Generalization of Pauli channels through mutually unbiased measurements
Katarzyna Siudzińska
quant-ph math-ph math.MP





Institute of Physics, Faculty of Physics, Astronomy and Informatics 
  Nicolaus Copernicus University, Grudziądzka 5/7, 87–100 Toruń, Poland


We introduce a new generalization of the Pauli channels using the mutually unbiased measurement operators. The resulting channels are bistochastic but their eigenvectors are not unitary. We analyze the channel properties, such as complete positivity, entanglement breaking, and multiplicativity of maximal output purity. We illustrate our results with the maps constructed from the Gell-Mann matrices and the Heisenberg-Weyl observables.




Generalization of Pauli channels through mutually unbiased measurements
    Katarzyna Siudzińska
    December 30, 2023
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§ INTRODUCTION


The concept of mutual unbiasedness was first considered in regard to orthonormal vector bases. Two orthonormal bases are called mutually unbiased if the probability of transition between any of their vectors is constant. The d-dimensional Hilbert space admits at most d+1 mutually unbiased bases (MUBs), and the maximum is reached for d being a prime power <cit.>. In any dimension d, one can always construct at least three MUBs <cit.>. A new approach to unbiasedness has been introduced by Kalev and Gour <cit.>, who generalized the notion of mutually unbiased bases to the mutually unbiased measurements (MUMs). These are the sets of positive operators that sum up to identity and contain the projectors onto MUB vectors as a special case of projective measurements. Interestingly, one can always construct d+1 MUMs, regardless of the dimension d.

The applications of mutually unbiased measurements have been widely studied in uncertainty relations and entanglement detection. In particular, the MUMs were used to derive state-dependent <cit.>, state-independent <cit.>, and fine-grained <cit.> entropic uncertainty relations. The last type helped to find new separability conditions for bipartite system <cit.>. Moreover, it was shown that there is an equality between the amounts of uncertainty for MUMs and entanglement of the measured states quantified by the conditional collision entropy <cit.>. New separability criteria were given for arbitrary d-dimensional bipartite <cit.> and multipartite systems <cit.>. Liu, Gao, and Yan <cit.> provided the criteria whose experimental implementation does not require a full state tomography. In another paper <cit.>, they also presented the conditions for k-nonseparability detection of multipartite qudit systems. Graydon and Appleby generalized projective 2-designs to conical t-designs <cit.> and applied them to describe a connection between designs and entanglement <cit.>. Recently, the MUMs have also been used to find more operational Einstein-Podolsky-Rosen (EPR) steering inequalities <cit.>. Finally, Li et al. <cit.> used the MUMs to introduce new positive quantum maps and entanglement witnesses, which generalize the constructions from <cit.>.

In this paper, we construct a new class of bistochastic quantum channels. These channels generalize Nathanson and Ruskai's diagonal Pauli channels constant on axes (also known as generalized Pauli channels), whose definition includes the mutually unbiased bases. Our construction method uses the mutually unbiased measurement operators. It is valid for any finite dimension d, as one can always find the maximal number of d+1 MUMs. We find the necessary and sufficient conditions for complete positivity of the channels. We also analyze how the properties of Nathanson and Ruskai's channels change after replacing the MUBs with MUMs. Finally, we provide examples of bistochastic channels whose eigenvectors are not unitary operators.



§ MUTUALLY UNBIASED MEASUREMENTS


Following the work by Kalev and Gour <cit.>, let us introduce the notion of mutually unbiasedness for measurement operators.
In quantum mechanics, a measurement is determined by a set of measurement operators (POVMs) M_k that are positive and sum up to identity, {M_k|M_k≥ 0,∑_kM_k=𝕀_d}. The probability of the k-th outcome is (M_kρ), where ρ is the density operator of a quantum system. As a special case, one considers the projective measurement, where the measurement operators M_k are orthogonal projectors. Clearly, mutually unbiased bases can be used to perform projective measurements. Consider N orthonormal bases {ψ_k^(α),k=0,…,d-1} in ℂ^d that are numbered by α=1,…,N. These bases are mutually unbiased if and only if |ψ_k^(α)|ψ_l^(β)|^2=1/d for α≠β. Therefore, the set of projectors P_k^(α)=|ψ_k^(α)ψ_k^(α)| onto the α-th basis forms a measurement with the following properties,

    (P_k^(α))=1,
       (P_k^(α)P_l^(β))=δ_αβδ_kl+1/d (1-δ_αβ).

Note that P_k^(α) can be regarded as either states or measurement operators. Therefore, two projective measurements are mutually unbiased if the probability of measuring one with the other is constant. This notion can be generalized to POVMs. Namely, the measurements {P_k^(α)|P_k^(α)≥ 0,∑_k=0^d-1P_k^(α)=𝕀_d} are mutually unbiased if and only if <cit.>

    (P_k^(α))=1,
       (P_k^(α)P_l^(β))=1/d +dκ-1/d-1δ_αβ(δ_kl
    -1/d),

where 1/d<κ≤ 1. For κ=1, the above conditions reproduce eq. (<ref>). It is important to note that one can always construct the maximal number of d+1 mutually unbiased measurements. Moreover, MUMs form an informationally complete set, and any state can be written as

    ρ=1/d𝕀_d+ d-1/dκ-1∑_α=1^d+1∑_k=0^d-1P_k^(α)[(ρ P_k^(α))-1/d].

In their seminal paper, Kalev and Gour <cit.> proposed a method of constructing d+1 MUMs from an orthonormal basis {𝕀_d/√(d),F_α,k|α=1,…,d+1,k=1,…,d-1}, where F_α,k are traceless Hermitian operators. Namely, one has

    P_k^(α)=1/d𝕀_d+tF_k^(α),

where

    F_k^(α)=∑_l=1^d-1F_α,l-√(d)(1+√(d))F_α,k,    k≠ 0,
    
    (1+√(d))∑_l=1^d-1F_α,l,     k=0,

and t≠ 0 is a free real parameter such that P_k^(α)≥ 0. The relation between t and κ reads

    κ=1/d +(d-1)t^2(1+√(d))^2.




§ GENERALIZATION OF PAULI CHANNELS


A mixed unitary evolution of a qubit is described by the Pauli channel

    Λ[ρ]=∑_α=0^3 p_ασ_αρσ_α,

which is the most general form of a bistochastic quantum channel <cit.>. In the above formula, p_α denote the probability distribution, and σ_α are the Pauli matrices. One has

    Λ[σ_α]=λ_ασ_α,

where λ_0=1 and

    λ_α = 2(p_0 + p_α) - 1

for α=1,2,3. Now, Λ is completely positive if and only if its eigenvalues λ_α satisfy the Fujiwara-Algoet conditions <cit.>

    -1 ≤∑_α=1^3λ_α≤ 1+2min_αλ_α.


An interesting feature of the Pauli channels is that the eigenvectors of their Kraus operators σ_α are mutually unbiased. This property was used by Nathanson and Ruskai <cit.> to introduce the generalized Pauli channels

    Λ=dp_0-1/d-1+d/d-1∑_α=1^d+1p_αΦ_α,

where p_α is the probability distribution,  denotes the identity map, and

    Φ_α[X]=∑_k=0^d-1P_k^(α)(XP_k^(α))

are the quantum-classical channels constructed from the projectors P_k^(α) onto the MUB vectors. It has been shown that <cit.>

    Φ_αΦ_β=Φ_0,  α≠β,
       Φ_αΦ_α=Φ_α,
       ∑_α=1^d+1Φ_α=dΦ_0+,

where Φ_0[X]=𝕀_d(X)/d is the completely depolarizing channel. Now, the eigenvalue equations for Λ read Λ[𝕀_d]=𝕀_d and

    Λ[U_α,k]=λ_α U_α,k

with the unitary operators

    U_α,k=∑_l=0^d-1ω^klP_l^(α),    ω=e^2π i/d.

This indicates that Λ is a bistochastic channel. It is also self-dual (Λ=Λ^†), so its (d-1)-times degenerated eigenvalues 

    λ_α=1/d-1[d(p_α+p_0)-1]

are real. Note that the generalized Pauli channels can be equivalently written as

    Λ=p_0+1/d-1∑_α=1^d+1p_α𝕌_α,

where

    𝕌_α[X]=∑_k=1^d-1U_α,kXU_α,k^†

satisfies 𝕌_α=dΦ_α-.
The complete positivity conditions for Λ are the generalized Fujiwara-Algoet conditions <cit.>

    -1/d-1≤∑_α=1^d+1λ_α≤ 1+dmin_αλ_α.


Our goal is to generalize the generalized Pauli channels from eq. (<ref>). We achieve this by replacing the mutually unbiased bases with mutually unbiased measurements. After this procedure, the bistochastic quantum-classical channels from eq. (<ref>) no longer satisfy properties (<ref>) but instead

    Φ_αΦ_β=Φ_0,  α≠β,
       Φ_αΦ_α[P_l^(β)]=Φ_0,  α≠β,
       Φ_αΦ_α[P_l^(α)]=dκ-1/d-1Φ_α[P_l^(α)]
                   +d(1-κ)/d-1Φ_0[P_l^(α)],
       ∑_α=1^d+1Φ_α=d(d-κ)/d-1Φ_0+dκ-1/d-1.

The generalized Pauli channel has the form (<ref>), and its eigenvectors are again U_α,k defined in eq. (<ref>), as

    Φ_α[U_β,l]=dκ-1/d-1δ_αβU_β,l.

This time, however, there is no simple relation between 𝕌_α and Φ_α. Now, U_α,k form an orthogonal basis with

    [U_α,kU_β,l^†]=d(dκ-1)/d-1δ_αβδ_kl.

Observe that, in terms of the Hermitian orthonormal basis, these operators read

    U_α,k=√(d)t∑_l=1^d-1F_α,l[1-(√(d)+1)ω^kl],

and they are no longer unitary by definition. Therefore, our method allows one to construct bistochastic channels whose eigenvectors are not unitary. 
The eigenvalues of Λ are given by

    λ_α=1/d-1[d(p_0+dκ-1/d-1p_α)-1],

where the inverse relation is

    p_0=1/d^2(d-κ)[(d-1)^2∑_α=1^d+1λ_α-d(dκ-1)+d^2-1],


    p_α=(d-1)^2/d^2(dκ-1)(d-κ)[dκ-1   +d(d-κ)λ_α
       -(d-1)∑_β=1^d+1λ_β].

Sufficient complete positivity conditions for the generalized Pauli channels read p_0≤ 1/d and p_α≥ 0, which is equivalent to

    dκ-1/d-1≤∑_α=1^d+1λ_α≤1/d-1[dκ-1+d(d-κ)min_βλ_β].



Note that the operators U_α,k can be used to generate d+1 mutually unbiased measurements. Namely, let us take any orthogonal basis {𝕀_d,U_α,k} with U_α,k^†=U_α,d-k, a normalization given by eq. (<ref>), and satisfying an additional condition 𝕀_d+∑_l=1^d-1ω^-klU_α,l≥ 0. Then, by inverting formula (<ref>), we get

    P_k^(α)=1/d[𝕀_d+∑_l=1^d-1ω^-klU_α,l].

It is straightforward to check that such P_k^(α) form a POVM and satisfy conditions (<ref>). If for U_α,k one chooses the operators that form a cyclic subgroup, then U_α,k are linearly proportional to the Weyl operators W_kl=∑_m=0^d-1ω^mk|m+lm|,
and P_k^(α) are rank-1 projectors onto the mutually unbiased bases.




§ PROPERTIES


In this section, we show which properties of the generalized Pauli channels constructed from MUBs transfer over after one replaces MUBs with MUMs. First, in general, the generalized Pauli channels Λ are no longer covariant with respect to all U_α,k <cit.>, which means that

    Λ[U_α,kXU_α,k^†]=U_α,kΛ[X]U_α,k^†

does not hold for an arbitrary operator X <cit.>. However, one still has ΛΦ_α=Φ_αΛ for every α=1,…,d+1. Next, the necessary and sufficient conditions ∑_α=1^d+1λ_α≤ 1 for entanglement breaking of Λ with λ_α≥ 0 <cit.> become only necessary. New sufficient conditions are established in the following theorem.


The generalized Pauli channel with λ_α≥ 0 is entanglement breaking if ∑_α=1^d+1λ_α≤dκ-1/d-1.



Recall that a quantum channel Λ is entanglement breaking if and only if it can be written in the Holevo form Λ[X]=∑_j R_j(ρ E_j), where R_j are density operators and {E_j} form a POVM <cit.>. Observe that the generalized Pauli channels can be equivalently rewritten as

    Λ=(1-∑_α=1^d+1μ_α)Φ_0
    +∑_α=1^d+1μ_αΦ_α

with

    μ_α=d-1/dκ-1λ_α.

Now, it is easy to show that

    Λ=∑_α=1^d+1∑_k=0^d-1P_k^(α)[X E_k,α]

with

    E_k,α=1-∑_β=1^d+1μ_β/d(d+1)𝕀_d
    +μ_α P_k^(α).

By definition, P_k^(α) is a density operator, and E_k,α is a sum of two positive operators, so it is also positive. Finally,

    ∑_α=1^d+1∑_k=0^d-1E_k,α=𝕀_d,

and hence {E_k,α} form a POVM.


There are several important measures that have been analyzed for Nathanson and Ruskai's generalized Pauli channels. One of them is the minimal and maximal channel fidelity f_min/max(Λ) on pure states <cit.> that measures the distance between the input P and the output state Λ[P]. Unluckily, the formulas for f_min/max(Λ) do not carry over to the generalized Pauli channels constructed from MUMs. The reason is that the extremal fidelities for the channels constant on axes are reached on the projectors onto MUB vectors. However, the mutually unbiased measurements are not projectors, and the formula for the channel fidelity on mixed states is much more complicated than on pure states <cit.>. Another important measure is the maximal output p-norm ν_p(Λ) that measures optimal output purity. In other words, ν_p(Λ) determines how close the channel output Λ[ρ] is to a pure state. If Λ is constructed from MUBs, the analytical formulas for ν_2(Λ) and ν_∞(Λ) are known <cit.>. However, only the former generalizes in a straightforward manner.


The maximal output 2-norm of Λ is equal to

    ν_2^2(Λ)=1/d[1+(dκ-1)max_αλ_α^2].




Starting from eq. (<ref>), we calculate the Frobenius norm

    Λ[ρ]_2^2   =(Λ[ρ]^2)
       =1/d +d-1/dκ-1∑_α=1^d+1λ_α^2[∑_k=0^d-1(ρ P_k^(α))^2-1/d].

Observe that Λ[ρ]_2^2 achieves its maximal value for ρ=P_l^(α_∗), where λ_α_∗=√(max_αλ_α^2). We find that

    ∑_k=0^d-1( P_k^(α) P_l^(β))^2=
    1/d[1+(dκ-1)^2/d-1δ_αβ],

and hence eq. (<ref>) follows.


By definition, the maximal output 2-norm of Λ is strongly multiplicative if ν_2(Λ⊗Ω)=ν_2(Λ)ν_2(Ω) for any quantum channel Ω. From the theorem by Fukuda, Holevo, and Nathanson <cit.>, it follows that ν_2(Λ) is strongly multiplicative if and only if

    ν_2^2(Λ)=1/d[1+(d-1)max_α|λ_α|^2].

Observe that ν_2(Λ) in eq. (<ref>) satisfies the above condition only for κ=1.



§ EXAMPLES




 §.§ Pauli matrices


Let us consider d=2 and construct the mutually unbiased measurements from the rescaled Pauli matrices F_1,α=σ_α/√(2). Now, using eq. (<ref>), we find that the MUMs are given by

    P_k^(α)=1/d𝕀_d+(-1)^k t1+√(2)/√(2)σ_α,

where

    -√(2)-1/√(2)≤ t≤1/2+√(2)

and t≠ 0. Interestingly, the corresponding parameter κ belongs to the full range 1/2<κ≤ 1. The channel eigenvectors U_α,1 are again rescaled Pauli matrices,

    U_α,1=P_0^(α)-P_1^(α)=2t1+√(2)/√(2)σ_α.

Therefore, regardless of the choice of κ, Λ is the Pauli channel.



 §.§ Gell-Mann matrices


Now, consider the higher-dimensional (d≥ 3) Hermitian generalization of the Pauli matrices, known as the Gell-Mann matrices. They are defined as follows,

    σ_kl:=1/√(2)(|kl|+|lk|),
       σ_lk:=i/√(2)(|kl|-|lk|),
       σ_kk:=√(1/k(k+1))(∑_j=0^k-1|jj|-k|kk|),

for all 0≤ k<l≤ d-1 and 0≤ k≤ d-1, respectively. Together with σ_00=𝕀_d/√(d), they form an orthonormal Hermitian operator basis. After Kalev and Gour <cit.>, we group the operators into the following sets,

    {F_α,k|k=1,…,d-1}={σ_k,α-1|k≠α-1},
       {F_d+1,k|k=1,…,d-1}={σ_kk|k=1,…,d-1}.

In the next step, we construct

    F_0^(d+1)=(√(d)+1)∑_l=1^d-1σ_ll,
       F_k^(d+1)=-√(d)(√(d)+1)σ_kk+∑_l=1^d-1σ_ll

for k=1,…,d-1, as well as

    F_k^(α)=-√(d)(√(d)+1)σ_k,α-1+∑_l≠α-1σ_l,α-1,
       F_α-1^(α)=(√(d)+1)∑_l≠α-1σ_l,α-1

for k≠α-1 and α=1,…,d.
The corresponding MUMs are given by

    P_k^(α)=1/d𝕀_d+tF_k^(α),

where t is a free parameter such that P_k^(α)≥ 0. Let us take the optimal

    t=√(2)/d(√(d)+1)√(d-1),

which follows from the optimal κ=d+2/d^2 via eq. (<ref>) and guarantees that P_k^(α)≥ 0 <cit.>. One easily finds

    U_d+1,k=√(d)t∑_l=1^d-1σ_ll[1-(√(d)+1)ω^kl],


    U_α,k=   √(d)t∑_l=0^α-2σ_l,α-1[1-(√(d)+1)ω^k(l+1)]
       
    +√(d)t∑_l=α^d-1σ_l,α-1[1-(√(d)+1)ω^kl].

For d=3, these operators explicitly read

    U_1,1=U_1,2^†=3/2√(2) (1+√(3))t
    [    0 -1-i  1-i;  1+i    0    0; -1+i    0    0 ],


    U_2,1=U_2,2^†=3/2√(2) (1+√(3))t
    [    0  1-i    0;  1-i    0  1-i;    0 -1+i    0 ],


    U_3,1=U_3,2^†=3/2√(2) (1+√(3))t
    [   0   0 1+i;   0   0 1-i; 1+i 1-i   0 ],


    U_4,1=U_4,2^†=√(3)t diag[       2+√(3)-i;    i(2+√(3)+i); -(1+i)(1+√(3)) ]^T.

Observe that, among the above U_α,k, the only operators that become unitary after rescaling are U_4,1 and U_4,2. These are also the only ones that mutually commute. Therefore, the generalized Pauli channel Λ constructed from the Gell-Mann matrices is an example of a bistochastic channel whose eigenvectors are not unitary.



 §.§ Heisenberg-Weyl observables


An alternative generalization of the Pauli matrices is provided by the Weyl operators

    W_kl=∑_m=0^d-1ω^km|m+lm|.

They form an orthogonal unitary operator basis, so they cannot be used to generate mutually unbiased measurements. However, it is possible to use W_kl in order to introduce a Hermitian basis. Consider the case with d=3 and let us define, after <cit.>, the Heisenberg-Weyl observables

    V_kl=1-i/2√(d)W_kl+1+i/2√(d)W_kl^†,   k≤ l,
       V_kl=1+i/2√(d)W_kl+1-i/2√(d)W_kl^†,   k>l.

Notably, such defined V_kl are orthonormal traceless Hermitian operators. Now, in analogy to the previous example, we group V_kl into the sets

    {F_α,k|k=1,…,d-1}={V_k,α-1|k≠α-1},
       {F_d+1,k|k=1,…,d-1}={V_kk|k=1,…,d-1},

so that eqs. (<ref>–<ref>) still apply if one replaces σ_kl with V_kl. Numerical calculations show that the optimal values of t and κ read

    t≃ 0.112,    κ≃ 0.522.

For comparison, the optimal κ for the MUMs constructed from the Gell-Mann operators in d=3 is κ=5/9≃ 0.556. Therefore, the Heisenberg-Weyl operators are a worse choice for the operator basis, as their optimal parameter κ is further from its maximal value κ=1. 

The mutually unbiased measurements are again given by eq. (<ref>), and the associated operators U_α,k in d=3 read

    U_1,1=U_1,2^†=√(3)(1+√(3))t
    [   1   0   0;   0   ω   0;   0   0 ω^2 ],


    U_2,1=   U_2,2^†=√(3)/2 (1+√(3))t
       ×[            0 iω^2(√(3)-1)      ω(2i-ω);    -ω^2(1+i)            0        -iω^2; -i(ω^2+√(3))     -(ω^2+i)            0 ],


    U_3,1=   U_3,2^†=√(3)/2 (1+√(3))t
       ×[            0         -ω^2 i(√(3)-1)ω^2; -i(ω^2+√(3))            0  -(ω^2+√(3));  (√(3)-1)ω^2        -iω^2            0 ],


    U_4,1=U_4,2^†=√(3)/2 (1+√(3))t
    [       0      -ω   √(3)ω; √(3)ω^2       0      -1;    -ω^2    √(3)       0 ].

Once again, there are only two operators U_α,k that become unitary after rescaling: U_1,1 and U_1,2. Moreover, U_1,1 and U_1,2 are linearly proportional to the Weyl operators W_10 and W_20, respectively. This time, however, there are two pairs of mutually commuting operators: the aforementioned {U_1,1,U_1,2}, but also {U_4,1,U_4,2}. 
Therefore, the generalized Pauli channel Λ constructed from the Heisenberg-Weyl observables is another example of a bistochastic channel with non-unitary eigenvectors.



§ CONCLUSIONS


We introduced a new generalization of the Pauli channels, whose construction is based on the concept of mutually unbiased measurements. The resulting channels are bistochastic, but they not necessarily have unitary eigenvectors. We found  sufficient conditions for the generalized Pauli channels to be completely positive, as well as the conditions under which they break quantum entanglement. Further work is needed to establish the necessary and sufficient conditions for complete positivity. Also, we showed that their maximal output 2-norm is strongly multiplicative if and only if the MUMs are rank-1 projectors onto the MUB vectors.  The next step is to analyze the evolution of open quantum systems provided by the generalized Pauli dynamical maps. It would be interesting to check the κ-dependence of quantum Markovianity.



§ ACKNOWLEDGEMENTS


This paper was supported by the Polish National Science Centre project No. 2018/31/N/ST2/00250. The data that supports the findings of this study are available within the article.


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