Detecting Entanglement can be More Effective with Inequivalent Mutually Unbiased Bases

Mutually unbiased bases (MUBs) provide a standard tool in the verification of quantum states, especially when harnessing a complete set for optimal quantum state tomography. In this work, we investigate the detection of entanglement via inequivalent sets of MUBs, with a particular focus on unextendible MUBs. These are bases for which an additional unbiased basis cannot be constructed and, consequently, are unsuitable for quantum state verification. Here, we show that unextendible MUBs, as well as other inequivalent sets in higher dimensions, can be more effective in the verification of entanglement. Furthermore, we provide an efficient and systematic method to search for inequivalent MUBs and show that such sets occur regularly within the Heisenberg-Weyl MUBs, as the dimension increases. Our findings are particularly useful for experimentalists since adding optimal MUBs to an experimental setup enables a step-by-step approach to detect a larger class of entangled states.

Introduction.-Quantum entanglement is one of the key ingredients responsible for many of the recent advances in quantum technologies, however, the detection of this fundamental property, even with full knowledge of the quantum state, is in general a NP-hard problem [1]. In this contribution we exploit an experimentally feasible protocol to detect entanglement in which the subsequent addition of measurement settings detect a larger class of entangled states. The essential feature of this scheme is for the measurement settings to exhibit complementarity, namely, that the measurements form a set of mutually unbiased bases (MUBs), i.e. the overlap of any pair of vectors from different bases is constant [2]. Physically, this means that exact knowledge of the measurement outcome of one observable implies maximal uncertainty in the other.
Utilising this property, MUBs play an important role in many information processing tasks, such as quantum state tomography [3,4], quantum key distribution [5], signal processing [6], and quantum error correction [7], to name just a few. Unfortunately, the existence of maximal sets of these highly symmetric bases is a difficult unresolved question: an equivalent formulation of the problem in terms of orthogonal decompositions of the algebra sl d (C) dates back almost forty years [8,9]. Whilst it is known that maximal sets of d+ 1 MUBs in C d exist when d is a prime or prime-power, e.g. by a construction based on the Heisenberg-Weyl group [10], it is conjectured that fewer MUBs exist in all other dimensions [11,12]. The classification of subsets of MUBs is an open problem for d > 5, and a rich structure of inequivalent MUBs (up to unitary transformations) exists [13]. Of particular rele-vance to this work is the property of unextendibility, i.e. sets of MUBs which cannot be completed to a maximal set.
One striking feature of the entanglement witness, first introduced in [14], is that the value of the upper bound (which is violated by entanglement) depends only on the number of MUBs and not on the choice between inequivalent sets. For example, the experimenter is free to use extendible or unextendible MUBs. The protocol has since been applied to two photons entangled in their orbital angular momentum, and bound entangled (PPTentangled) states have been verified experimentally for the first time [15,16]. Further modifications of the witness have also been considered, e.g. with 2-designs and orthogonal rotations [17][18][19], as well as in applications to other scenarios [20,21].
Recently, entanglement witnesses were also shown to have a lower bound [22,23], which often turns out to be non-trivial, i.e. entanglement is detectable. In this contribution, we compute the lower bounds of the MUBwitness and reveal that, in contrast to the upper bound, the values depend strongly on the choice of MUBs, as well as the number of measurements. One consequence of this sensitivity, as we will see, is that unextendible MUBs can be more effective at entanglement detection than extendible ones. Furthermore, this operational distinction between MUBs reveals the existence of inequivalent sets and hence the witness provides a way to classify these sets. We use this criterion to establish inequivalences within the Heisenberg-Weyl MUBs when d = 5, 7, 9, as well as for some continuous families and unextendible sets.
This contribution is perhaps the first application of MUBs in which unextendible sets may be the preferred measurement choice over Heisenberg-Weyl subsets. In fact, most applications exhibit no preference for a particular subset of MUBs, and hence the physical differences between inequivalent sets have not been fully realised. Our results complement some recent observations on measurement incompatibility [24] and quantum random access codes (QRAC) [25,26]. In particular, inequivalent MUBs contain varying degrees of incompatibility when quantified by their noise robustness, meaning that some sets require additional noise to become jointly measurable. Furthermore, measuring different subsets of MUBs in a QRAC protocol reveals "anomalies" in the average success probability, which appear to coincide with our different lower bounds of the witness. Here, by considering entanglement witnesses, we provide a physically significant application which exploits these inequivalences.
Inequivalent and unextendible MUBs.-Formally, we say a pair of bases in For prime dimensions the standard Heisenberg-Weyl group provides the essential building blocks for the construction of a complete set of d + 1 MUBs, while for prime-powers d = p n the generalised tensor product Heisengerg-Weyl group (as described in the Appendix) is used. However, in almost all dimensions, these provide only a small subset of all possible cases. In order to classify MUBs, we need to introduce the notion of equivalence classes. Two sets of m MUBs are equivalent, if one set can be transformed into the other by a unitary or antiunitary transformation, permutations within (or of) bases, and phase factor multiplications. The task of classification is challenging, with success currently limited to d ≤ 5 [27,28]. When d = 2, 3, 5 the Heisenberg-Weyl MUBs exhaust all possibilities: no inequivalence occurs when d = 2, 3 and only two inequivalent triples appear among all possible subsets in d = 5. In contrast, d = 4 yields a one-parameter family of pairs, and a three-parameter family of triples, inequivalent to all Heisenberg-Weyl subsets [29]. In higher dimensions d ≥ 6 the situation is complicated and closely related to the (very old [30,31]) problem of searching for Hadamard matrices. In some instances, constructions of MUBs produce cases which do not extend to complete sets, and are aptly named unextendible MUBs [32,33]. In dimensions d = p n , the first examples appear when d = 4 [28] and d = 7 [34], and more generally d = p 2 [35,36].
Finally, we fix some notation. It is often convenient to represent MUBs as sets of unitary matrices, where the   [14] an experimentally friendly protocol was introduced that can verify the entanglement of a bipartite state ρ on C d ⊗ C d , by subsequently measuring a set of MUBs. Both parties, which may be locally separated, and usually called Alice and Bob, each possess the (same) set of m MUBs in C d . Applying identical projections from each basis, they calculate where P (i, i|B k , B k ) = tr(|i k i k | ⊗ |i k i k |ρ) denotes the joint probability of Alice and Bob each obtaining outcome i of the basis measurement B k , given the state ρ.
The authors of Ref. [14] showed that the correlation function M m is bounded above for all separable states by U m (see Eq. (3)). This is easily seen when we assume that the source produces the separable state |i l i l , with |i l ∈ B l .
Here, the joint probability P (i, i|B l , B l ) is obviously maximal, i.e. 1, and for any other basis choice k = l, the joint probabilities result in i P (i, i|B k , B k ) = 1 d due to the unbiasedness condition. The bound follows by exploiting the arithmetic mean and the convexity of separable states. Recently, it was shown in [22] that the very definition of an entanglement witness also yields a non-trivial second bound, which for the MUB-witness results in a lower bound. Thus, the quantity M m has two bounds, and any violation of the above inequality detects entanglement of the state ρ. The upper bound has the simple form and is independent of the choice of MUBs, which implies that only the unbiasedness property is exploited [14]. One should also note that although the upper bound is independent of the ordering of the MUBs, the results depend on the initial correlation of the considered quantum state ρ. This in turn can be compensated by Alice or Bob applying local unitaries to the state. Further note that observing entanglement in any pure quantum state via the above inequality only requires measurements in (any) two MUBs. For a complete set, the bounds are L d+1 = 1 and U d+1 = 2, which follow from the 2-design property of MUBs [17], and the witness is most effective at detecting entangled states. Detection with inequivalent MUBs.-In striking contrast with the upper bound of the function M m (ρ), the lower bound L m depends not only on the dimension d and the number of MUBs, m, but also on the choice of MUBs. In other words, the bound is highly sensitive to the particular set of MUBs chosen, and therefore inequivalent sets often yield different values of L m . In the following, we analyse this dependence to understand its effect on entanglement detection. We restrict our analysis to dimensions d < 10 to ensure that the numerical optimisations of the bounds are reliable, where we exploit the composite parameterization of unitaries introduced by [37]. Let us first formulate three main observations in the form of theorems, which we will further discuss below:  H(y, z) is a family of Hadamard matrices defined in the Appendix with x, y, z ∈ [0, π]. These cover all pairs and triples in d = 4 and coincide with the Heisenberg-Weyl case when x = y = z = π 2 (for which it extends to a complete set). The triple is unextendible for all other parameter choices. As summarized in Table I  = 1/2 is reached when x = π/2 and y = z = 0, and is achieved for any pure separable state ρ = |a, b a, b| with a|b =0. In contrast, the Heisenberg-Weyl triple saturates the lower bound only for particular separable states, e.g.
. We note that L 2 = 0 for all inequivalent pairs. Unextendible MUBs are more efficient.-Let us now study whether unextendible MUBs are more efficient at detecting entanglement (Theorems B and C ), which is suggested by the stricter lower bounds. However, this need not be the case since the witness itself is also altered and may act differently on any given state.
Let us consider a state from the magic simplex [38][39][40], e.g. ρ α,β = (1 − α − β)½/d 2 + αP 0,0 + βP 0,1 (it applies for any two Bell states), where the entanglement properties are also fully known. Here P 0,0 = 1 d d−1 s,t=0 |ss tt| denotes a Bell state and any other Bell state P k,l can be obtained by locally applying in one subsystem a unitary Wely operator W (k,l) = The results for d = 4 and α = β are summarized in Fig. 1 and show the unextendible sets are more efficient in detecting entanglement, as described in Theorems B & C.
A second example is provided by Werner states ρ W , which are invariant under any local unitary U ⊗ U [41]. The lower bound of (2) detects all entangled Werner states for a complete set of MUBs, and becomes less effective as the number of measurements is reduced. Due to U ⊗ U invariance, the quantity M m (ρ W ) is independent of the choice of MUBs, therefore the value of the bound plays a fundamental role in its effectiveness. For three MUBs, the unextendible triple with the strictest lower bound L unext 3 = 1/2 detects the largest set of entangled states.
Inequivalent subsets.-We now analyse the lower bound L m for all subsets of Heisenberg-Weyl MUBs when d = 5, . . . , 9, to search for inequivalent sets. Dimension d = 5: Here, a full classification is already known [27,28], and our search recovers all equivalence classes. All subsets of equal cardinality are equivalent except for triples, which can be grouped into one of two classes: Computing the lower bound L 3 over all 6 3 = 20 triples, we find two different bounds, as summarized in Table II. No other inequivalent sets appear from the 2 6 permutations, in agreement with previous results.
Dimension d = 7: Searching over all subsets, we find only two inequivalent quadruplets These appear by observing L Q1 = 0.1514 and L Q2 = 0.20101. There are 8 4 = 70 combinations of size four, with 42 sets achieving the first bound and 28 the second (higher bound). To explain this distribution, note that no inequivalent triples exist, i.e. any triple is equivalent to {B 1 , B 2 , B 3 }. Hence, there are only five possible extensions to a quadruplet, namely {B 1 , B 2 , B 3 , B k }, k = 4, . . . , 8. If k = 4, 6, 8/5, 7 the quadruplets are equivalent to Q 1 /Q 2 , therefore the distribution is split as above rather than evenly. We note that two inequivalent quadruplets were also found by analysing the incompatibility content of these subsets [24] and their success in a QRAC protocol [26]. This case is also interesting due to the existence of an unextendible triple, {B 1 , B 2 , A 7 }, defined in the Appendix. For this set, the lower bound L 3 = 0.0557 is smaller than the Heisenberg-Weyl bound (0.0698). Again, we have examples of the above theorems. The results are summarized in Table III  . We note that these results depend heavily on numerical optimizations. The inequivalences are in full agreement with those found in [24,26], except when d = 9, m = 3, where we detect no inequivalent triples (L 3 = 0) unlike the two cases found in [24] and the anomaly in [26]. Furthermore, we observe that L m < L m+1 does not always hold, which is of particular importance for experimental realisations.
Dimension d = 6: This is the first dimension where a maximal set has not been found, and it is conjectured four MUBs do not exist [42]. The Heisenberg-Weyl construction yields only three bases, although many inequivalent pairs and triples exist [43,44]. There is only one known unextendible pair, {B 1 , S 6 }, where S 6 is the Tao matrix [45,46] defined in the Appendix. The lower bound for any pair (extendible and unextendible) results in L 2 = 0. The lower bound for the unextendible Heisenberg-Weyl triple is L 3 = 0.1056.
Summary & Outlook.-We have studied the role that inequivalent MUBs play in the detection of entanglement, as well as providing a method to systematically distinguish inequivalent sets. The dual bounds of the witness exhibit contrasting behaviours, as the choice of MUBs plays a fundamental role in the effectiveness of the lower bound. This is a crucial observation for experimentalists who want to maximise their success in detecting entanglement.
Little is known about the physical significance of inequivalent MUBs, and our witness is perhaps the first application where unextendible MUBs are the preferred choice. This leads to questions of whether other applications exist which prioritise one set over another (as is the case for QRACs [26]), or if inequivalent MUBs have other properties responsible for their varying degrees of usefulness. One such possibility is their incompatibility content, which also distinguishes between equivalence classes of MUBs [24]. Exploring connections between incompatibility, inequivalent MUBs, and entanglement detection, such as the role incompatibility plays in the effectiveness of entanglement witnesses, may reveal new insights into these topics. Finally, we point out that our findings provide an alternative method to study the structure of the convex set of separable states and subsequently, the rich structure of entanglement.