Linking Entanglement Detection and State Tomography via Quantum 2-Designs

We present an experimentally feasible and efficient method for detecting entangled states with measurements that extend naturally to a tomographically complete set. Our detection criterion is based on measurements from subsets of a quantum 2-design, e.g., mutually unbiased bases or symmetric informationally complete states, and has several advantages over standard entanglement witnesses. First, as more detectors in the measurement are applied, there is a higher chance of witnessing a larger set of entangled states, in such a way that the measurement setting converges to a complete setup for quantum state tomography. Secondly, our method is twice as effective as standard witnesses in the sense that both upper and lower bounds can be derived. Thirdly, the scheme can be readily applied to measurement-device-independent scenarios.

We present an experimentally feasible and efficient method for detecting entangled states with measurements that extend naturally to a tomographically complete set. Our detection criterion is based on measurements from subsets of a quantum 2-design, e.g., mutually unbiased bases or symmetric informationally complete states, and has several advantages over standard entanglement witnesses. First, as more detectors in the measurement are applied, there is a higher chance of witnessing a larger set of entangled states, in such a way that the measurement setting converges to a complete setup for quantum state tomography. Secondly, our method is twice as effective as standard witnesses in the sense that both upper and lower bounds can be derived. Thirdly, the scheme can be readily applied to measurement-device-independent scenarios. For quantum information applications, it is often more interesting to learn if multipartite quantum states are entangled than to identify quantum states themselves, e.g., [1][2][3]. This is in fact what direct detection of entanglement executes, which aims to find if quantum states are entangled even before identifying quantum states. Entanglement witnesses (EWs) that work with individual measurements followed by post-processing of the outcomes [4] provide an experimentally feasible approach for this purpose in general [5,6]. Entanglement detection under less assumptions, for instance, when detectors are not trusted [7][8][9] or dimensions are unknown [10], is of practical significance for cryptographic applications.
For the practical usefulness of entanglement detection, it is worth exploring the experimental resources. If a priori information about a quantum state is given, a set of EWs may be constructed accordingly and exploited for entanglement detection. With no a priori information multiple EWs may be required. One possible method is quantum state tomography (QST) which verifies a ddimensional quantum state with O(d 2 ) measurements. Then, theoretical tools such as positive maps [11], e.g. partial transpose, or numerical tests involving semidefinite programming [12][13][14] can be applied. For EWs, however, little is known about the minimal measurements for their realization. In fact, it may happen that repeating experiments for multiple EWs may be less cost effective than QST [15], and quite possible that no useful information is obtained, neither for entanglement detection nor for quantum state identification. This raises questions on the usefulness of EWs, in particular when a priori information about a particular state is not available.
A useful experimental setup for entanglement detection may distinguish the largest collection of entangled states with as few measurements as possible. It is note-worthy that a tomographically complete measurement can ultimately identify a quantum state so that theoretical tools may completely determine whether it is entangled or separable. From a practical point of view, it would be therefore highly desirable that measurements for entanglement detection are constructive, i.e., they can be extended to a tomographically complete set by augmenting more detectors.
In this work we establish a feasible and practical framework of entanglement detection by applying a subset of measurements taken from a quantum 2-design, namely mutually unbiased bases (MUBs) [16] and symmetric informationally complete states (SICs) [17]. The connections between entanglement detection, MUBs, and quantum 2-designs have first been explored in Refs. [18,19], and subsequent results were found in, e.g. [20][21][22]. Let us emphasize here that the detection via MUBs is in some cases more powerful than the Peres-Horodecki criterion since also bound entangled states, those mixed entangled states from which no entanglement can be distilled, are detected. Furthermore, measurement setups with MUBs are very experimentally friendly, indeed the MUB criterion [18] resulted in the first experimental demonstration of bipartite bound entanglement [23,24], predicted in 1998 [25]. Here we present a unifying approach to these connections with a three-fold advantage. First, by using incomplete sets of MUBs and SICs, the entanglement detection scheme then extends naturally to an optimal reconstruction of the quantum state [26,27]: once direct detection of entanglement fails, additional detectors are applied in the measurement scheme to distinguish a larger set of entangled states, and can be ultimately utilised to find its separability via state tomography. This demonstrates in a natural framework that larger sets of detectors are more useful for distinguishing entan-gled states. Next, our results have twice the efficiency of standard EWs, in the sense that both a lower and upper bound for separable states exist, whereas EWs have only the zero-valued lower bound. Finally, the scheme can be readily applied to a measurement-device-independent (MDI) scenario for which the assumptions on the detectors are relaxed. This can be achieved by converting the measurement into the preparation of a quantum 2-design.
Let us begin with a brief summary on the implementation of EWs in practice. EWs correspond to observables that have non-negative expectation values for all separable states as well as negative values for some entangled states. They can be factorized into local observables in general, which are then decomposed by positiveoperator-valued-measure (POVM) elements. A witness W can be written with POVMs denoted by {M = I X where I X denotes the identity operator on system X, as with constants {c i }. In implementation, a POVM element can be realized by projective measurements with ancillary systems, see e.g., [28]. Although the factorization with local measurements in Eq. (1) is not necessary to realize EWs, it provides a natural framework for converting standard EWs to the MDI scenario that closes all loopholes arising from detectors. In such a scenario two parties Alice and Bob, who want to learn if an unknown quantum state ρ AB is entangled, prepare a set of quantum states, after which a measurement is performed by untrusted parties. A standard witness in Eq. (1) can be used to construct an MDI-EW as follows, where the transpose is performed in a chosen basis of H Y for Y = A, B [8]. The separable decomposition in Eq. (2) shows which quantum states the two parties must prepare, { M ] correspond to the quantum states. Let us reiterate that EWs with local measurements in Eq. (1) are readily converted to their counterparts in an MDI scenario, where entangled states are detected with less assumptions. We also note that, to the best of our knowledge, there is no general and systematic way of finding the factorization with a minimal number of local measurements. The decomposition with a minimal number of POVMs is essential, as mentioned, to take the advantage of EWs which can detect entangled states without QST.
We now introduce a particular set of POVMs called a quantum 2-design. A set of quantum states {|ψ i } k in a d-dimensional Hilbert space, |ψ i ∈ H d , or their corresponding rank-one operators, is called a quantum 2-design if the average value of any second order polynomial over the set {|ψ i } k is equal to the average f (ψ) over all normalized states given a suitable measure, such as the Haar measure. This holds true if and only if the average of |ψ i ψ i | ⊗2 over the entire 2-design is proportional to the symmetric projection onto H d ⊗ H d . A complete set of (d + 1) MUBs, and a SIC-POVM containing d 2 elements, are both quantum 2-designs. In fact, the existence of (d + 1) MUBs and d 2 SIC states in all dimensions have been long-standing open problems in quantum information theory [29,30]. For instance, complete sets of MUBs are known to exist in prime-power dimensions [26,[31][32][33][34][35][36] but have not been found in in any other composite dimension. For example, when d = 6, it is conjectured that only 3 MUBs exist [37][38][39][40][41], but no proof exists. While it is conjectured that a SIC-POVM exists for any d, the largest dimension for which an example has been found is d = 323 [42].
k=1 denote a SIC-POVM in the same Hilbert space. The two sets satisfy the equations respectively, for all k = l. It is well known that a full set of (d + 1) MUBs and a SIC-POVM are tomographically complete: measurements from either set determine a quantum state uniquely. Furthermore, the sets are both optimal and simple for QST, in that they minimize the error of the estimated statistics while at the same time having exceptionally simple state reconstruction formulas [26,27]. Note that both MUBs and SIC-POVMs are experimentally feasible, and have been implemented for the purpose of QST. A recent demonstration has been given in [43]. We now consider tomographically incomplete sets of MUBs and SICs for detecting entangled states. We denote by I where S m denotes a collection of m states out of d 2 SICs, and Pr(α, β|A, B) the probability of obtaining outcome (α, β) given a measurement in A and B. To be explicit, for state ρ, and Pr(j, j|S m , S m ) = tr[|s j s j | ⊗ |s j s j | ρ] [44]. These probabilities can be obtained simply by preparing local measurements in MUBs or SICs. Note that we have Our strategy for detecting entangled states via MUBs and SICs is illustrated, where X = M, S and n = m, m, see inequalities in Eqs. (9) and (12) satisfied by all separable states. Violation of the bounds implies detection of entangled states. Once the measurement outcomes are collected, they are exploited twice to find if the upper or lower bound is violated, in which case entangled states are detected. m ≤ d + 1 and m ≤ d 2 , where the equality corresponds to cases that the measurement setting is tomographically complete. Then, from the measurements one can construct the quantum state for which one can apply all theoretically known criteria to detect entanglement.
Since the set of all separable states forms a convex set, the quantities I where the optimisation is taken over all separable states σ sep and all possible collections of m MUBs, {B k } m k=1 , that exist in dimension d. It is clear that L m,d , and the gap between the bounds is due to different sets of m MUBs having different overlaps with the set of separable states.
Unfortunately, we do not find a systematic and general method of obtaining these bounds but had to consider all possible sets of m MUBs minimizing I In Ref. [18], it has been shown that the upper bound does not depend on selections of MUBs, and is given by In our first main result, using Table I and Eq. (8), we can construct the inequalities with optimization over m MUBs in Eq. (4) as that are satisfied by all separable states in H d ⊗ H d . A quantum state must be entangled if it violates one of the inequalities above, see also Fig. 1. It is also worth mentioning that these inequalities detect bound entangled states when m = d + 1, as shown in [23,24]. In a similar way, lower and upper bounds for SICs are denoted as follows, with g = ±, and opt + = max and opt − = min, Lower Bounds Upper Bounds   In contrast to MUBs, we find that U where S m is a set of m SICs. Then, the full set of SICs is denoted by S d 2 . Again, we do not find a systematic and general method of computing upper and lower bounds. However, having explored all possible subsets of SICs in d = 2, 3, for a given SIC-POVM, we present these bounds in Table II. Suboptimal bounds for d = 4 are also presented in the Appendix. We observe that U i.e., differences in the subsets of SICs give rise to the gap between these upper bounds. Therefore, the inequalities which are satisfied by all separable states are constructed in our second main result as   [18] and [21], respectively. Lower bounds are shown in Ref. [19] and later in Ref. [22]. As mentioned earlier, when the full measurement set of a quantum 2-design is used, it is more efficient to exploit the measurements for QST, and use theoretical tools to solve the separability problem that is known to be N P -hard.
To illustrate the effectiveness of the inequalities in Eqs.   (12), consider the isotropic and Werner states, where Π sym and Π asym denote the normalized projections onto the symmetric and anti-symmetric subspaces, respectively, and 1 d = 1/d, the normalized identity operator in dimension d. It is known that ρ W is entangled iff p < 1/2 and ρ iso iff q > (d + 1) −1 . In Fig. 2, the capability of entanglement detection with I (M) m,3 is shown for m = 2, 3, 4. The capability of entanglement detection via SICs is given in the Appendix.
Due to the linearity of Eqs. (4) and (5), with respect to the state ρ, one may expect that the inequalities in Eq. (13) are closely connected to standard EWs. Here we point out the equivalence between the lower bounds in Eq. (13) and the partial transpose criterion, by considering the so-called structural physical approximation (SPA) [1]. For recent reviews on the SPA see [2,3], as well as the Appendix for further details. The Choi-Jamiolkowski (CJ) operator for the transpose map corresponds to an EW, denoted by W , i.e., tr[σ sep W ] ≥ 0, and tr[ρW ] < 0 for some entangled states ρ which include the entangled Werner states in Eq. (14). By applying the SPA to the transpose map, the resulting CJ operator denoted by W is given by W = Π sym . The condition tr[σ sep W ] ≥ 0 then translates to tr[σ sep W ] ≥ [d(d + 1)] −1 , see Ref. [19], which is equivalent to the lower bounds in Eq. (13).
Finally, we can see that I |s j s j | ⊗ |s j s j |.
As described in Eq. (2) To conclude, let us recall the problem addressed at the outset. How do we learn efficiently if an unknown quantum state is entangled, with a measurement that is tomographically incomplete? We also assume that, for practical purposes, the setup is constructive in that it can be easily extended to that of QST. While EWs are useful for direct detection of entanglement, it is highly non-trivial to compare and connect their measurements to those which are useful for QST. However, this is a crucial requirement when experimentalists decide whether to perform direct detection of entanglement or ultimately add more detectors to identify the separability problem via state reconstruction. Our results achieve this objective with a measurement setup which can detect entangled states with cost effective measurements, and which extend naturally to the tomographically complete setup of a quantum 2-design which allows for optimal state reconstruction. Furthermore, they offer double the efficiency of standard and non-linear EWs by providing both upper and lower bounds. One consequence of our analysis is that certain sets of MUBs are more 'useful' for entanglement detection than others. For instance, in dimension d = 4, the set of 3 MUBs which extends to a complete set provides the minimal (weakest) lower bound and therefore detects a smaller set of entangled states than unextendible MUBs. Thus, one might expect that unextendible MUBs are more useful in other dimensions too. We also note that the results can be generalized to weighted 2-designs [45], which would allow for entanglement detection and QST in dimensions where the existence of MUBs and SICs is not yet known.
We envisage directions in entanglement detection beyond standard EWs and towards related problems in quantum information theory. While we have already shown some links between standard EWs and the MUBinequality (9) and the SIC-inequality (12), we expect further connections to also hold true. For example, recently it has been shown that MUBs can be used to construct positive but not completely positive maps, which lead to a class of EWs [46]. Further relations in this direction may reveal additional capabilities of EWs at an even deeper level. It would also be interesting to consider nonlinearity, e.g., in Ref. [47], to improve the inequalities.
We also hope that the presented framework of entanglement detection may offer insightful hints towards a solution of the existence problem for MUBs and SICs from an entanglement perspective [29,30]. In addition, MUBs and SICs have quite recently been generalized by relaxing the rank-1 condition to so-called mutually unbiased measurements (MUMs) and symmetric informationally complete measurements (SIMs), which exist in all finite dimensions [48,49]. Both MUMs and SIMs, as well as other similar measurements, could be applied to our framework in similar ways, leading to more experimentally feasible entanglement detection methods in arbitrary dimensions.  In these appendices we review known results on quantum 2-designs, mutually unbiased bases (MUBs), symmetric informationally complete measurements (SICs), and entanglement witnesses (EWs). The main results are presented, including a derivation of the lower and upper bounds for inequalities which detect entangled states via collections of MUBs and SICs. We analyse the capability of our criterion, and show that as we apply more measurements, i.e., as the number of MUBs and SICs increase, the criterion detects larger sets of entangled states. When we apply a quantum 2-design, i.e., a full set of (d + 1) MUBs or d 2 SICs, the inequalities provide a necessary and sufficient condition for the separability of a certain class of quantum states, namely the symmetric states. We also show for quantum 2-designs how our detection criterion is related to EWs.
Let us begin with a discussion on quantum 2-designs, also known as complex projective 2-designs, and two well known examples, a complete set of (d + 1) MUBs and a SIC-POVM consisting of d 2 elements. An ensemble of n normalized d-dimensional vectors D = {|ψ k } ⊆ C d is a quantum 2-design if the average value of any second order polynomial f (ψ) over the set D is identical to the average of f (ψ) over the unitarily invariant Haar distribution of unit vectors |ψ ∈ C d . To be precise, f (ψ) is a homogenous polynomial of degree two in the coefficients of |ψ and of degree two in the complex conjugates of these coefficients. In other words, D is a quantum 2-design if it has the first two moments equal to those of the Haar distribution. It can be shown that such an ensemble of vectors is a quantum 2-design if and only if where Π sym is the projector onto the symmetric subspace of C d ⊗ C d . We write the symmetric and anti-symmetric projectors, respectively, where 1 d denotes the identity operator in d-dimensional Hilbert space, and Π corresponds to the permutation operator in B(C d ⊗ C d ). Note the useful relation that Π Γ = d|Φ + Φ + |, with Γ the partial transpose and |Φ |ii the maximally entangled state.
Well known examples of quantum 2-designs are complete sets of (d + 1) MUBs and a SIC-POVM.
where Π sym denotes the normalized projection onto the symmetric subspace, Note that the existence of a complete set of MUBs and a SIC-POVM has been a long-standing open problem in quantum information theory and is related to several other unsolved problems in mathematics such as orthogonal decompositions of Lie algebras. It is conjectured that there exist (d + 1) MUBs if and only if the dimension d is a prime-power, while a set of d 2 SICs is conjectured to exist for all d [50]. So far, it is known that complete sets of MUBs exist in all prime-power dimensions [26,[31][32][33][34][35][36], while only significantly smaller sets have been found in other composite dimensions. In particular, for dimension d = 6, numerical calculations suggest that there exist only 3 MUBs [37][38][39]. On the other hand, it is known that a SIC-POVM exists in all dimensions d ≤ 323 [42,51]. Let us now consider incomplete sets of MUBs and SICs for entanglement detection. We will formulate the inequalities in terms of probabilities, having both upper and lower bounds, which are satisfied by all separable states. Since the structure of MUBs and SICs is not fully understood, it is a non-trivial task to derive these bounds. For instance, in certain dimensions d, different equivalence classes of MUBs exist, and the bounds can often depend on the choice of a particular class. Furthermore, the bounds do not appear to have a simple analytical expression, behaving differently as the dimension changes. In the following, we will first consider entanglement detection with measurements corresponding to MUBs, and then apply similar techniques to derive bounds for SICs. Finally, we show the relationship between quantum 2-designs and EWs.
We denote by I   [52]. Thus, the bounds above may also have a dependence on the choice of MUBs, and hence we also classify these additional bounds as follows, We note that for m = d + 1, i.e., the quantum 2-design case, the upper bound is given by U (M) d+1,d = 2 and is clearly independent of the dimension d.
We also observe that removing a single basis from the set of m MUBs decreases the upper bound uniformly by 1/d, i.e., and the bound is not influenced by which basis is subtracted from the set of MUBs. The bounds for d = 2, 3, 4 are summarized in Table III. Recall that a separable state can be decomposed by a convex combination of product states. This means that it suffices to consider the minimization over only product states, as follows, The converse, however, does not hold true in general. It therefore suffices to consider distinct equivalence classes in the optimization of Eqs.  Table IV. The case m = 3, i.e., a quantum 2-design, for which L (M) 3,2 = 1 will be shown later using a connection to EWs.
In d = 3, the complete set of four MUBs in matrix form are, m,d (σ sep ) over all normalized states |e , |f ∈ C 3 , we obtain the bounds given in Table IV. For d = 4, it is no longer true that there is a unique equivalence class of MUBs for each m, thus, we have in general, For pairs of MUBs, i.e., m = 2, there exists a oneparameter family of equivalence classes, denoted by P(x) = {B 1 , B 2 (x)}, and for triples of MUBs, i.e., m = 3, there exists a three-parameter family of equivalence classes, namely, Here, the parameters take the values x, y, z ∈ [0, π], and in matrix form, the bases can be expressed as where the columns correspond to the basis vectors [54]. Then, for x = x the two sets P(x) and P(x ) are inequivalent. Similarly, the two sets T (x, y, z) and T (x , y , z ) for ( These bounds are achieved for the triples T (π/2, 0, 0) and T (π/2, π/2, π/2), respectively. The only triple which extends to a larger set of MUBs is T (π/2, π/2, π/2). All other members of the three-parameter family are examples of unextendible MUBs. Hence, the unextendible MUBs detect more entanglement than the extendible triple since they provide tighter lower bounds. There is only one equivalence class of MUBs for each m = 4, 5, given by T (π/2, π/2, π/2) ∪ {B 4 } and T (π/2, π/2, π/2) ∪ {B 4 , B 5 }, respectively, where, Thus, since it is not necessary to optimize over collections of MUBs, we perform a minimization over product states to find L where S d 2 is a given SIC-POVM in dimension d. Note that these optimizations can only be applied when the explicit form of the SIC-POVM is known, which is not the case in large dimensions. As shown below, it turns out that I For the optimizations in Eqs. (B17) and (B18) over separable states, it suffices to consider only product states due to the convexity of the set of separable states. Hence, given a set S m of m SICs, where |e , |f ∈ C d . We have not yet found a systematic method to find these minimal and maximal bounds in general. In the following we derive the bounds for d = 2, 3 and optimize over all subsets of m SICs from a given SIC-POVM. However, for d = 4, we only find suboptimal bounds.
a. Upper bounds U (S) m,d As previously mentioned, the bounds we derive will depend explicitly on the given SIC-POVM. Here, we will only consider Heisenberg-Weyl SICs which are constructed from the Heisenberg-Weyl group, generated by the phase and cyclic shift operators (modulo d), which are defined as where ω = e 2πi/d and {|j } d−1 j=0 is the standard basis of C d . A Heisenberg-Weyl SIC can then be constructed by taking the orbit of a fiducial vector |ψ f , i.e., for a, b = 0, . . . , d − 1.
For dimension d = 2, there is a unique SIC-POVM, found in [17,50], which can be generated from the fiducial vector This SIC-POVM can be written more simply as the four vectors |s 1 = |0 , where {|0 , |1 } is the standard basis of C 2 . The states, which form a quantum 2-design, also form a tetrahedron in the Bloch sphere. It turns out that the upper bounds we calculate do not depend on which choice of m SICs from Eq. (B28) we take. In particular, for both m = 2, 3, the two bounds U  We note that it is also possible to find the vector |e in Eq. (B25) which attains these bounds. For example, given the set {|s 1 , |s 3 }, then |e max = κ(|s 1 +e πi/3 |s 3 ), where κ is a normalization factor. In fact, in all of the dimensions we investigate, given a set of m SICS, {|s j }, the vector achieving the maximum takes the form |e max = κ( j e πiλj |s j ), where the summation is taken over all SICs from the set {|s j }.
where the values L −(M) m,d are given in Table IV  which only holds true if we restrict the choice of MUBs to a specific set. When m = 3, this triple of MUBs is given by T (π/2, π/2, π/2) ∈ T (x, y, z), as defined in Eq. (B14). From an experimental perspective, this triple of MUBs is more useful for entanglement detection since it detects a larger set of entangled states than any other member of the family T (x, y, z).
The situation is more complicated for SICs. First we must specify which SIC-POVM we apply in dimension d. We then show in dimensions d = 2, 3, that for any subset of m SICs, the quantity I (S) m,d is bounded above and below, for all separable states, by where the values L as sumarized in Table VI. However, Eq. (C3) only applies for a specific set of m SICs, which we have specified explicitly in the derivations above. For d = 4 we are unable to find upper and lower bounds which apply for any subset of m SICs, however, we do find suboptimal bounds which apply for a specified set of m SICs, namely where the bounds are given in Table VII. To apply these bounds experimentally, it is required that the measurements correspond to the specified set of SICs. We will also prove later that for a complete set of (d+1) MUBs and d 2 SICs, which correspond to quantum 2designs, the bounds simplify to and We now show that the inequalities above, for I

Examples: Symmetric States
To demonstrate the observation made in Remark 1, we consider a particular class of bipartite (d×d)-dimensional quantum states, the so-called symmetric states, and analyse their behaviour with respect to our detection criterion. The first set of states we investigate are the Werner states, where p ∈ [0, 1]. Werner states are separable for p ≥ 1/2 and entangled if p < 1/2. We also consider the bipartite isotropic states which are invariant under U ⊗ U * , where we note that Pr(i, i|B k , d+1,d (ρ W (p)) = 2p. From Eq. (C5), it follows that the Werner states ρ W (p) violate the lower bound if p < 1/2, and hence the criterion coincides with the exact separability conditions. We also remark that the upper bound from Eq. (C5) has been used to detect bound entangled states [23,24].
In Fig. 2, the inequalities of Eq. (C1) for I · · · · · · p 6 p 9 q 9 q 3 For SICs, it is also straightforward to compute the following quantities, Then, the detection scheme with local measurements is given by the relation, tr[W ρ] = a,b,x,y c x,y a,b Pr(a, b|x, y) , where Pr(a, b|x, y) = tr[M x a ⊗ M y b ρ] and the parameters {c x,y a,b } can be found from the witness W . We now introduce an equivalent scheme for detecting entangled states by modifying an EW as follows. Let X ∈ B(H ⊗ H) denote a non-negative, full-rank and unittrace operator. Then, for a witness W , and an operator X, we define the following transformation, with parameter 0 ≤ p ≤ 1. Note that we have the relation, W = W X (p = 0). Since X is non-negative and of full-rank, it holds that for all separable states σ sep , it sufficed to consider product states since mixing does not decrease the norm of the above quantity. Inequalities satisfied by separable states can therefore be constructed from Eq. (D6) as follows. Assume that W X (p) has a separable decomposition, for some fixed p, and let P (a, b, |x, y) = tr[M x a ⊗ M y b ρ], for a given state ρ. Then it follows directly from Eq. (D6) that the inequality x,y,a,b c x,y a,b P (a, b|x, y) ≥ p m s (X) , is satisfied for all separable states. A violation of the inequality leads to the conclusion that the given quantum state ρ is entangled. where |Φ + = d i=1 |ii / √ d, is the maximally entangled state in C d ⊗C d . Note that W = |Φ + Φ + | Γ , where Γ denotes the partial transpose, corresponds to the permutation operator Π, i.e., W = d −1 (Π sym −Π asym ) where Π sym and Π asym denote the projectors onto the symmetric and antisymmetric subspaces. We also fix p * = d(d + 1) −1 so that W X0 (p * ) is non-negative. This is called the structural physical approximation (SPA) of the witness [ |s j s j | ⊗ |s j s j | .
Since the left-hand-side of Eq. (D11) can be decomposed in terms of SICs, the inequality can be rewritten as, Next, let us consider the case when the quantum 2design in Eq. (D10) is decomposed using a set of (d + 1) MUBs, i.e., The left-hand-side of Eq. (D11) is then written in terms of a set of MUBs, so that  [18,21]. For the upper