Two-unitary complex Hadamard matrices of order 36

1 Introduction

A unitary matrix U of order d 2 is called two-unitary (2-unitary), if the partially transposed matrix U Γ and the reshuffled matrix U R are also unitary. Operations of reshuffling and partial transpose of any square-size matrix X ∈ C d 2 × d 2 , addressed by a four-index j k ; l m , are defined as

where X j k ; l m = ⟨ j k ∣ X ∣ l m ⟩ is a representation of X in a local basis { ∣ j k l m ⟩ } of four copies of the Hilbert space: ⨂ j = 1 4 C d = A ⊗ B ⊗ C ⊗ D [14]. We use Dirac notation, where ∣ a b ⟩ = ∣ a ⟩ ⊗ ∣ b ⟩ . Two-unitary matrices play a significant role in the theory of quantum information. They are related to quantum orthogonal Latin squares (OLS) [21], perfect tensors [22], and absolutely maximally entangled states [14]. In this article, we put the main focus on the last class of objects. In fact, any 2-unitary matrix U ∈ U ( d 2 ) corresponds to an absolutely maximally entangled state of four qudits (quantum systems with d degrees of freedom), written ∣ ψ ⟩ ∈ AME ( 4 , d ) . Formally, state ∣ ψ ⟩ is defined by the formula Tr Q ′ ∣ ψ ⟩ ⟨ ψ ∣ ∝ I , where Q denotes any balanced bipartition of four parties: Q ∪ Q ′ = { A , B , C , D } , Q ∩ Q ′ = ∅ . Such states contain maximal entanglement with respect to any bipartition and provide important resource for many practical applications.

It is known that 2-unitary matrices do not exist for d = 2 [18]. For larger dimensions, d ≠ 6 , 2-unitary permutation matrices are implied by OLS of size d . Recent result [1,27] concerning the particular case d = 6 shows that such objects exist for any d > 2 . The original solution of the problem for a local dimension d = 6 was shown not to be unique [2], which encourages us to search for further simplifications.

Throughout the article, if not stated otherwise, the global dimension N is always a square of the local dimension d with d 2 = N , and we mostly focus on d = 6 . By N k = { k , k + 1 , k + 2 , … } we denote the set of natural numbers starting with k , where k ∈ { 0 , 1 , 2 , … } . The three main sets of matrices we use are as follows:

1. the set of 2-unitary matrices, U 2 ( d 2 ) ,

2. complex Hadamard matrices (CHMs) H ( d 2 ) = { H ∈ U ( d 2 ) : 1 d 2 H H † = I d 2 , ∣ H j k ∣ = 1 } , and

3. its proper subclass of Butson-type CHM [9,10],

   (2) B H ( d 2 , q ) = H ∈ H ( d 2 ) : H j k = exp i 2 π m j k q ,

   for some q ⩾ 2 and m j k ∈ N 0 . Finally, let us define H 2 ( d 2 ) = H ( d 2 ) ∩ U 2 ( d 2 ) .

Here P 9 denotes a permutation matrix of order 9, which determines the AME ( 4 , 3 ) state of four subsystems with three levels each [17]. An analogous construction can be used to construct 2-unitary matrix of dimension d 2 for any dimension d for which OLS( d ) exist. Such combinatorial designs are known for any d > 3 apart from the six-dimensional Euler case [8,25].

In this article, we answer the question about existence of a 2-unitary CHM of order 36 affirmatively. This solution can be considered interesting from the quantum physics point of view, as it leads to a four-party state with a large coherence with respect to a generic locally equivalent basis. As an additional benefit, we provide not only a single representative of such a matrix, ℋ , but also a multidimensional affine family stemming from ℋ . This internal parameterization is different from the one considered in [2] as it preserves both properties of being 2-unitary and Hadamard at the same time and might serve as an independent tool in classification CHMs of square size and 2-unitary matrices.

This article is structured as follows: We start with recalling the algorithm that produces numerically 2-unitary matrices of arbitrary dimension. In Section 3, we present analytical form of a 2-unitary CHM ℋ and its properties. The full form of a family stemming from ℋ is relegated to Appendix A, due to its algebraically overcomplex form. Finally, in Section 4, we compare our result with recent work on biunimodular vectors which can also be used to construct representatives of the set H 2 ( 36 ) . Conclusions and future prospects are envisioned in Section 5.

2 Modified Sinkhorn algorithm

Let us briefly recall that originally a 2-unitary representation U ∈ U ( 36 ) of the “golden” AME ( 4 , 6 ) state was obtained numerically in [1] using the iterative procedure X t + 1 = ℳ ( X t ) for t ∈ N 0 with

where Π denotes polar decomposition projection onto the manifold of unitary matrices of order N = 36 . This hardly converging procedure starts working nicely when ℳ is supplied with very particular seeds X 0 – initial matrix points. A collection of potential seeds consists of slightly perturbed permutation matrices. For example, X 0 = P exp ( i η G ) , where G is a random matrix whose entries are drawn according to the normal distribution. Matrix P should be close^[1] to the permutation matrix P * defined as the best classical approximation of two OLS [11]. Consequently, adding a non-zero “noise” controlled by small values of η ∈ ( 0 , 1 ) guarantees the uniqueness of polar decomposition because the tiny perturbations provide an input for Π that remains a full-rank matrix after operations of reshuffling and partial transpose. Then, after several dozens of steps, e.g., t ∈ N 256 , one obtains a fixed point of this map, which is the numerical approximation of a 2-unitary matrix within the limits of the machine precision. Final analytic shape of U was a result of the tedious work of searching for local unitary operations V j ∈ U ( 6 ) such that ( V 1 ⊗ V 2 ) U ( V 3 ⊗ V 4 ) took sparse enough form and allowed for its elements to be expressed as roots of unity located at the particular concentric circles around the origin.

Although the details and the general behavior of this algorithm are still not understood completely, it is possible to amend this procedure to obtain even more interesting results. One possible modification consists of adding additional step which might be considered as purging the matrix elements from the noise. To this end, we define a chopping procedure by means of the map c ε : C N × N ∋ X ↦ c ε ( X ) ∈ C N × N , where c ε ( X j k ) = 0 if ∣ X j k ∣ ⩽ ε , otherwise matrix elements stay intact. This means that near-zero entries are set to zero, which in general obviously breaks unitarity, however, starting with a relatively small value of ε ≈ 0 and gradually increasing it to ε ↗ 1 , one can smoothly steer the form of the final matrix. This is because above some threshold ε > ε * , the operation c ε stops affecting values of the matrix and, additionally, in some cases, the map ℳ does not disturb its zero values either. Formal explanation of these facts is currently beyond the scope of this report. Here, we take the numerical behavior as a strong although obscured evidence of a yet-to-be-discovered feature of this algorithm and define a new iterative procedure as

for t ∈ N 0 and ε ∈ ( 0 , 1 ] . We shall use a short notation Y = ℳ ε ( X ) to denote output Y for a seed X . This additional modification can significantly change possible outputs of the original map ℳ and result in new analytical representatives of AME ( 4 , 6 ) states. One must remember that not every seed provides a solution, and still a kind of fine-tuning must be performed to make this recipe, based on the original algorithm of Sinkhorn [23], Sinkhorn and Knopp [24], converge quickly.

From now on, we fix d = 6 and N = d 2 = 36 . Again, the best seeds for ℳ ε are seemingly those which are close to the permutation matrix P * [11]. Provided that there exists Y = ℳ ε ( X ) , for some seed X (meaning that the procedure is convergent), the closer to P * the larger probability to obtain a more sparser form of Y , which is usually permutationally equivalent to the block diagonal matrix with three blocks, each of size 12. In other words, there exist two permutation matrices P L and P R such that P L Y P R = B 1 ⊕ B 2 ⊕ B 3 , where B j is one such block of size 12.

Having a collection of outputs, Y = { Y : Y = ℳ ε ( X ) with X ∈ C 36 × 36 } , we ask whether some of them might be representatives of a CHM or become so when subjected to local unitary rotations. Paradoxically, this time we do not intend to make further simplifications but, in some sense, we are going to slightly “complicate” the matrix form, turning all entries (including zeros) into unimodular complex numbers. The objective function reads

where optimization is performed over two local unitary matrices V 1 , V 2 ∈ U ( 6 ) . Mildly interesting fact is that we do not need to introduce all four local unitaries, setting V 3 = V 4 = I . Matrices V 1 and V 2 are initially drawn at random and during the optimization process at each step they are being converted to unitary matrices via polar decomposition; V j ↦ Π ( V j + Δ j ) for j = 1 , 2 and perturbation Δ j proportional to Z ( Y ) . Many numerical investigations have not exposed the necessity of using additional pair, so only two of them, on either side, form a good enough structure.

Another more important observation is that not every Y ∈ Y can be used to minimize the objective function Z ( Y ) . Actually, exhaustive numerical search (for many different seeds) revealed only a tiny fraction of 2-unitary matrices Y for which Z → 0 , assisted by particular V j ∈ U ( 6 ) . Moreover, in some cases one must additionally realign entries of Y by means of the operations R or Γ to make Z ( Y ) tend to zero. One particular example of a matrix Y is shown in Figure 1 (left panel). This matrix plugged into (6) can be transformed into another one, Y → L U ℋ for which ∣ ℋ j k ∣ = 1 for any value of j and k . Since local unitaries do not affect 2-unitarity, at this stage we can tentatively and numerically confirm the fact that there exists a CHM ℋ of order 36, which after rescaling by 1/6 becomes 2-unitary.

Figure 1

Left panel: Matrix Y of order 36, 2-unitary after rescaling by 1/6; result of the map ℳ ε ( X ) for some X . Two different colors represent two amplitudes ∈ { 3 , 3 } ∕ 6 of 216 non-zero elements. Remaining blank entries denote zeros. Right panel: The same matrix reshuffled and transformed by some two permutation matrices, P L Y Γ R P R , in order to form a highly symmetric structure at the price of loss of 2-unitarity. In both cases, phases are omitted as irrelevant at this stage.

3 Structure of the two-unitary CHM

At first glance, the matrix ℋ does not present any simple structure. However, dephasing (operation that brings any CHM to the normalized form in which its first row and first column are all-one vectors) transforms a number of entries into sixth root of unity. Unfortunately, dephasing also destroys 2-unitarity. But such a form of dephased ℋ suggests that this matrix might actually be a Butson-type Hadamard one [9,10] shifted by an internal parameterization. This assumption is supported by the fact that the defect^[2] [26] of ℋ does not vanish; depending on the result of optimization it might take different values, e.g., 61 (see below), but it never equals to zero. Indeed, putting some effort, using unimodularity and orthogonality conditions, one can fully recover the analytical form of ℋ , which manifests itself as a Butson-type CHM with all entries being sixth roots of unity, as presumed: ℋ ∈ B H ( 36 , 6 ) . An array of integer-valued phases of ℋ is presented in Appendix A. Now we can formally arrive at the following observation, which solves the open problem formulated in [7].

Non-vanishing defect suggests that ℋ might not be an isolated point in the space B H ( 36 , 6 ) of Butson-type matrices. In fact, we show below that ℋ admits internal parameterization in the form of 19 affine parameters plus 5 non-affine ones. Affinity (vs non-affinity) indicates that the character of variability of phases in ℋ j k as functions of orbit parameters, α j k , is only linear, i.e.,

So, in the most general form matrix ℋ reads ℋ = ℋ ( α , η ) with α ∈ R 19 and η ∈ R 5 . However, for the purpose of this article, we shall focus only on affine orbits and properties that can be derived from these additional degrees of freedom, leaving detailed description of the non-affine dependence for a possible future investigation.

In the following, let 0 denote a vector of zeros 0 ∈ R 19 and ℋ = ℋ ( 0 ) . Hence, we consider ℋ ( α ) , with α ∈ [ 0 , 2 π ) × 19 ⧹ { 0 } . First of all, any non-zero value of α does not change the properties of being 2-unitary and CHM, in contrary to local unitary operations which immediately move ℋ outside the set H ( 36 ) . This makes this result quite important, providing a flexible family of very special AME states, the form of which can be controlled by fine-tuning its degrees of freedom, if one would want to realize such an object experimentally.

Moreover, special choices of the vector of phases α can simplify the form ℋ ( α ) further. For the sake of simplicity in presentation in the next two examples we assume that α j ∈ { 0 , 1 , 2 , 3 , 4 , 5 } and actual parameter α = exp { i π α j ∕ 3 } . The vector of phases

turns ℋ ( σ ) into a symmetric form, ℋ ( σ ) = ℋ ( σ ) T . This fact supports the observation from [6], where we noted that vast majority of CHM can be brought to the symmetric or Hermitian form (depending on the dimension). The problem of symmetrizability of CHM extends the same problem for real Hadamard matrices, known in the literature since at least 30 years and as such it requires independent studies, see [12] and references therein. In particular, new examples of symmetric real-valued Hadamard matrices of orders 188, 292, and 452 are constructed in [4].

One can confirm that for any φ ∈ [ 0 , 2 π ) and γ = 3 φ ∕ π the following vector of parameters depending on a single phase

provides ℋ ( δ ) with a constant-valued diagonal, i.e., arg ( ℋ ( δ ) j j ) = φ . In particular, suitable values of φ allow us to obtain matrices from the class B H ( 36 , 6 k ) with k ∈ N 1 . Despite many numerical and analytical attempts, no simpler Butson-type matrix ℋ was found and currently six is the smallest possible root of unity that a 2-unitary CHM of order 36 can admit.

By appropriately tuning α , it is possible to impose many different forms of ℋ ( α ) depending on the actual requirements. Since internal phases cover entire matrix with different intensity, one can adjust a selected subset of entries, optimizing over 19-dimensional space.

4 Alternative construction via biunimodular vectors

In a recent publication [3], the authors provide independent construction of a 2-unitary CHM of order 36 based on the approach via biunimodular vectors [13]. Such vectors remain unimodular under the transformation by F 6 ⊗ F 6 , where F 6 is the standard unitary Fourier matrix of order six. Three biunimodular vectors { Λ j } j = 1 , 2 , 3 of size 36, listed in [3], read as follows:

They are building blocks for three matrices U j as follows:

where the permutation P is a generalized CNOT gate (i.e., addition modulo 6)

and matrices K and L are both B H ( 36 , 6 ) such that K = K † , L = L T and K ( X A ⊗ I ) = L , where A is anti-diagonal permutation matrix. It is straightforward to confirm that indeed all three matrices U j are 2-unitary CHM.^[3]

As all U j are characterized by a non-vanishing value of defect, they all might possibly admit internal parameterizations too. Subsequently, appropriate choice of parameters might cause their orbit to intersect. A quick instrument to examine whether it is possible to introduce internal affine parameterization is to check the existence of particularly related columns in a given matrix – “ER-pairs” [15] – a method that necessarily works for objects of even order. This might be accompanied by the method of introducing internal parameterization described in [5]. To this end, one must exactly solve a system of linear equations associated with a given matrix, the solution of which indicates the form of an additional matrix R containing affine parameterization. Using these tools, we were able to obtain conclusive results only for U 3 .

Interestingly, the matrix U 3 seems the “closest” to the one introduced in this article due to the value of its defect, d ( U 3 ) = 185 .

Having all these matrices, one question that remains is how much different they are. In other words, whether it is possible to transform one into another via either local unitary (LU) or global Hadamard-like operations. In the latter case, we say that two (complex) Hadamard matrices H 1 and H 2 are H -equivalent^[4], if one matrix can be unitarily rotated into the other one by means of two monomial unitary matrices M 1 and M 2 , written H 1 ≃ H H 2 ⇔ H 1 = M 1 H 2 M 2 [16,19]. Problem of H -equivalence is also addressed in [20] for a special subset of real Hadamard matrices, called Sylvester matrices. In that case it reduces to permutations and changes the sign of rows and columns of H .

In order to check LU-equivalence one should use the set of invariants of an analyzed matrix A , denoted by A ( ρ 1 , ρ 2 , ρ 3 , ρ 4 ) described in [2]. Appropriately chosen permutations ρ j ∈ S n , applied to rearrange 4-index in matrix A ∈ C ( d 2 ) , can be used to determine the LU-class it belongs to. In other words, different values of invariants assigned to different matrices imply that they are representatives of different LU-classes. Numerical calculations suggest that this is the case for ℋ ( 0 ) and U 1 , 2 , 3 ; however, numerical complexity implied by the fully non-zero form of the matrices (no vanishing entries) prevents us from any formal statement regarding the general case of ℋ ( α ) and U 3 .

As for the H -equivalence we will use another numerical signature – the aforementioned defect of a matrix. Value of the defect, being invariant with respect to monomial transformations, can also tell whether two Hadamard matrices belong to different (Hadamard) classes; however, one must remember that this criterion works in one direction only. Several attributes of every matrix are collected in Table 1.

Armed only with a numerical evidence we conclude in the following.

5 Summary

In this article, we constructed a 19-dimensional affine family of 2-unitary CHM of order 36, which can be further extended over non-affine subspace. Such a matrix is a new unitary representation of absolutely maximally entangled state of four quhexes. The advantage of this object over previous solutions is two-fold. First, it admits only one amplitude while having no zeros. It is a kind of trade-off with comparison with the original matrix representing the “golden” AME state [1], which contains rather complicated 20th-root of unity of three different amplitudes spread over the matrix with as many as 1184 vanishing entries. Second, being a member of a multidimensional family preserving both: Hadamardness and the property of being a 2-unitary matrix, it might serve as a flexible object in practical performance. The problem of preparation an associated quantum circuit with four pure quhexes at the input and ℋ or U 1 , 2 , 3 as output, designed for a potential experimental realization, is currently subjected to an examination.