma, inasmuch as the right-hand side follows after observing that the sum on the left commutes with all $V_{j}^{i}$ and must therefore be a multiple of the identity. Schwinger calls such statements about equal-weight averages over the whole phase space ergodic relations.

Equation (2.46) is the discrete analog of the familiar Baker–Campbell–Hausdorff relation for exponentiated position and momentum operators that we encountered in (1.34). An immediate consequence of (2.46) is

where $(\ )_{0}$ has the same meaning as in (2.23). In particular, (2.54) is fulfilled if $i\odot k=j\odot l$, which we note for later reference.

4.2 Abelian subgroups

Up to a global phase, (2.46) looks like a group composition law. Indeed, one can show that there is a true analog of what we observed in Sec. 1.1.6 for prime $N$: The $N^{2}$ unitary operators $V^{j}_{i}$ with $i,j=0,1,\dots,N-1$ make up $N+1$ commuting sets (abelian subgroups of the Heisenberg–Weyl group) of $N$ elements each that have only the identity $V_{0}^{0}$ in common. For each of these commuting sets, there is a basis of joint eigenkets of all $V_{i}^{j}$s in the set. The $N+1$ bases thus identified are pairwise MU. In passing, we note that this property can be shown, following an alternative approach developed in Ref. 35, to be a consequence of the fact that the $V_{i}^{j}$ operators form what is called “a maximally commuting basis of orthogonal unitary matrices.”

It is expedient to introduce a fitting notation and terminology before we proceed. We shall denote by $U^{i}_{l}$ the elements of these abelian subgroups, where $i$ labels the subgroup and runs from to $N$ to account for $N+1$ subgroups, while $l$ labels the $N$ elements in the subgroup and runs from to $N-1$. For the basis kets associated with the subgroups we use the convention that the $k$th basis ket for the $i$th subgroup is denoted by $|e^{i}_{k}\rangle$.