States that "look the same" with respect to every basis in a mutually unbiased set
A complete set of mutually unbiased bases in a Hilbert space of dimension defines a set of orthogonal measurements. Relative to such a set, we define a MUB-balanced state to be a pure state for which the list of probabilities of the outcomes of any of these measurements is independent of the choice of measurement, up to permutations. In this paper we explicitly construct a MUB-balanced state for each prime power dimension for which (mod 4). These states have already been constructed by Appleby in unpublished notes, but our presentation here is different in that both the expression for the states themselves and the proof of MUB-balancedness are given in terms of the discrete Wigner function, rather than the density matrix or state vector. The discrete Wigner functions of these states are “rotationally symmetric” in a sense roughly analogous to the rotational symmetry of the energy eigenstates of a harmonic oscillator in the continuous two-dimensional phase space. Upon converting the Wigner function to a density matrix, we find that the states are expressible as real state vectors in the standard basis. We observe numerically that when is large (and not a power of 3), a histogram of the components of such a state vector appears to form a semicircular distribution.
Consider any energy eigenstate of a simple harmonic oscillator. Given a system in such a state, suppose we measure the observable , where is real and and are the position and momentum operators scaled so that the Hamiltonian is proportional to . We find that the resulting probability distribution is independent of . This property is closely related to a symmetry of the state’s Wigner function(1); (2): again with suitable scaling of the position and momentum axes, the Wigner function of any harmonic oscillator eigenstate is circularly symmetric around the origin of the two-dimensional phase space.
Our aim in this paper is to find state vectors in a finite-dimensional Hilbert space that have properties analogous to the above properties of harmonic oscillator eigenstates. In finite dimension, the closest analog of the set of measurements of the form is the set of measurements defined by a complete set of mutually unbiased bases (MUBs).(3) Two orthonormal bases in dimension are called mutually unbiased if the inner product between any vector in one of the bases and any vector in the other basis has magnitude . That is, for all and , where is the vector in the basis. Such bases are “as different as possible” from each other.(4); (5); (6) It is known that in any dimension , one can find at most bases that are pairwise mutually unbiased, and this bound can be achieved when is a power of a prime.(6); (7); (8) It is not known whether the bound can be achieved for any other value of , though there is strong evidence against this possibility for the smallest such value, .(9); (10); (11); (12); (13); (14); (15); (16)
To identify an analog of the first of the above properties of harmonic oscillator eigenstates, we consider a Hilbert space in which a complete set of MUBs exists. Let the bases be labeled by . Relative to such a set, we will say that a pure state in the Hilbert space is “MUB-balanced” if the list of probabilities, , is independent of , up to permutations. That is, given a MUB-balanced state, if one were to perform on the state the measurement corresponding to one of the MUBs, it would not be possible to tell, just from the set of probabilities of the outcomes, which measurement was being performed.
The interest in such states comes partly from a desire to understand the relation between discrete and continuous quantum mechanics. We will see that there are intriguing differences between the two cases. On a more practical note, complete sets of mutually unbiased bases have been used in the construction of quantum key distribution and secret-sharing schemes(17); (18); (19); (20); (21); (22); (23); (24), for which MUB-balanced states could play a useful conceptual role. In thinking about intercept-resend eavesdropping attacks, for example, Brierley has noted that when the legitimate participants in a quantum cryptographic scheme are using a complete set of mutually unbiased bases, there is no orthogonal measurement an eavesdropper could use that is “halfway” between all of these bases.(23) However, if there exists a MUB-balanced state, it could be used to define a non-orthogonal measurement that would be related in essentially the same way to each of the mutually unbiased bases (see Section IV below) and would in this sense be an analog of the “halfway-between” measurement in the BB84 scheme.(25) The concept of a MUB-balanced state is also closely related to the concept of a “minimum uncertainty state,” which has been studied in a number of earlier papers(27); (30); (31); (32); (33) and has been connected to symmetric measurements(31) and quantum random-access codes.(33) (See Section VI below for a discussion of the relation between “MUB-balanced” and “minimum uncertainty.”)
Finally, there is always a certain mathematical interest in finding states that, according to parameter-counting arguments, have no right to exist. In this case, one can reasonably argue that, by asking that a state vector be MUB-balanced, we are imposing constraints on the vector: for the first measurement, the independent probabilities are unconstrained at first, but then for each of the other measurements, the probabilities have to match those of the first measurement. (The permutation freedom is discrete and does not change the number of parameters.) But a pure state is specified by only real numbers. So we seem to be over-constraining the state by a factor of . Of course the equations that have to be satisfied are nonlinear; so we cannot assume that the parameter-counting argument is reliable. Still, if such a state exists, this fact tells us that there is something special about the structure of the complete set of MUBs, that allows the state to “beat the odds.”
For the case , one can see immediately that several MUB-balanced states exist. In that case a convenient complete set of mutually unbiased bases consists of the bases of eigenstates of the three Pauli operators , , and . A MUB-balanced state would be any pure state that, on the Bloch sphere, makes equal angles with the , , and axes. Of course it becomes much harder to imagine a MUB-balanced state as the dimension increases. In this paper we explicitly construct a MUB-balanced state for each prime power dimension for which (mod 4). Moreover, once one such state has been identified, it can be used to generate several others, as we will see in Section IV.
We could pose the question of the existence of a MUB-balanced state for any dimension for which a complete set of MUBs exists. So we could consider any value of that is a power of a prime. The case has in effect already been treated in earlier papers.(20); (26); (27); (28); (29) Though those papers were addressing slightly different questions—the cyclic generation of mutually unbiased bases or the existence of minimum uncertainty states—the arguments given there show directly that MUB-balanced states exist for every . The case we consider here, with being a prime power equivalent to 3 (mod 4), has in fact also been considered before, in unpublished notes by Appleby(34), again addressing the closely related concept of a minimum uncertainty state. It follows from Appleby’s argument—which is similar to a more specialized argument by Sussman(30)—that the minimum uncertainty state he constructs for any such dimension is also a MUB-balanced state. (See also the new paper by Appleby, Bengtsson and Dang.(35)) However, the proof we present here is self-contained and is different from Appleby’s, though it is certainly related. One unusual feature of our proof is that it is based entirely on the discrete Wigner function of the special state (see below) rather than its state vector or density matrix. It turns out that our argument does not work at all for (mod 4), and it appears to be an open question whether a MUB-balanced state exists in any of those cases.
Just as a harmonic oscillator eigenstate has a circularly symmetric Wigner function, the states we identify as MUB-balanced have a kind of circular symmetry in a discrete phase space. Here we take the discrete phase space to consist of the elements of , that is, the two-dimensional vector space over the finite field with elements. The phase space can be pictured as a array of points, labeled by two coordinates and that take values in . The discrete Wigner function is a representation of a quantum state as a real function on this phase space. For our special MUB-balanced state, the Wigner function is constant on each “circle,” defined as the set of solutions of an equation of the form with . It is in this sense that the Wigner function is circularly symmetric. (A different analog of an energy eigenstate state of a harmonic oscillator has been investigated by other authors.(36))
Of particular interest for our purpose is the connection between the discrete Wigner function and a complete set of mutually unbiased bases. As we will discuss in greater detail in the following section, in the discrete phase space we can speak of “lines” and “parallel lines,” each line consisting of exactly points. There are possible slopes of a line, and for each value of the slope, the points of the phase space can be partitioned into parallel lines having that slope. We call such a set of parallel lines a “striation.” Moreover, each striation is associated with one of the bases in a complete set of MUBs, in the following sense: given a quantum state represented by its Wigner function, if we sum the Wigner function over the lines of a striation, we obtain the probabilities of the outcomes of the orthogonal measurement associated with that striation. Thus a state is MUB-balanced (relative to the set of MUBs associated with the discrete Wigner function) if and only if its Wigner function yields the same list of numbers (up to permutation) when summed over any striation. The notion of circular symmetry enters the argument as a way of achieving this invariance, as we explain in Section IV. We will see that the role of circular symmetry is somewhat more subtle than in the case of a continuous phase space.
For the class of ’s we consider, and for the representation of MUBs we use, we find that the MUB-balanced state we identify is representable as a real vector in the standard basis. Numerical evaluation of the components of this vector reveals an intriguing feature: for large , a histogram of the values of the components typically appears to form a semicircular distribution (though not when is a power of 3). That this appearance reflects a genuine limiting behavior has in fact now been proved, and in greater generality, in a recent paper by Katz.(37)
Though it has been proved that for each of the dimensions through 5 there is only one complete set of MUBs up to unitary equivalence,(38); (39) it is known that in many higher dimensions unitarily inequivalent complete sets exist.(40) In the following sections, we consider only a specific class of MUBs associated with a discrete phase space as described above. However, our results directly imply the existence of MUB-balanced states relative to any equivalent set of MUBs. (Our results say nothing about unitarily inequivalent sets of MUBs.) The question of whether and in what sense the observed semicircular distribution carries over to such equivalent sets of MUBs is more subtle and we do not explore that question here.
We begin in the following section by defining the Wigner function and explaining more fully its relation to mutually unbiased bases. We conclude that section by writing down an expression for the Wigner function of our special state. In Section III we prove that this expression does indeed define a pure quantum state, and in Section IV we prove that the state is MUB-balanced. Then in Section V we write down the density matrix of the state and present a histogram of the values of the components of the state vector for a typical large value of . One sees there the approximate semicircular distribution mentioned above. The final section summarizes our results and makes a connection with minimum uncertainty states.
Ii The discrete Wigner function
Again, the discrete Wigner function is a representation of a quantum state as a real function on discrete phase space. The state could be pure or mixed—the Wigner function contains exactly the information normally expressed in the density matrix. (For example, multiplying a state vector by an overall phase factor does not change its Wigner function.) For our work here, it will be convenient to use the term “Wigner function” somewhat more broadly, to refer to a representation in phase space of an arbitrary Hermitian operator on the -dimensional Hilbert space, not just a density operator.
Several different discrete Wigner functions have been defined in the literature(44); (41) (see also the references cited in those two papers). In this paper we use the version of the Wigner function that seems to have first appeared in papers by Klimov and Muñoz(42) and by Vourdas(43); it is a particularly simple and natural case of a broad class of generalized discrete Wigner functions based on finite fields.(44) For odd prime dimensions, this Wigner function is equivalent to discrete Wigner functions(3); (45) that have been shown to be especially useful for the analysis of quantum computing.(45); (46); (47); (48)
Both our discrete Wigner function and our later arguments are couched in terms of finite fields, so we begin by recalling a few basic facts about such fields.(49) First, there exists a field with elements if and only if is a power of a prime, and for any such value there is only one field up to isomorphism. We are calling it . When is a prime number, is the same as , that is, the set with addition and multiplication mod , but there is no such equivalence for other prime powers. For with prime, the elements of can be written as
where is a specific element of and each is identified with an element of . We can think of as a root of an degree polynomial with coefficients in that does not factor in . (In a similar way, we construct the complex numbers by defining to be a root of , which does not factor in the reals.) Eq. (1) shows that we may regard as an -dimensional vector space over . Of course it is much more than that, since its elements can also be multiplied. In this paper we will make essential use of the notion of the trace of a field element, which can be used to map a general element of into an element of . The trace of , with , is defined by
(We use the lower-case “tr” to distinguish the field trace from the trace of a matrix.) Though it is not obvious from the definition, is a field element for which all the ’s in Eq. (1) are equal to zero except possibly . Eq. (2) therefore identifies an element of . The trace has the following properties:
and, again regarding as a vector space over ,
That is, the trace defines a linear map from the -dimensional vector space to itself.
We can now define our discrete Wigner function. Let be an odd prime and let be equal to for some positive integer . For any complex Hermitian matrix , the discrete Wigner function associated with is a real function of the phase space point , where again and take values in . (We will think of as the horizontal coordinate and as the vertical coordinate.) is defined as(42); (43)
where is the , unit-trace, Hermitian matrix given by
Here is the root of unity and the indices and take values in . The arithmetic in the argument of the Kronecker delta and in the exponent is in , and the trace is to be interpreted as an ordinary integer exponent. The usefulness of the choice (6) of the matrices will become clear later in this section.
We will frequently use the following identity, which generalizes a familiar fact about roots of unity: for any ,
One consequence of this identity is that the ’s are orthonormal in the sense that
That is, the numbers are the coefficients in the expansion of as a linear combination of the ’s.
Two properties of the Wigner function will be particularly useful for our purposes. First, because of the orthonormality of the matrices, we have, for any Hermitian and ,
The other property is the rule for finding the Wigner function of a product , given the Wigner functions of and . From the above definitions one can work out that this rule is given as follows:
As we have said, in discrete phase space one can speak of lines and parallel lines: A line is the set of solutions of an equation of the form , where , , and are elements of with and not both zero. Two lines are called parallel if they can be expressed by two such linear equations differing only in the value of . There are in total lines, which can be grouped into sets of parallel lines; these sets of lines are the striations of the discrete phase space. The striations correspond to the possible slopes of the lines, that is, the possible values of . These slope values include all the elements of along with (infinite slope corresponding to the case ). Note that two lines that are not parallel intersect in exactly one point.
Now we make the connection with mutually unbiased bases. For each line, consider the function on phase space that is nonzero only on that line, where it has the constant value . Starting from Eq. (9), it is not hard to show that this function is the Wigner function of a pure-state density matrix. This property is in fact the main motivation for choosing the form (6) of the matrices. Thus every line in phase space corresponds to a pure state. It then follows from Eqs. (8) and (10) that the states corresponding to parallel lines are orthogonal. Since there are parallel lines in a striation, each striation corresponds to an orthogonal basis for the Hilbert space.
Now consider two state vectors, corresponding to two lines that belong to different striations. Let and be the density matrices of the two states, and let and be the corresponding lines in phase space. Then we have
But the ’s are orthogonal, and there is exactly one point that is common to both lines, so there is only one nonzero term in the sum. According to Eq. (8) the value of this term is . Therefore
Thus the orthogonal bases associated with two distinct striations have the property that if we choose any two vectors, one from each basis, their inner product will always have the same magnitude, —the bases are mutually unbiased.
Earlier we claimed that the sums of the Wigner function over the lines of a striation are the probabilities of the outcomes of the measurement associated with that striation. To see why this is true, let be the density matrix of the pure state associated with the line , and let be the density matrix of the state being measured, whose Wigner function is . Then
which is indeed the probability of obtaining the outcome corresponding to the pure state .
We now specialize to the case (mod 4) and write down the Wigner function of the state we claim is MUB-balanced. We call this Wigner function , anticipating that it is indeed the Wigner function corresponding to a legitimate density matrix , but we will have to prove this. We arrived at via methods developed by Sussman(30) and Appleby(32), but our proofs will not depend on how the state was derived. We will be able to show from the form of the Wigner function itself that it satisfies the conditions of a MUB-balanced state. The Wigner function is
where consists of all the nonzero elements of , and the function is the quadratic character: for , is defined by
We will never encounter . In Eq. (16), the argument of will never equal zero because negative one has no square root in . (See item 3 in the list below.)
Notice that is a real function: in the sum over the terms and , which are conjugates of each other, are multiplied by the same factor. Therefore is the Wigner function of some Hermitian operator in accordance with Eq. (9). We need to show (i) that is the density matrix of a pure state, and (ii) that this state is MUB-balanced. We prove these statements in the next two sections.
Before we get into the proofs, it may be helpful to gather at this point a few algebraic facts that we will use in the following sections.(49) These first two facts apply to any odd with prime:
For all , .
. (See Lidl and Niederreiter(49), p. 230.)
In addition, when is equivalent to 3 (mod 4), we have the following:
The field element is not the square of any element. That is, .
It follows that multiplying by changes the sign of .
For , . (See Lidl and Niederreiter(49), p. 218.) Note that when (mod 4), the exponent in must be odd. So this sum is purely imaginary.
Iii Proof that is the Wigner function of a pure state
To show that is a pure-state density matrix, it is sufficient to show that and that . The latter condition will be true if the sum of over all phase-space points is equal to 1 (as follows from Eq. (9) and the fact that has unit trace). Let us evaluate this sum for as given in Eq. (16):
Here we have used the fact that is the square of a single sum. Now, for (mod 4), the sum over in Eq. (18) is equal to . Thus when we square this sum we get simply . This leaves the sum over , that is, , which is equal to . We therefore have
So the Hermitian matrix represented by does indeed have unit trace.
Showing that requires more work. We prove it by proving the equivalent statement for the Wigner function, based on Eq. (11). That is, we will show that
Let be the sum on the right-hand side of Eq. (20). We want to show that . We will do the sum by breaking it into parts. The Wigner function of Eq. (16) has three terms inside the square bracket; let us call them , , and —that is, , , and is the sum over . We define the following functions that arise from these terms when we do the operations in Eq. (20):
In terms of the ’s, the desired sum is
For the cross terms, we get twice the real part because interchanging with has the effect of complex conjugating . (It will turn out, though, that each is already real.) For four of the terms in Eq. (22) the evaluation is straightforward and we simply present the results here:
We now go through the details for the other two terms, and .
Here the two delta functions have the effect of setting and equal to zero; so the remaining sum is
By completing the squares and shifting the summation variables and , one can write this as
The last step can be justified by changing the summation variable to . Then the argument of becomes . But . So .
These sums can be done by completing squares. One has to distinguish two cases: (i) , and (ii) . In the first case, the result comes out to be
And in the second case, one gets
We now plug these expressions back into Eq. (26). Let us write , where the first part includes all the terms with , and the second includes all the rest. In the sum over and , there are terms with , and they all have the same value. So we have
The remaining part is
For brevity, we now use the symbol for the combination . Fortunately, the value of is the same as the value of :
so that the ratio is a perfect square. We can therefore write
Now, for a given value of , how many allowed pairs yield that value of ? For the special case , there are such pairs, namely, all those for which . For any other value of , there are such pairs. To see this, solve for in terms of and :
If and , then there is exactly one allowed value of that gives the desired . If or , there is no such value. So the number of terms is .
We can therefore rewrite Eq. (33) as
The whole term can thus be written as
Putting the pieces together
Iv Proof that the state is MUB-balanced
We now want to show that when the Wigner function of Eq. (16) is summed over the lines of any striation, we always get the same list of values up to permutations. This will show that the state represented by is MUB-balanced. For this proof we need only two facts about : (i) is of the form , and (ii) the function has the property that . To see that the latter property holds, note that in Eq. (16), we can cancel a factor of in the exponent of by changing the summation variable to .
Let us first consider a striation with a slope that is not infinity. Let the lines of the striation be defined by the linear equations
where the “vertical displacement” can take any value in . The lines of the striation are distinguished from each other only by the value of . Let be the line in this striation with vertical displacement . We write our special Wigner function simply as
When we sum this function over , we get
By completing the square and shifting the summation variable , we can bring this expression to the form
If is equal to 1—that is, if for some nonzero —then we can define a new summation variable , so that
Now as ranges over the values in , the term takes the value zero once, and it takes each nonzero value that is a perfect square exactly twice. Thus every value of for which yields the same list of probabilities (up to permutations). On the other hand, if , we can use the symmetry of the function to rewrite Eq. (41) as
But now is equal to 1, and so we get the same set of values as before. Thus for every value of other than infinity, we get the same set of probabilities of the outcomes of the corresponding measurement.
It is not hard to check that also yields the same set of values. In that case, let the lines of the striation be defined by the equations with . Summing over a line simply means summing over . Then
which is the same as Eq. (41) with . So yields the same probability values.
We thus see that summing our special Wigner function over the lines of any striation yields the same list of probabilities up to permutation. So represents a MUB-balanced state.
which is zero because the summand is odd in .
It is worth pointing out an interesting difference between the continuous phase space and our discrete phase space with (mod 4). In the former case we can define a circle to be the set of solutions to an equation of the form , where is some nonzero real constant. Of course cannot be just any nonzero real constant if the equation is to have a solution: it must be positive. In our discrete phase space, we can again define a circle to be the set of solutions to an equation of the form (with arithmetic in ), but now any nonzero value of allows a solution, and there are two kinds of circle: those for which , and those for which . (Note that as an alternative to the polynomial in our definition of “circle,” we could, with just as much justification, use any other non-factorable homogeneous polynomial of degree 2 in and . There is no natural notion of “distance” in . However, the polynomial is convenient because when is equal to 3 (mod 4) it is guaranteed to be non-factorable.) To prove that a state is MUB-balanced, it is not enough to know that its Wigner function is constant on every circle (as, in the continuous case, an energy eigenstate of a harmonic oscillator is constant on every circle). That is, it is not enough that depends on and only through the combination . In the above argument we also needed the fact that takes the same value on the circle as it does on the circle . That is, for each circle of one kind, there needed to be a circle of the other kind with a matching value of the Wigner function.
Now that we have identified one MUB-balanced state for each of our values of , it is not hard to generate others. Let be a unit-determinant matrix with entries in . Then takes each phase space point into a phase space point according to
Now let the Wigner function be defined by
Then also represents a MUB-balanced state, as we now show. First, that represents a pure state is guaranteed by a correspondence between unit-determinant linear transformations on phase space (for odd prime-power ) and certain unitary operators.(50); (45); (32) In effect, we are simply performing a unitary transformation on our original MUB-balanced state; so the result is certainly a pure state. Second, the linear transformation preserves lines in phase space and preserves the notion of parallel lines. So the new Wigner function can be pictured as a permutation in phase space of the values of the original Wigner function, but it is a permutation that respects the striation structure. Thus the list of probabilities arising from summing over any set of parallel lines matches those arising from summing over a (possibly different) set of parallel lines. It follows that is MUB-balanced if is MUB-balanced. This use of linear transformations is analogous to a squeezing operation on the continuous phase space.
In a similar way, we can generate yet more MUB-balanced states through translations of the phase space, which are likewise associated with unitary transformations on the Hilbert space.(32); (42); (43) (The continuous analog would be a displacement in the continuous phase space.) Starting from a single MUB-balanced state, the collection of states generated by applying to that state all possible phase-space translations defines a non-orthogonal measurement (that is, a positive-operator-valued measure), each of whose outcomes can be identified with a MUB-balanced state. This measurement thus bears essentially the same relation to each of the mutually unbiased bases.
Note that all of the transformations we have mentioned here leave the set of probability values unchanged. We have found no set of MUB-balanced states that would be analogous to a set of distinct energy eigenstates of a harmonic oscillator, whose probability distributions would also be quite distinct. It is conceivable that the set of probabilities associated with the special state defined in Eq. (16) is the only set of probability values that can arise from a MUB-balanced state in dimension .
V The density matrix and the state vector
Eq. (9) tells us how to construct the density matrix corresponding to a given Wigner function. For our special Wigner function specified in Eq. (16), this formula gives us the components of , that is, , where is the standard basis.
The sum over is straightforward, and the sum over can be done by completing the square in the exponent. The result is
This density matrix is entirely real: the factor is imaginary, and the terms in the sum corresponding to and are the negative complex conjugates of each other, since . We know from Section III that is of the form for some normalized state vector . This fact has also been proved directly by Evans through an argument reproduced in Katz’s paper(37) (an argument that does not involve the Wigner function). We now see that can be taken to have only real components in the standard basis.
We can obtain the vector from the above expression for . For any fixed value of , we can say that , and as long as this last vector (with fixed) is not the zero vector, we can obtain by normalizing it. (We cannot use the value in this way, because is indeed the zero vector. This follows from the fact—seen in the preceding section—that the sum of over the line is equal to zero. The measurement outcome associated with the vertical line is the one whose probability is .)
It turns out to be interesting to look at a histogram of the values of the components . We show an example in Fig. 1; the dimension in that example is (the 2500 prime) and we have plotted a histogram of the components of the larger vector . The distribution of the values of the components appears approximately semicircular. We see a similar shape for prime-power values of (but not when is a power of 3). As we mentioned in the Introduction, it has been proved by Katz that the limiting distribution is indeed semicircular(37), as long as avoids the values . (A key element in his proof is the construction of an explicit expression for not obtained simply by normalizing a column of .) From the semicircularity we can estimate the maximum magnitude of a component of . Treating the histogram as if it were a continuous distribution , with the number of components of having values in the interval between and , let us take the distribution to have a semicircular form:
The values of and can be determined by insisting (i) that the total number of components is , and (ii) that the sum of the squares of the components is 1. That is, we insist that
From these conditions we find that and . Thus we expect the largest magnitude of a component to be around . So in a histogram of the components of the rescaled vector , we expect the range of values to extend roughly from to , as indeed seems to be the case in Fig. 1. The proof by Katz shows, in fact, that the values of the components of this rescaled vector are confined to the interval .
In this paper we have defined the notion of a state that is “balanced” with respect to a complete set of mutually unbiased bases. For the complete set of MUBs constructed from the operators of Eq. (6), we have identified, for each prime-power equivalent to 3 (mod 4), one special state that is both MUB-balanced and “circularly symmetric” if circles are defined by , and we have indicated how this state can be used to generate other MUB-balanced states. For the purpose of both specifying the state and proving that it has the desired properties, we found it easiest to work directly with the state’s discrete Wigner function, rather than with its density matrix or state vector. From this Wigner function we obtained an expression for the density matrix and used it to plot a histogram of the component values of the state vector, which typically approximates a semicircle.
As we mentioned in the Introduction, a number of previous papers have addressed the existence of minimum uncertainty states(27); (30); (31); (32); (33), which are likewise defined relative to a complete set of MUBs and which are closely related to our MUB-balanced states. Given a complete set of MUBs , every pure state satisfies the inequality(51)
where and is the Rényi entropy of order 2:
A state is called a minimum uncertainty state if equality holds in Eq. (52). This will happen whenever the sum is independent of . Evidently, then, any MUB-balanced state is automatically a minimum uncertainty state, since not only the sum but the whole set of probabilities is independent of . In unpublished notes, Appleby has proven the existence of at least one minimum uncertainty state in every odd prime power dimension .(34) However, for the case (mod 4), he has also shown that his construction does not yield a MUB-balanced state. It seems to be unknown whether such states exist in these dimensions.
As we suggested in our opening paragraph, there is a sense in which every energy eigenstate of a harmonic oscillator is like a MUB-balanced state. Its probability distribution for the observable is independent of . However, in that case the “balanced” property follows directly from the circular symmetry of the Wigner function. (The observables , like the measurements defined by our MUBs, can be associated with striations of the phase space.(3)) We have seen in Section IV that for the discrete case, the analog of circular symmetry is not a sufficient condition to guarantee MUB-balancedness. We also needed a symmetry between pairs of “circles” of the form and . This fact may partly explain why, for a given dimension , we were able to identify only a single circularly symmetric MUB-balanced state, rather than a set of MUB-balanced states analogous to all the energy eigenstates of a harmonic oscillator. The property of being MUB-balanced in finite dimensions appears to be more stringent than the analogous property in the continuous case.
We are grateful for discussions and email correspondence with Marcus Appleby. We also thank Steven Miller, Ron Evans, and Nick Katz for their interest in the semicircular distribution suggested by our numerical results and for pursuing an explanation. Research by WKW is supported in part by the Foundational Questions Institute (grant FQXi-RFP3-1350).
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