ORIGINAL ARTICLE: David Raveh, Rafael I. Nepomechie (2024), q-analog qudit Dicke states, Journal of Physics A: Mathematical and Theoretical in press. https://doi.org/10.1088/1751-8121/ad1ea4
Article Abstract: Dicke states are completely symmetric states of multiple qubits (2-level systems), and qudit Dicke states are their level generalization. We define here deformed qudit Dicke states using the quantum algebra . We show that these states can be compactly expressed as a weighted sum over permutations with factors involving the so-called inversion number, an important permutation statistic in Combinatorics. We use this result to compute the bipartite entanglement entropy of these states. We also discuss the preparation of these states on a quantum computer, and show that introducing a dependence does not change the circuit gate count.
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Introduction + review of qudit Dicke states
Introduction + review of qudit Dicke states
In this article (Reference [1]) on the arxiv we define the q-deformation of the perfectly symmetric qudit Dicke states using the quantum algebra (d). We prove that these states satisfy a simple formula involving the inversion number, an important combinatorial statistic, and leverage this formula to compute the bipartite entanglement entropy and formulate an efficient quantum computing algorithm for preparing these states. Here, we provide the relevant Mathematica code that allowed us to find and check these formulas. We begin with a review of the ordinary qudit Dicke states. Let (,,…,) and , where is a non-negative integer. For a each , we let denote the set of permutations with i’s for . For example, for (1,2,1), the permutation is in .The cardinality of is given by the multinomial
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For each we define a corresponding qudit Dicke state , which we denote by , as the equal superposition of all permutations . So
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For example, the qudit Dicke state with (1,2,1) is .
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(1,2,1)(|0112〉+|1012〉+|0121〉+|1102〉+|0211〉+|1021〉+|1120〉+|1201〉+|2011〉+|1210〉+|2101〉+|2110〉)
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Here , where the computational basis is understood.
|0112〉|0〉⊗|1〉⊗|1〉⊗|2〉
Let us now write code that constructs the qudit Dicke states, with an example:
In[]:=
(*Tensorproductforvectors*)KP[x_,y__]:=Flatten[KroneckerProduct[x,y]](*Generatesetofallpermutationsfromidentitypermutation,whichisthepermutationinweakly-increasingorder*)idperm[k_]:=Map[ConstantArray[#-1,k[[#]]]&,Range[Length[k]]]//FlattenPerm[k_]:=Permutations[idperm[k]](*GenerateforafixedpermutationpthecorrespondingketbyapplyingtensorproductsonthecomputationbasisstatessHat*)sHat[i_,d_]:=Table[KroneckerDelta[i+1,j],{j,1,d}]oneperm[p_,k_]:=KP@@(sHat[#,Length[k]]&/@p)(*Sumoverallpermutationsandnormalize*)DickeSum[k_]:=Total[oneperm[#,k]&/@Perm[k]]DickeSumNorm[k_]:=1/Sqrt[Multinomial@@k]DickeSum[k]
In[]:=
DickeSumNorm[{1,2,1}]
Out[]=
0,0,0,0,0,0,0,0,0,0,0,0,0,0,,0,,0,0,0,0,0,,0,0,0,0,0,0,0,0,0,,0,,0,0,0,,0,0,0,,0,0,0,,0,,0,0,0,0,0,0,0,0,0,,0,0,0,0,0,,0,,0,0,0,0,0,0,0,0,0,0,0,0,0,0
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Each non-zero entry in this vector with a corresponds to one of the 12 permutations in this state. We can show this explicitly by defining a function that, given a permutation p, will output the string p (removing braces, commas, etc.). We then insert these strings into the sum formula (we ignore the trivial entries):
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In[]:=
name[p_]:=StringDelete[TextString[p],{"{",",","}"," "}]Dickename[k_]:=Total[name[#]*oneperm[#,k]&/@Perm[k]]DickeSumname[k_]:=Dickename[k]*DickeSumNorm[k]
In[]:=
DeleteCases[DickeSumname[{1,2,1}],0]
Out[]=
,,,,,,,,,,,
0112
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3
0121
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3
0211
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1012
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1021
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1102
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1120
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1201
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1210
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2011
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2101
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2110
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This will be very useful later, when there is a weighted sum of q-factors.
Defining the q-qudit Dicke states via the suq(d) algebra
Defining the q-qudit Dicke states via the (d) algebra
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We now wish to define a q-analog of the qudit Dicke states, which we call q-qudit Dicke states , for . Being a q-analog, it must reduce to the regular qudit Dicke state upon taking the limit . Therefore, we expect that the q-qudit Dicke states will be a weighted sum over the permutations p with i’s, where the weights are functions (q) that satisfy (q)1. So there is ambiguity how to define the q-qudit Dicke states. Generalizing [2], where the qubit case () was considered, we make use of the (d) quantum algebra. This algebra motivates defining which is a q-analog of . For non-negative integers , we define the q-factorial , and the q-multinomial
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We define the q-qudit Dicke states by the “operator formula” in terms products of lowering operators acting on the state with ’s. For more details, see Section 2 and the definition of the q-qudit Dicke states in Section 3 in the article (Eq. 3.1). We give the code that performs this, with an example for (1,2,1):
|0…0〉
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In[]:=
Com[x_,y_]:=x.y-y.x;id[n_,d_]:=IdentityMatrix[];(*Elementarydxdmatrices*)e[i_,j_,d_]:=Table[KroneckerDelta[i,a]KroneckerDelta[j,b],{a,d},{b,d}](*q-bracket,q-factorial,q-multinomial*)brq[x_,1]:=xbrq[x_,q_]:=(q^x-q^-x)/(q-1/q)facq[0,q_]:=1facq[x_,q_]:=brq[x,q]*facq[x-1,q]multinomq[k_,q_]:=Expand[FullSimplify[facq[Total[k],q]/Product[facq[k[[i]],q],{i,1,Length[k]}]]](*Generatorsofalgebraatasinglequditlocation*)H[i_,d_,q_,1]:=e[i,i,d]-e[i+1,i+1,d];Xp[i_,d_,q_,1]:=e[i,i+1,d];Xm[i_,d_,q_,1]:=e[i+1,i,d];(*Generatorsofalgebraatnquditlocations;theycanbedefinedrecursively*)H[i_,d_,q_,n_]:=KroneckerProduct[H[i,d,q,n-1],id[1,d]]+KroneckerProduct[id[n-1,d],H[i,d,q,1]];Xp[i_,d_,q_,n_]:=KroneckerProduct[MatrixExp[H[i,d,q,1]Log[q]/2],Xp[i,d,q,n-1]]+KroneckerProduct[Xp[i,d,q,1],Fold[KroneckerProduct,ConstantArray[MatrixExp[-H[i,d,q,1]Log[q]/2],n-1]]]Xm[i_,d_,q_,n_]:=KroneckerProduct[MatrixExp[H[i,d,q,1]Log[q]/2],Xm[i,d,q,n-1]]+KroneckerProduct[Xm[i,d,q,1],Fold[KroneckerProduct,ConstantArray[MatrixExp[-H[i,d,q,1]Log[q]/2],n-1]]](*Statewillallzeros*)zerostate[n_,d_]:=Prepend[Table[0,-1],1](*Matrixtonthpowerallowingn=0*)mymatpow[x_,n_]:=If[n==0,IdentityMatrix[Length[x]],MatrixPower[x,n]](*Requirednomalizationfactor*)factorq[k_,q_]:=Product[facq[k[[i]],q],{i,2,Length[k]}](*M[i,d,q,n]foreachiissequenceofnestedcommutators.EachM[i,d,q,n]canbethoughtasloweringthestate0tothestatei*)M[1,d_,q_,n_]:=Xm[1,d,q,n]M[i_,d_,q_,n_]:=Com[Xm[i,d,q,n],M[i-1,d,q,n]](*Finally,wedefinetheq-quditDickestatebyproductsofpowersoftheloweringoperatorsM[i,d,q,n],wherethepowerofM[i,d,q,n]isk[[i+1]]*)DickeOpq[k_,q_]:=FullSimplify[With[{d=Length[k],n=Total[k]},Dot@@Map[mymatpow[M[#-1,d,q,n],k[[#]]]&,Range[2,d]].zerostate[n,d]]/factorq[k,q]]DickeOpNormq[k_,q_]:=1/Sqrt[multinomq[k,q]]DickeOpq[k,q]
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Sum formula for q-qudit Dicke states
Sum formula for q-qudit Dicke states
Guessing the formula
Guessing the formula
We now use Dickename to associate the q-factors in the operator formula with the corresponding permutations p (ignore trivial entries and overall q-multinomial normalization for now):
Generalized sum formula
Generalized sum formula
The normalization is given by the q-multinomial; this is because of the well-known combinatorial identity (see, for example, Eq. (1.68) in [2])
Mathematica allows us to make an educated guess; at this point, a formal proof can be formulated and is provided in Section 3 of the article. We now give code to construct the q-qudit Dicke states by summing over the permutations with corresponding q-factors:
We can check that the sum formula and the operator formula are consistent:
Recursion of q-qudit Dicke states
Recursion of q-qudit Dicke states
The q-qudit Dicke states satisfy the important recursion
Dual symmetry
Dual symmetry
Using the sum formula, it isn’t difficult to see the dual symmetry of the q-qudit Dicke states,
We check this:
This symmetry can be used to improve the algorithm for the preparation of the q-qudit Dicke states on a quantum computer.
Entanglement Entropy
Entanglement Entropy
where the eigenvalues of the reduced density matrix are given by
Then the Von Neumann bipartite entanglement entropy is given by
WARNING: These plots take around an hour to run!
References
References
1
.D. Raveh and R. I. Nepomechie, “q-analog qudit Dicke states,” arXiv:2308.08392 [quant-ph].
https://arxiv.org/trackback/2308.08392
https://arxiv.org/trackback/2308.08392
2
.Z.-H. Li and A.-M. Wang, “Entanglement entropy in quasi-symmetric multi-qubit states,” Int. J. Quant. Inf. 13 no. 02, (2015) 1550007, arXiv:1310.3089 [quant-ph].
3
.R. P. Stanley, Enumerative Combinatorics, vol. 1. Cambridge University Press, 2011. 2nd edition.
4
.V. Popkov, M. Salerno, and G. Schutz, “Entangling power of permutation-invariant quantum states,” Phys. Rev. A 72 no. 3, (Sept., 2005) 032327, arXiv:quant-ph/0506209 [quant-ph].
5
.J. A. Carrasco, F. Finkel, A. Gonzalez-Lopez, M. A. Rodriguez, and P. Tempesta, “Generalized isotropic Lipkin–Meshkov–Glick models: ground state entanglement and quantum entropies,” J. Stat. Mech. 1603 no. 3, (2016) 033114, arXiv:1511.09346 [quant-ph].
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
q -analog qudit Dicke states
by David Raveh and Rafael Nepomechie
Wolfram Community, STAFF PICKS, September 6, 2023
https://community.wolfram.com/groups/-/m/t/3007383
by David Raveh and Rafael Nepomechie
Wolfram Community, STAFF PICKS, September 6, 2023
https://community.wolfram.com/groups/-/m/t/3007383