Eigenvalues of Two-Parameter Complex Hadamard Matrices of Order Six

Characterizing complex Hadamard matrices of order n is a fundamental problem in quantum physics, as it is a way to find the complete mutually unbiased bases (MUBs). In general, k MUBs in Hilbert space ℂⁿ are k orthogonal bases, the inner product of any two vectors from different bases has a modulus of 1/n1/\sqrt n . When there are n + 1 MUBs in ℂⁿ, then they are referred to as complete MUBs. So far, it has been proven that complete MUBs exist in ℂⁿ when n is a prime power [1]. The complete classification of CHMs gives us a new way to find the maximal MUBs in the unsolved dimensions. In particular, the dimension six is the smallest one. Paper [2] investigated that the average distance between four bases in dimension six, providing a strong evidence against the existence of four MUBs in ℂ⁶. Paper [3] showed that if complete MUBs in dimension six exist, it cannot include more than one product basis. The examination of the number of product vectors in a set of four MUBs in dimension six showed that each of the remaining three MUBs contains at most two product vectors [4]. The existence and limitations of MUB product bases in dimension six were explored in [5–7]. The skills such as Schmidt ranks, Lie algebra, semidefinite programming, entanglement theory and so on are used in [8–14].

The complete classification of 6 × 6 CHMs is also unsolved. Karlsson presented a three-parameter CHMs family in ℂ⁶ [15], and named the family “the H₂-reducible matrices”. As far as we know, most known 6 × 6 CHMs belong to the H₂-reducible matrices such as the Haagerup matrix [16], but the Tao matrix is an exception [17]. The paper [18] presented a four-parameter 6 × 6 CHM family, though the analytic form of these matrices is still unknown. Another idea is analyzing the eigenvalues of the CHM. It was shown that CHMs of order six with dephased form have eigenvalues 6\sqrt 6 , – 6\sqrt 6 , and for the remaining four eigenvalues, at most two of them are the same [19]. More facts investigating the classification problem have been presented in [20–25].

CHMs in dimension six are also bipartite unitary matrices from the point of view in quantum information theory. Such matrices can be applied to create quantum entanglement and to implement quantum circuits and cryptography [26]. The so-called controlled-unitary (say controlled-NOT) gate, which is a fundamental bipartite unitary operator, serves as both the primary entanglement resource and a critical component of universal quantum computation [27]. Finding the decompositions by controlled-unitary gates of bipartite unitary operators generated by CHMs will provide a simple physical realization of complex operations. Schmidt rank of bipartite unitary matrices was shown to be closely connected to the existence of this decomposition [28,29].

In this paper, we investigate CHMs with certain type of eigenvalues. In Theorem 1 we prove that CHMs with only two sorts of eigenvalues must be complex equivalent to a family we will refer to as K(d, x) in (8) with two parameters d, x. Lemma 1 gives the solutions of parameters in Theorem 1, and gives the suitable value range of the parameters. The main technical treatment in the proof of Theorem 1 is the construction in (A4), which exhausts the form of CHMs with only two sorts of eigenvalues. Next we analyze the CHMs in dephased form whose eigenvalues are 6\sqrt 6 , – 6\sqrt 6 , λ₁, λ₁, λ₂, λ₂. We give a complete classification of these CHMs, containing the Tao matrix in (3) and a family K′ (d, x) in (21) with parameters d, x. That is the result in Theorem 2. The key equation in the proof of Theorem 2 is shown in (A35), which gives the main restriction of this class of CHMs. In Corollary 1 we present the outcome of a special case of Theorem 2, we claim that if λ₂ = –λ₁, then such CHM must be complex equivalent to the Hermitian matrices. In Theorem 3 we show that a family of K(d, x) has Schmidt rank at most three, and this family can be carried out as controlled unitary gates.

The rest of this paper will be presented as follows. In Section 2 we introduce some facts used in this paper, such as the definition of MUBs and CHMs, the eigenvalues of some famous CHMs. In Section 3, we classify all the CHMs with the specified eigenvalue structure. In Section 4, we find the application of some CHMs in implementable unitary gate. We conclude in Section 5.

In this section, we present the definition of CHMs and MUBs, as well as the eigenvalues of some special CHMs.

We also introduce the definition of complex equivalence. We refer to the monomial unitary matrix as a unitary matrix that every row and column have exactly one nonzero element. Then two matrices U and V are complex equivalent when U = PVQ where P, Q are monomial unitary matrices.

Next we list some famous CHMs and their eigenvalues. The Haagerup matrices [16] are 1H6q=[ 1111111−1ii−i−i1i−1−iq−q1i−i−1−qq1−i−1q1q−1i1−i1q−1qi−1 ],H_6^q = \left[ {\matrix{ 1 & 1 & 1 & 1 & 1 & 1 \cr 1 & { - 1} & i & i & { - i} & { - i} \cr 1 & i & { - 1} & { - i} & q & { - q} \cr 1 & i & { - i} & { - 1} & { - q} & q \cr 1 & { - i} & { - {1 \over q}} & {{1 \over q}} & { - 1} & i \cr 1 & { - i} & {{1 \over q}} & { - {1 \over q}} & i & { - 1} \cr } } \right], where |q| = 1. The eigenvalues of H6qH_6^q are 6\sqrt 6 , – 6\sqrt 6 , –1 + 5i\sqrt 5 i, –1 + 5i\sqrt 5 i, –1 – 5i\sqrt 5 i, –1 – 5i\sqrt 5 i.

Second, the Diţă matrix D₀ [30] is equivalent to H6qH_6^q for q = ±i. D₀ has the same eigenvalues as H6qH_6^q.2D0=[ 1111111−1i−i−ii1i−1i−i−i1−ii−1i−i1−i−ii−1i1i−i−ii−1 ].{D_0} = \left[ {\matrix{ 1 & 1 & 1 & 1 & 1 & 1 \cr 1 & { - 1} & i & { - i} & { - i} & i \cr 1 & i & { - 1} & i & { - i} & { - i} \cr 1 & { - i} & i & { - 1} & i & { - i} \cr 1 & { - i} & { - i} & i & { - 1} & i \cr 1 & i & { - i} & { - i} & i & { - 1} \cr } } \right].

Third, the Tao matrix is isolated and belongs to the Buston matrices [31,32].3S6(0)=[ 11111111ωωω2ω21ω1ω2ω2ω1ωω21ωω21ω2ω2ω1ω1ω2ωω2ω1 ],S_6^{(0)} = \left[ {\matrix{ 1 & 1 & 1 & 1 & 1 & 1 \cr 1 & 1 & \omega & \omega & {{\omega ^2}} & {{\omega ^2}} \cr 1 & \omega & 1 & {{\omega ^2}} & {{\omega ^2}} & \omega \cr 1 & \omega & {{\omega ^2}} & 1 & \omega & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^2}} & \omega & 1 & \omega \cr 1 & {{\omega ^2}} & \omega & {{\omega ^2}} & \omega & 1 \cr } } \right], where ω=e2πi3\omega = {e^{{{2\pi i} \over 3}}}. The eigenvalues of S6(0)S_6^{(0)} are 6,−6,3+15i2,3+15i2,3−15i2,3−15i2\sqrt 6 , - \sqrt 6 ,{{3 + \sqrt {15} i} \over 2},{{3 + \sqrt {15} i} \over 2},{{3 - \sqrt {15} i} \over 2},{{3 - \sqrt {15} i} \over 2}.

The Hermitian matrices, by exchanging rows and columns, can take the following form [33], 4H(t)=[ 1111111−11x−y−1xy1x−1t−t−x1−1y1t−11y−1t1−x−1ty11z11y−1x−tz1 ],H(t) = \left[ {\matrix{ 1 & 1 & 1 & 1 & 1 & 1 \cr 1 & { - 1} & {{1 \over x}} & { - y} & { - {1 \over x}} & y \cr 1 & x & { - 1} & t & { - t} & { - x} \cr 1 & { - {1 \over y}} & {{1 \over t}} & { - 1} & {{1 \over y}} & { - {1 \over t}} \cr 1 & { - x} & { - {1 \over t}} & y & 1 & {{1 \over z}} \cr 1 & {{1 \over y}} & { - {1 \over x}} & { - t} & z & 1 \cr } } \right], where θ ∈ [–π, – arccos(−1+32)\left( {{{ - 1 + \sqrt 3 } \over 2}} \right)] ∪ [arccos(−1+32)\left( {{{ - 1 + \sqrt 3 } \over 2}} \right), π], and the parameters x, y, t, z are given by 5y=eiθ,z=1+2y−y2y(−1+2y+y2);y = {e^{i\theta }},z = {{1 + 2y - {y^2}} \over {y\left( { - 1 + 2y + {y^2}} \right)}}; 6x=1+2y+y2−2(1+2y+2y3+y4)1+2y−y2;x = {{1 + 2y + {y^2} - \sqrt {2\left( {1 + 2y + 2{y^3} + {y^4}} \right)} } \over {1 + 2y - {y^2}}}; 7t=1+2y+y2−2(1+2y+2y3+y4)−1+2y+y2.t = {{1 + 2y + {y^2} - \sqrt {2\left( {1 + 2y + 2{y^3} + {y^4}} \right)} } \over { - 1 + 2y + {y^2}}}.

The eigenvalues of the Hermitian matrix H(t) are 6,6,6,−6,−6,−6\sqrt 6 ,\sqrt 6 ,\sqrt 6 , - \sqrt 6 , - \sqrt 6 , - \sqrt 6 .

In this section, we will present our results in this paper. We will discuss CHMs with special type of eigenvalues up to complex equivalence. Though eigenvalues are not invariant with respect to the equivalence relation, we can simplify the result by this relation and at the same time, no results will be omitted. We first consider a family of CHMs with exactly two sorts of eigenvalues. Paper [19] claimed that there is no 6 × 6 CHM with four identical eigenvalues. So the two distinct eigenvalues must be of algebraic multiplicity three. The following theorem shows the complete classification of such CHMs.

The proof of Lemma 1 will be presented in Appendix B. Noting that the solutions of (9) can be expressed by parameter ϕ₁ below (11) or x. So the notation K(d, x) means a class of two-parameter CHMs, here d controls the eigenvalues, d and x control the solutions of restriction (9).

Recall that the CHM of order-six with dephased form has eigenvalues 6,−6\sqrt 6 , - \sqrt 6 [19]. If the remaining four eigenvalues have at most two distinct elements, then [19] claims that the remaining four eigenvalues are λ₃, λ₃, λ₄, λ₄ with λ₃ ≠ λ₄. Next we present another theorem to classify such CHMs.

We present the brief version of proof of Theorem 2 below. More details can be found in Appendix C. The proof shows the connection between Theorems 1 and 2. In fact, K(d, x) and K′ (d, x) are equivalent by the equality 18K′(d,x)=d·p·K(d,x)·p{K^\prime }(d,x) = d\cdotp\cdotK(d,x)\cdotp where 19P=[ d¯00000010000001000000100000010000001 ].P = \left[ {\matrix{ {\bar d} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \cr } } \right].

Here 22H2=6|u1〉〈u1|−6|u2〉〈u2|=15[ 5555555−1−1−1−1−15−1−1−1−1−15−1−1−1−1−15−1−1−1−1−15−1−1−1−1−1 ].{H_2} = \sqrt 6 \left| {{u_1}} \right\rangle \langle {u_1}| - \sqrt 6 \left| {{u_2}} \right\rangle \langle {u_2}| = {1 \over 5}\left[ {\matrix{ 5 & 5 & 5 & 5 & 5 & 5 \cr 5 & { - 1} & { - 1} & { - 1} & { - 1} & { - 1} \cr 5 & { - 1} & { - 1} & { - 1} & { - 1} & { - 1} \cr 5 & { - 1} & { - 1} & { - 1} & { - 1} & { - 1} \cr 5 & { - 1} & { - 1} & { - 1} & { - 1} & { - 1} \cr 5 & { - 1} & { - 1} & { - 1} & { - 1} & { - 1} \cr } } \right].

The equation (21) can be rewritten as follows.23H=6|u1〉〈u1|−6|u2〉〈u2|+λ3∑i=34|ui〉〈ui|+λ4∑j=56|uj〉〈uj|H = \sqrt 6 \left| {{u_1}} \right\rangle \langle {u_1}| - \sqrt 6 \left| {{u_2}} \right\rangle \langle {u_2}| + {\lambda _3}\sum\limits_{i = 3}^4 {\left| {{u_i}} \right\rangle } \langle {u_i}| + {\lambda _4}\sum\limits_{j = 5}^6 {\left| {{u_j}} \right\rangle } \langle {u_j}| 24=6|u1〉〈u1|−6|u2〉〈u2|+λ3(I−|u1〉〈u1|−|u2〉〈u2|)+(λ4−λ3)∑j=56|uj〉〈uj| = \sqrt 6 \left| {{u_1}} \right\rangle \langle {u_1}| - \sqrt 6 \left| {{u_2}} \right\rangle \langle {u_2}| + {\lambda _3}\left( {I - \left| {{u_1}} \right\rangle \langle {u_1}| - \left| {{u_2}} \right\rangle \langle {u_2}|} \right) + \left( {{\lambda _4} - {\lambda _3}} \right)\sum\limits_{j = 5}^6 {\left| {{u_j}} \right\rangle } \langle {u_j}| 25=15[ 5555555−1+4λ3−1−λ3−1−λ3−1−λ3−1−λ35−1−λ3−1+4λ3−1−λ3−1−λ3−1−λ35−1−λ3−1−λ3−1+4λ3−1−λ3−1−λ35−1−λ3−1−λ3−1−λ3−1+4λ3−1−λ35−1−λ3−1−λ3−1−λ3−1−λ3−1+4λ3 ]+(λ4−λ3)∑j=56|uj〉〈uj|. = {1 \over 5}\left[ {\matrix{ 5 & 5 & 5 & 5 & 5 & 5 \cr 5 & { - 1 + 4{\lambda _3}} & { - 1 - {\lambda _3}} & { - 1 - {\lambda _3}} & { - 1 - {\lambda _3}} & { - 1 - {\lambda _3}} \cr 5 & { - 1 - {\lambda _3}} & { - 1 + 4{\lambda _3}} & { - 1 - {\lambda _3}} & { - 1 - {\lambda _3}} & { - 1 - {\lambda _3}} \cr 5 & { - 1 - {\lambda _3}} & { - 1 - {\lambda _3}} & { - 1 + 4{\lambda _3}} & { - 1 - {\lambda _3}} & { - 1 - {\lambda _3}} \cr 5 & { - 1 - {\lambda _3}} & { - 1 - {\lambda _3}} & { - 1 - {\lambda _3}} & { - 1 + 4{\lambda _3}} & { - 1 - {\lambda _3}} \cr 5 & { - 1 - {\lambda _3}} & { - 1 - {\lambda _3}} & { - 1 - {\lambda _3}} & { - 1 - {\lambda _3}} & { - 1 + 4{\lambda _3}} \cr } } \right] + \left( {{\lambda _4} - {\lambda _3}} \right)\sum\limits_{j = 5}^6 {\left| {{u_j}} \right\rangle } \langle {u_j}|.

Let the vectors |uj〉=[ g1j,…,g6j ]T,j=5,6\left| {{u_j}} \right\rangle = {\left[ {g_1^j, \ldots ,g_6^j} \right]^T},j = 5,6, and we can deduce the expressions of all elements in H. The details will be present in Appendix C.

By discussing three distinct cases, we obtain the relations of λ₃, λ₄. Further construction shows that for given CHM in dephased form with eigenvalues 6,−6,λ3,λ3,λ4,λ4\sqrt 6 , - \sqrt 6 ,{\lambda _3},{\lambda _3},{\lambda _4},{\lambda _4}, we will deduce the Tao matrix in (3) or a CHM with two sorts of eigenvalues, that is K(d, x). By Theorem 1, we have finished the proof of Theorem 2. The details will be present in Appendix C.

In this section, we introduce an application of our results. For a bipartite Hilbert space ℋ_A ⊗ ℋ_B, a controlled unitary gate up to local unitary equivalence takes the form 26U=∑i=1dA|i〉〈i|A⊗UB(i).U = \sum\limits_{i = 1}^{{d_A}} {\left| i \right\rangle } {\left\langle i \right|_A} \otimes U_B^{(i)}. where {|i〉_A} is an orthonormal basis for subsystem A and UB(i)U_B^{(i)} are unitary operators acting on subsystem B. d_A is the dimension of subsystem A.

It is known that every bipartite unitary matrix can be regarded a non-local operation from the viewpoint of quantum information. For example, the so-called controlled-NOT and other types of controlled unitary gates play a key role in quantum computing and so on, as every bipartite unitary gate can be decomposed into the product of some controlled unitary gates [34].

Further, it has been proven that a bipartite unitary gate of Schmidt rank at most three is actually a controlled unitary gate [28], which is more easily implementable in experiments. Here, for a bipartite unitary matrix U ∈ ℋ_A ⊗ ℋ_B, the Schmidt rank is the smallest integer r such that 27U=∑i=1rλiMi(A)⊗Ni(B)U = \sum\limits_{i = 1}^r {{\lambda _i}} M_i^{(A)} \otimes N_i^{(B)} with linearly independent sets {M_i} and {N_i} and all λ_i > 0.

In the following, we show that a family of CHMs K(d, x) are complex equivalent to some CHMs of Schmidt rank three. In particular, all order-six Hermitian CHMs are included in this family. Hence, such CHMs can be actually carried out as a controlled unitary gate, and they can be applied in experiments more easily. Our result help realize more non-local operations in quantum system with less coherence time and lower error rates.

We denote M=[ ABCD ]M = \left[ {\matrix{ A \hfill & B \hfill \cr C \hfill & D \hfill \cr } } \right] and A, B, C, D are four 3 × 3 submatrices of M. Then we have 30A+D=(d−d¯)I3,A + D = (d - \bar d){I_3}, and 31B+C=[ 2−y−z−y¯−z¯−y¯−z¯2x¯(y¯+z¯)−y−zx(y+z)2 ].B + C = \left[ {\matrix{ 2 & { - y - z} & { - \bar y - \bar z} \cr { - \bar y - \bar z} & 2 & {\bar x(\bar y + \bar z)} \cr { - y - z} & {x(y + z)} & 2 \cr } } \right].

If y = –z, then B + C = 2I₃, and A + D, B + C are linear dependent. It means that the CHM M has Schmidt rank at most three. Using the Lemma 1 we know that if y = –z, then x = d or x=d¯x = \bar d and {y,z}={ix¯,−ix¯}\{ y,z\} = \{ i\bar x, - i\bar x\} . Hence we have finished the proof.

For example, if d = 1, then x = 1 and K(d, x) is indeed Hermitian. We shall take y = i, z = –i, then M in (29) turns to be 32[ 1111ii11−1i1−i1−11i−i11−i−i−1−1−1−i1i−1−11−ii1−11−1 ]\left[ {\matrix{ 1 & 1 & 1 & 1 & i & i \cr 1 & 1 & { - 1} & i & 1 & { - i} \cr 1 & { - 1} & 1 & i & { - i} & 1 \cr 1 & { - i} & { - i} & { - 1} & { - 1} & { - 1} \cr { - i} & 1 & i & { - 1} & { - 1} & 1 \cr { - i} & i & 1 & { - 1} & 1 & { - 1} \cr } } \right] and we can rewrite M as follows: 33M=[ 100−1 ]⊗[ 11111−11−11 ]+[ 0100 ]⊗[ 1iii1−ii−i1 ]+[ 0010 ]⊗[ 1−i−i−i1i−ii1 ]M = \left[ {\matrix{ 1 & 0 \cr 0 & { - 1} \cr } } \right] \otimes \left[ {\matrix{ 1 & 1 & 1 \cr 1 & 1 & { - 1} \cr 1 & { - 1} & 1 \cr } } \right] + \left[ {\matrix{ 0 \hfill & 1 \hfill \cr 0 \hfill & 0 \hfill \cr } } \right] \otimes \left[ {\matrix{ 1 & i & i \cr i & 1 & { - i} \cr i & { - i} & 1 \cr } } \right] + \left[ {\matrix{ 0 \hfill & 0 \hfill \cr 1 \hfill & 0 \hfill \cr } } \right] \otimes \left[ {\matrix{ 1 & { - i} & { - i} \cr { - i} & 1 & i \cr { - i} & i & 1 \cr } } \right]

We take 34Q1=I2⊗16[ 231−23−1−202 ].{Q_1} = {I_2} \otimes {1 \over {\sqrt 6 }}\left[ {\matrix{ {\sqrt 2 } & {\sqrt 3 } & 1 \cr { - \sqrt 2 } & {\sqrt 3 } & { - 1} \cr { - \sqrt 2 } & 0 & 2 \cr } } \right].

Then 35Q1+MQ1=[ −11−2i1+2i1 ]⊗|0〉〈0|+[ 21+i1−i−2 ]⊗(|1〉〈1|+|2〉〈2|).Q_1^ + M{Q_1} = \left[ {\matrix{ { - 1} & {1 - 2i} \cr {1 + 2i} & 1 \cr } } \right] \otimes |0\rangle \langle 0| + \left[ {\matrix{ 2 & {1 + i} \cr {1 - i} & { - 2} \cr } } \right] \otimes (|1\rangle \langle 1| + |2\rangle \langle 2|).

Where |0〉 = [1,0,0]^T, |1〉 = [0,1,0]^T and |2〉 = [0,0,1]^T. So we know that up to local unitary equivalence, M is a controlled unitary gate. Thus it can be applied in experiments more easily.

In this paper, we have presented the complete classification of CHMs with two sorts of eigenvalues λ₁, λ₂, these CHMs are included in a two-parameter family K(d, x) in (8). Moreover, we have classified all CHMs with dephased form whose eigenvalues are exactly 6,−6,λ1,λ1,λ2,λ2\sqrt 6 , - \sqrt 6 ,{\lambda _1},{\lambda _1},{\lambda _2},{\lambda _2}, this class consists of K′(d, x) in (21) and Tao matrix in (3), here K′(d, x) is complex equivalent to K(d, x) for the same d, x. One interesting thing is that some known CHMs such as Diţă matrix, Haagerup matrices and Hermitian matrices, are included in the class K′(d, x) and all CHMs in this class are H₂-reducible. We have shown that the Schmidt rank of some CHMs including all Hermitian CHMs in K(d, x) is at most three up to complex equivalence. Our future research will focus on the Schmidt rank of every CHM in K(d, x) and more CHMs with three or more sorts of eigenvalues to find more connections between the classification of CHMs and their eigenvalues.