Realization of mutually unbiased bases for a qubit with only one wave plate: Theory and experiment

We consider the problem of implementing mutually unbiased bases (MUB) for a polarization qubit with only one wave plate, the minimum number of wave plates. We show that one wave plate is sufficient to realize two MUB as long as its phase shift (modulo $360^\circ$) ranges between $45^\circ$ and $315^\circ$. {It can realize} three MUB (a complete set of MUB for a qubit) if the phase shift of the wave plate is within $[111.5^\circ, 141.7^\circ]$ or its symmetric range with respect to 180$^\circ$. The systematic error of the realized MUB using a third-wave plate (TWP) with $120^\circ$ phase is calculated to be a half of that using the combination of a quarter-wave plate (QWP) and a half-wave plate (HWP). As experimental applications, TWPs are used in single-qubit and two-qubit quantum state tomography experiments and the results show a systematic error reduction by $50\%$. This technique not only saves one wave plate but also reduces the systematic error, which can be applied to quantum state tomography and other applications involving MUB. The proposed TWP may become a useful instrument in optical experiments, replacing multiple elements like QWP and HWP.


Introduction
Measurement bases with a special geometry structure in Hilbert space, such as 2-designs [1,2], weighted 2-designs [3], tight frames [4] and Platonic solids [5], have attracted wide attention in the community of quantum information science in recent years. A typical example of 2designs is mutually unbiased bases (MUB) [6,7]. The special geometry for MUB is the equal Hilbert-Schmidt overlap between any two projective operators corresponding to two MUB.
A complete set of MUB in a multi-qubit system involves nonlocal measurements that are difficult to be realized in the laboratory. In many scenarios such as quantum state tomography [18][19][20][21], the tensor product of a complete set of single-qubit MUB is a preferred choice as the measurement bases for multi-qubit systems [5,17]. Therefore, it is important to implement MUB for a single qubit in experiments [22][23][24].
In the context of polarization optics, a specific type of polarization transformation poses constraints on the minimum number of optical elements that must be used. Three wave plates are the minimum number to realize any SU(2) polarization transformations [25] and the visual tool kit can be found in [26]. As for transformations between a given pair of nonorthogonal polarization states, two QWPs are enough (see a sketched proof in [27] and a recent constructive demonstration by Zela in [28]). When the initial state is linearly polarized, a quarter-wave plate (QWP) and a half-wave plate (HWP) can realize any polarization state [27]. The combination of QWP and HWP is used in most of current optical experiments [29] where the linear polarization is generated by a polarizing beam splitter (PBS). In the setting composed of a QWP, a HWP and a PBS, the PBS acts as σ z and by adjusting the optic axis angles of the QWP and HWP (a twoparameter setting), the Bloch vector of σ z can be unitarily rotated to any point on the unit Bloch sphere [27]. However, when the problem is restricted to realize MUB rather than arbitrary bases, is one wave plate sufficient to transform the initial σ z to a set of MUB, especially a complete set of three MUB? If the answer is positive, not only one wave plate is saved but also the systematic error [29,30] can be expected to decrease as the parameter uncertainties of the devices (i.e., wave plates in our case) are the sources of systematic error. Intuitively, the less measurement devices, the smaller the systematic error. So far, people know how to construct two MUB with one HWP or one QWP. For example, one HWP itself can be used to realize σ z and σ x by setting its rotation angles as 0 • and 22.5 • ; one QWP can be used to realize σ z and σ y by setting its rotation angles as 0 • and 45 • . One wave plate is also used to perform Fourier transform tomography [31] and the optimal phase of the wave plate is numerically calculated in [32]. As to a complete set of three MUB, this problem is barely considered. In this paper, we show that it is indeed feasible to employ only one wave plate to realize a complete set of single-qubit MUB with reduced systematic error.
Here is the organization for the rest of the paper. In section 2, the conditions to realize two MUB and three MUB using only one wave plate are considered, respectively. Section 3 calculates the systematic error in the realization of MUB with only one wave plate. In section 4, third-wave plates are used to perform MUB measurements in both single qubit and twoqubit tomography experiments and the results demonstrate an error reduction by 50%. Section 5 concludes the paper.

Transforming polarization states with QWP-HWP setting
In optical experiments, arbitrary projective measurements on polarization qubits are often implemented by a QWP-HWP setting; see Fig. 1(a). Define |H = (1, 0) T , |V = (0, 1) T and their corresponding eigenvalues are ±1, where T denotes transpose. Then the PBS acts as σ z or (0, 0, 1) T in the Bloch representation. This setup transforms |H to where q, h are, respectively, the optic axis angles of QWP and HWP deviated from the horizontal direction, and U is a unitary transformation operator on polarization by a phase plate with phase δ and rotation angle θ , Or equivalently this setup transforms the original operator σ z to the Pauli operator r r r(q, h) · σ σ σ with r r r(q, h) = (sin 2q cos(4h − 2q), sin(4h − 2q), cos2q cos(4h − 2q)) T and σ σ σ = (σ x , σ y , σ z ), where σ x , σ y and σ z are three Pauli operators. Two variables of rotation angles q and h correspond to a plane and can rotate the initial Bloch vector (0, 0, 1) T to any direction or position on the unit Bloch sphere. Thus, this QWP-HWP setting is used to realize any one qubit projective measurement basis. In order to realize MUB for a qubit, only two or three specific Bloch vectors need to be implemented and it is not necessary to cover the Bloch sphere with two rotation angles of two wave plates. We will show below that one wave plate with appropriate phases is sufficient to realize single-qubit MUB.

Realizing two mutually unbiased bases with one wave plate
Two sets of orthogonal bases Hilbert space are mutually unbiased if | ψ 1 i |ψ 2 j | 2 = 1 d for any i and j. There are at most d + 1 bases with every two of them mutually unbiased, which are called a complete set of MUB. So far, how to construct a complete set of MUB is known only in systems with dimensions which are powers of primes [6,7]. For general cases, the existence of such a set is still an open problem even in the simple case of d = 6 [6,33]. For a qubit with dimension d = 2, the bases {|ψ are the eigenvectors of a unit Pauli operator r r r j · σ σ σ with r r r j = 1. In the Bloch representation, the jth basis is directly related to its Bloch vector r r r j as |ψ j i ψ j i | = I±r r r j ·σ σ σ 2 and ± corresponds to i = 1, 2. Thus, the requirement for two bases to be mutually unbiased is that their Bloch vectors are normal to each other r r r 1 · r r r 2 = 0.

Realizing a complete set of MUB with one wave plate
Three rotation angles θ i , i = 1, 2, 3, are chosen to realize three MUB, which consist of a complete set of MUB for a qubit, r r r(θ i ) · r r r(θ j ) = 0, where i, j = 1, 2, 3 and i = j. For a complete set of MUB, it is much more complicated and difficult to obtain theoretical solutions. Hence, the problem is investigated numerically in this paper. Borrowed from frame theory [2,34], the frame potential of a wave plate is defined as The solution of Eq. (6) exists only if the phase δ of the wave plate makes the frame potential Eq. (7) vanishing. The frame potential is numerically computed by a MATLAB solver lsqnonlin, which is intended to solve nonlinear least-squares problems. One hundred different initial points of θ 1 , θ 2 and θ 3 are taken for us to avoid local minimum and find all the possible solutions. The numerical result of the frame potential for different phases is shown in Fig. 2. From Eq. (5), Φ(δ ) has a period of 360 • . Thus, we only consider 0 With δ between 111.5 • and 141.7 • , the frame potential is zero and the solutions of Eq. (6) are shown in Fig. 3. From Eq. (5), r r r(θ ) = r r r(θ + 180 • ), we only consider rotation angles within Fig. 3) is the solution, the modules of 90 • ± θ i and 180 • − θ i (i = 1, 2, 3) by 180 • (represented as blue, green and black in Fig. 3 correspondingly) are also the solutions. Thus, all the four sets of MUB are considered as one class of solutions.
As shown in Fig. 3, solutions from different sets of MUB in the same class intersect around δ = 120 • , 126.3 • and 141.7 • . Using the symmetries represented by the colors, the four variables (i.e. δ , θ 1 , θ 2 and θ 3 ) reduce to two and we can theoretically calculate the solutions at these intersections. The phases at these intersections are also rigourously found to be δ = 120 • , 126.32 • and 141.76 • (see Appendix A.1 and A.2). Here we compare our results with those in [32]. The optimal phase was numerically calculated in [32] to be 7π/10 (i.e. 126 • ). The figure of merit times the total number of counts in [32] at this optimal phase equals 10.03, which is very close to 10, the bound achieved by MUB. This phase falls within our calculated range [111.5 • , 141.7 • ]. The reason why they only found one phase rather than an available interval of phases and the optimal performance at this phase was slightly worse than the bound is their restriction of six equally spaced rotation angles.
In the special case of δ t = 120 • , called third-wave plate (TWP), the Bloch vector is where t is the rotation angle of the optic axis of TWP deviated from horizontal direction.

Systematic error in the realization of MUB in one wave plate setting
Imperfect measurement devices are the main sources of the systematic error in the realization of MUB. Here we consider the systematic error due to the parameter uncertainties of wave plates. The realized bases are denoted by their Bloch vectors as r r r(δ , θ ), where δ is the real phase of the wave plate in the one wave plate setting in Fig. 1(b). The systematic error in the realization of r r r is where r r r ξ = ∂ r r r ∂ ξ , ξ = δ , θ . From Eq. (5), r r r δ 2 = sin 2 2θ = (r 2 ) 2 sin 2 δ , r r r θ 2 = 16 sin 2 δ 2 − 4 sin 2 δ sin 2 2θ = 16 sin 2 δ 2 − 4(r 2 ) 2 .
For a complete set of MUB (r r r j , j = 1, 2, 3), the systematic error sums up to As r r r j is orthogonal to each other, from Eq. (10), Thus, the systematic error in the one wave plate setting is As ε 2 in Eq. (12) is an increasing function of δ in the interval [111.5 • , 141.7 • ], the minimum and maximum systematic error in the realization of three MUB with one wave plate setting is 1.16(∆δ ) 2 + 28.80(∆θ ) 2 and 2.60(∆δ ) 2 + 38.83(∆θ ) 2 , achieved at δ = 111.5 • and δ = 141.7 • . For a third-wave plate with δ t = 120 • , Under the assumption that (∆δ h ) 2 = (∆δ q ) 2 = (∆δ t ) 2 = (∆δ ) 2 and (∆h) 2 = (∆q) 2 = (∆t) 2 = (∆θ ) 2 , the systematic error in the realization of MUB is 1.33(∆δ ) 2 + 32(∆θ ) 2 in the TWP setting and averaged as (2.5∆δ ) 2 + 68(∆θ ) 2 in the QWP-HWP setting in Eq. (26). Thus, the TWP setting outperforms the QWP-HWP setting by about a factor of two. Measurements based on single-qubit MUB are preferable choices in quantum state tomography. In qubit state estimation, a complete set of single-qubit MUB is used to extract information of the qubit optimally. In multi-qubit quantum state tomography, the product measurements of single-qubit MUB on each photon are used to reduce estimation error due to statistical fluctuation. When the copies of states ρ are infinite, the estimated stateρ based on the measurement data should be the same as the real state ρ. However, since single-qubit MUB are imperfectly realized, the estimated stateρ no longer converges to the real state ρ and tr(ρ − ρ) 2 is defined as the systematic error in state estimation. Generally, tr(ρ − ρ) 2 depends on ρ. Averaged over unitarily equivalent states, tr(ρ − ρ) 2 in both single-qubit and multi-qubit state estimation is proportionate to the systematic error of the realized bases [35]. As the systematic error in the realization of multi-qubit product bases is the sum over the systematic error of single MUB for each qubit, the systematic error in the realized multi-qubit product bases with one wave plate for each qubit is still a half of that with the QWP-HWP combination. This systematic error reduction effect in quantum state tomography is experimentally verified in both single-qubit and two-qubit tomography experiments in the next section.

Qubit tomography experiments
The experimental setup, shown in Fig. 4, includes two parts: state preparation and MUB measurement. A 40 mW, V-polarized beam at 404 nm from a semiconductor laser pumps a type I phase-matched β -barium borate (BBO) crystal. After the spontaneous parametric downconversion (SPDC) process, a pair of 808 nm H-polarized photons are created. One photon passes through a 3 nm interference filter and is detected by a single photon detector to herald the presence of its twin photon. The quantum state of the heralded photon is prepared by HWP0, HWP1, QWP1 and a 770λ quartz crystal which is much larger than the coherence length of about 270λ with λ = 808 nm. HWP0 with rotation angle h 0 and the quartz crystal with optic axis aligned horizontally together prepare the quantum state ρ wih Bloch vector s s s = cos 4h 0 (0, 0, 1) T ; HWP1 and QWP1 can transform s s s to arbitrary direction. This part is capable of preparing arbitrary qubit state. In the MUB measurement part, a complete set of MUB is performed with two methods: a TWP with rotation angles set as 0 • , 27.37 • and 117.37 • in Fig. 4(a); the conventional QWP-HWP setting with rotation angles set as (45 • , 22.5 • ), (0 • , 22.5 • ) and (0 • , 0 • ) in Fig. 4(b).
As phase errors are determined by the manufacture and wavelength, without a variable wavelength we can only experimentally verify the relationship between the systematic error of the estimated state and the angle errors of QWP, HWP and TWP for all these three states. In the experiment, MUB measurements are performed on 3 × 10 6 photons with two different settings in Figs. 4(a) and 4 (b). We first measure the state with the well-calibrated setting and assume the estimated state ρ as the real state. Then we intentionally mis-calibrate the optic axes of the wave plates with an angular error, and obtain an estimationρ. Thus, the systematic error due to this angular error is calculated as tr(ρ − ρ) 2 . The estimated states by the well-calibrated TWP setting have a fidelity of over 99.9% with those by the well-calibrated QWP-HWP setting for all the three states above, validating each other. In terms of systematic error, experimental results (dots) and the theoretical results (solid lines) are shown in Fig. 5, and they match very well. The performance of these two settings depend on ρ and neither always outperforms the other. For example, for states in Figs. 5(a) and 5(c), TWP beats the QWP-HWP combination while reversely for the state in Fig. 5(b). However, there are more states where TWP performs better. The total systematic error in the estimation of these three states in the TWP setting adds up to be about two times smaller than that in the QWP-HWP setting as shown in Fig. 5(d).

Two-qubit tomography experiments
In Fig. 6, a 100 mW, H-polarized beam at 404 nm from a continuous laser pumps a pair of type I phase-matched β -barium borate (BBO) crystals whose optic axes are normal to each other. After the spontaneous parametric down-conversion (SPDC) process, a pair of 808 nm photons are created. When the optic axis of half-wave plate (HWP0) at 404 nm is deviated 22.5 • from horizontal direction, the twin SPDC photons are maximally entangled. HWP1 and  HWP2 rotate H and V to the fast and slow axes of the single mode fibers. At the output ports of the fibers, HWP3 and HWP4 rotate the polarization direction back to horizontal and vertical. QWP1 is tilted to compensate the phase of the entangled states to a singlet state. In the MUB measurement part, a complete set of MUB on either photon is performed with two methods: TWP setting in Fig. 4(a) and QWP-HWP setting in Fig. 4(b).
In the two-qubit case, the singlet state ρ = |Ψ − Ψ − | is chosen for three reasons: firstly, systematic error in the estimation of product states is a direct sum of that of single-qubit states; secondly, entangled states reveal the peculiar features of quantum systems and are valuable quantum resources; the last reason is that the systematic error in the estimation of Werner states is proportionate to the sum of the systematic error in the realization of single-qubit MUB for either photon [35], which is similar to the systematic error averaged over unitarily equivalent states. The systematic error for Werner states [35] is where ρ = p |Ψ − Ψ − | + (1 − p)I/4, |Ψ − = 1 √ 2 (|HV − |V H ) and ε 2 i is the systematic error of the realized single-qubit MUB on photon i, i = 1, 2.
Similar to the qubit tomography experiment, we only experimentally measure the dependence of the systematic error on the angle errors of wave plates in the branches of both photons. Then product measurements of MUB are performed on 9 × 10 5 pairs of prepared singlet states. The estimated state by the well-calibrated TWP setting has a fidelity of over 99.8% with that by the well-calibrated QWP-HWP setting, agreeing well with each other. From Eqs. (13), (14) and (25), angle errors of wave plates for either photon theoretically contribute 8(∆t) 2 in the TWP setting and 12(∆h) 2 + 5(∆q) 2 in the QWP-HWP setting. However, numerical results in Fig. 7 (a) show that angle errors only cause 10.5(∆h) 2 + 4.6(∆q) 2 in the QWP-HWP setting and 6.9(∆t) 2 in the TWP setting. This gap is due to the positive semi-definite conditions of density matrices, which arises when states are singular. Both the experimental results and numerical results in Fig. 7 (b) show that the systematic error in the TWP setting is only about a half of that in the QWP-HWP setting. The experimental results are slightly smaller than the numerical results because the prepared state is not exactly the expected singlet state, which only has a fidelity of 98%.

Conclusion
We have found that one wave plate is sufficient to realize two MUB as long as its phase is within [45 • , 315 • ]. It is capable of realizing a complete set of MUB if the phase is within [111.5 • , 141.7 • ] or the symmetric interval about 180 • . The systematic error in the realization of MUB in one wave plate setting is calculated to be twice smaller than that in the conventional QWP-HWP setting. TWPs are applied to single-qubit and two-qubit quantum state tomography experiments and experimentally show an error reduction by 50% compared with the QWP-HWP combination. Other applications of TWP and arbitrary phase plates in the realization of any SU(2) and polarization state transformations need to be explored in the future.
The solutions of Eq.