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International Journal of Quantum Informationc (cid:13)
World Scientific Publishing Company
NON-GAUSSIANITY AND PURITY IN FINITE DIMENSION
MARCO G. GENONI , , and MATTEO G. A. PARIS , , Dipartimento di Fisica dell’Universit`a di Milano, I-20133, Milano, Italia. CNISM, UdR Milano Universit`a, 20133, Milano, Italia ISI Foundation, I-10133, Torino, Italia.
We address truncated states of continuous variable systems and analyze their statisticalproperties numerically by generating random states in finite-dimensional Hilbert spaces.In particular, we focus to the distribution of purity and non-Gaussianity for dimensionup to d = 21. We found that both quantities are distributed around typical valueswith variances that decrease for increasing dimension. Approximate formulas for typicalpurity and non-Gaussianity as a function of the dimension are derived.
1. Introduction
Quantum states of continuous variable (CV) systems with bounded occupationnumber N correspond to finite superpositions, or mixture, of Fock states ̺ N = P Nnk =0 ̺ nk | n ih k | and are usually referred to as truncated states. Truncated statesmay be obtained by heralding techniques from entangled sources 1 , , , d as trun-cated continuous variables states with maximum occupation number N = d − , ,
11 and cloning 12 of quantumstates may be improved. De-Gaussification protocols for single-mode and two-modestates have been indeed proposed 9 , , , ,
14 and realized 15. ovember 16, 2018 11:18 WSPC/INSTRUCTION FILE nonGNum Marco G. Genoni and Matteo G. A. Paris
In this sense, the nonG character of states and operations represents a resourcefor CV quantum information and a question arises on whether the nonG characteris a general feature of truncated states. In order to gain information about nonGproperties of finite-dimensional states we exploit a recently proposed nonG measure16. We then generate uniformly random quantum states for different dimension ofthe truncated Hilbert space 5 and analyze the distribution of nonG and purity ofthese states and their average values. We focus on the dependence on the dimen-sion of the Hilbert space and use our results to draw some conjectures about thebehaviour in higher dimensions.The paper is structured as follows. In the next section we briefly review thegeneration of uniformly distributed quantum states in a finite dimensional Hilbertspace. In Section 3 we review the basic properties of Gaussian states and of themeasure of the nonG character. Then, in Section 4 we evaluate the typical valuesand the distributions of nonG and purity for random quantum states in finite di-mensional Hilbert spaces. Section 5 closes the paper with some concluding remarks.
2. Random quantum states
In order to generate states randomly distributed in a d -dimensional Hilbert weconsider the spectral decomposition ̺ = P d − n =0 λ n P n , P d − n =0 λ n = 1, λ n ≥ P n form a complete set of orthogonal projectors. Therefore, we may view the set ofquantum states as the Cartesian product 5, S = P × ∆ where P denotes the familyof complete sets of orthonormal projectors and where ∆ denotes the simplex, i.e. the subset of the ( d − R N , definedby the trace condition P d − n =0 λ n = 1. This representation of quantum states in d -dimensional Hilbert spaces corresponds to the decomposition ̺ = U DU † , where U denotes a unitary matrix and D a diagonal matrix with trace equal to one.A uniform distribution of density matrices may be thus obtained by choosing theuniform distribution on the group of unitary transformations U ( N ) (Haar measure)and on the set of diagonal matrices D , i.e. the distribution on the simplex. Followingthese lines we have generated random quantum states upon employing an algorithmto generate random U ( N ) matrices according to the Haar measure 17 as well as analgorithm to generate random points on the simplex 5.
3. Gaussian states
In this section we will review the definition and the principal properties of Gaussianstates, by using the quantum optical terminology of modes carrying photons, thoughour theory applies to general bosonic systems. Let us consider a CV systems of n modes described by the mode operators a k , k = 1 . . . n , satisfying the commutationrelations [ a k , a † j ] = δ kj . A quantum state ̺ of the n modes is fully described byits characteristic function 18 χ [ ̺ ]( λ ) = Tr[ ̺ D ( λ )] where D ( λ ) = N nk =1 D k ( λ k )is the n -mode displacement operator, with λ = ( λ , . . . , λ n ) T , λ k ∈ C , andovember 16, 2018 11:18 WSPC/INSTRUCTION FILE nonGNum Purity and non-Gaussianity in truncated Hilbert spaces D k ( λ k ) = exp { λ k a † k − λ ∗ k a k } is the single-mode displacement operator. The canon-ical operators are q k = √ ( a k + a † k ), p k = i √ ( a k − a † k ) with commutation relationsgiven by [ q j , p k ] = iδ jk . Upon introducing the real vector R = ( q , p , . . . , q n , p n ) T ,we define vector of mean values X = X [ ̺ ] and the covariance matrix σ = σ [ ̺ ]as X j = h R j i and σ kj = h{ R k , R j }i − h R j ih R k i , where { A, B } = AB + BA denotes the anti-commutator, and h O i = Tr[ ̺ O ] is the expectation value of theoperator O . A quantum state ̺ G is referred to as a Gaussian state if its char-acteristic function has the Gaussian form χ [ ̺ G ]( Λ ) = exp n − Λ T σ Λ + X T ΩΛ o where Λ is the real vector Λ = (Re λ , Im λ , . . . , Re λ n , Im λ n ) T . Of course, oncethe covariance matrix and the vector of mean values are given, a Gaussian stateis fully determined. For a single-mode system the most general Gaussian statecan be written as τ = D ( α ) S ( ζ ) ν ( n t ) S † ( ζ ) D † ( α ) , D ( α ) being the displacementoperator, S ( ζ ) = exp[ ζ ( a † ) − ζ ∗ a ] the squeezing operator, α, ζ ∈ C , and ν ( n t ) = (1 + n t ) − [ n t / (1 + n t )] a † a a thermal state with n t average number ofphotons. Its matrix elements in the Fock basis are given by 19 h l | τ | m i = K ( l ! m !) / l,m ] X k =0 k ! (cid:18) lk (cid:19)(cid:18) mk (cid:19) ˜ A k ( 12 ˜ B ) ( l − k ) / ( 12 ˜ B ∗ ) ( m − k ) / × H l − k ((2 ˜ B ) − / ˜ C ) H m − k ((2 ˜ B ∗ ) − / ˜ C ∗ ) (1)where ˜ A = A (1 + A ) − | B | (1 + A ) − | B | ˜ B = C (1 + A ) − | B | ˜ C = (1 + A ) C + BC ∗ (1 + A ) − | B | K = [(1 + A ) − | B | ] − / exp (cid:26) − (1 + A ) | C | + [ B ( C ∗ ) + B ∗ C ](1 + A ) − | B | (cid:27) H n ( x ) denotes a Hermite polynomial and A = σ + σ − C = X + iX √ ℜ e[ B ] = σ − σ ℑ m[ B ] = − σ A measure of non-Gaussianity
The non-Gaussian character of a quantum state ̺ may be quantified as the squaredHilbert distance between ̺ and a reference Gaussian state τ , normalized by thepurity of ̺ itself, in formula 16 δ [ ̺ ] = D HS [ ̺, τ ] µ [ ̺ ] (2)where D HS [ ̺, τ ] denotes the Hilbert-Schmidt distance between ̺ and τ , i.e. D HS [ ̺, τ ] = Tr[( ̺ − τ ) ] = ( µ [ ̺ ] + µ [ τ ] − κ [ ̺, τ ]) with µ [ ̺ ] = Tr[ ̺ ] and κ [ ̺, τ ] = Tr[ ̺ τ ] denoting the purity of ̺ and the overlap between ̺ and τ re-spectively. The Gaussian reference τ is chosen as the Gaussian state with thesame covariance matrix σ and the same vector X of ̺ , that is X [ ̺ ] = X [ τ ] andovember 16, 2018 11:18 WSPC/INSTRUCTION FILE nonGNum Marco G. Genoni and Matteo G. A. Paris σ [ ̺ ] = σ [ τ ]. The nonG measure δ [ ̺ ] vanishes iff ̺ is a Gaussian state, it is invariantunder symplectic transformations and have been employed to analyze the evolutionof quantum states undergoing Gaussification and de-Gaussification protocols 16.
4. Non-Gaussianity and purity of random quantum states
We have generated 10 random quantum states ̺ N = P Nnk =0 ̺ nk | n ih k | in finitedimensional subspaces, dim( H ) = N + 1 ( N = { , . . . , } ), following the algo-rithm explained in Section 2. We have evaluated the vector of mean values X andthe covariance matrix σ for each generated state ̺ N , the corresponding referenceGaussian state τ , as well as parameters A , B and C . Then, using Eq. (1) we havereconstructed the density matrix elements of τ , truncating the Hilbert space uponchecking the normalization condition Tr[ τ ] = 1 up to an error of 10 − . We have eval-uated the purity of the state µ [ ̺ ], its nonG δ [ ̺ ] and its symplectic eigenvalue s [ ̺ ].The corresponding average values along with the standard deviations are reportedin Table 1, where we also report the average values and the standard deviations ofthe purity of the reference Gaussian state µ [ τ ] and of the overlap κ [ ̺, τ ]. N µ [ ̺ ] N µ [ τ ] N κ [ ̺, τ ] N δ [ ̺ ] N s [ ̺ ] N ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± As we expected, upon increasing the dimension of the Hilbert space the average pu-rity decreases and the average of the symplectic eigenvalue increases. If we ratherpoint our attention on the average overlap between the random states and theirreference Gaussian states and the average nonG, we observe that κ [ ̺, τ ] N decreaseswhile δ [ ̺ ] N increases. We also notice that, except for the symplectic eigenvalue, thevariances decrease with the dimension. We conclude that all these quantities areconcentrating around typical values. A more accurate analysis has been made onovember 16, 2018 11:18 WSPC/INSTRUCTION FILE nonGNum Purity and non-Gaussianity in truncated Hilbert spaces the distributions of the purity µ [ ̺ ] and nonG δ [ ̺ ]. Μ ð Μ ð Μ ð Μ ð Fig. 1. Histograms corresponding to the distributions of purities of 10 random quantum statesfor different dimensions of the Hilbert space. From left to right: N = 2, N = 5, N = 10, N = 20. In Fig. 1 and in Fig. 2 we show the distributions of purity and nonG of the 10 random quantum states for different dimensions of the Hilbert space. We noticethat both these quantities distribute according to Gaussian-like distributions and,as said before, they concentrate around typical values increasing the dimension. ∆ ð ∆ ð ∆ ð ∆ ð Fig. 2. Histograms corresponding to the distributions of nonG of 10 random quantum statesfor different dimensions of the Hilbert space. From left to right: N = 2, N = 5, N = 10, N = 20. We also analyzed the behaviour of the average values, in particular we have lookedfor fitting functions µ f = µ f ( N ) and δ f = δ f ( N ) able to describe the behaviourof both µ N = µ [ ̺ ] N and δ N = δ [ ̺ ] N as a function of the maximum number of pho-tons N , i.e. varying the dimension of the truncated Hilbert space d = N + 1. Thefollowing fitting functions have been obtained µ f ( N ) = 2 N + 2 δ f ( N ) = − N + 2) c + c (3)wit c = 1 .
560 and c = 0 . i.e. with maximum number of photons N ≫
1, we observe that the typical purity vanishes as 2 /N while the typical nonGapproaches a finite value δ ∞ ≈ c .In order to better understand the relationship between the purity and the nonGof a truncated quantum state, we report the purity and the nonG of the generatedstates as points in the plane ( µ, δ ). Results are shown in Fig. 4 where differentcolors denotes states generated in Hilbert subspaces with different dimensions. Asit is apparent from the plot, the points concentrate, at fixed dimension, in well-defined regions of the plane, whose area decreases for increasing the dimension.Since, as mentioned above, upon increasing the dimension the typical nonG δ N ovember 16, 2018 11:18 WSPC/INSTRUCTION FILE nonGNum Marco G. Genoni and Matteo G. A. Paris N Μ(cid:143) N ∆(cid:143) Fig. 3. (Left) Black points: typical purity as a function of the maximum occupation number N . Blue line: fitting function µ f for the typical purity as a function of N . (Right) Black points:typical nonG as a function of the maximum occupation number N . Blue line: fitting function δ f for the typical nonG as a function of N . increases and the typical purity µ N decreases, we have that higher typical nonGcorresponds to lower typical purities. On the other hand, if we rather focus to pointsfixed dimension, we have that large values of nonG correspond to large values ofpurity, this effect being more pronounced for higher values of N . Μ(cid:143) ∆(cid:143)
Fig. 4. Left: Purity and nonG of random states as points in the plane ( µ, δ ). Different colorscorrespond to different dimension of the Hilbert space. The black points correspond to the average(typical) purity and nonG at each dimension. blue: N = 2. green: N = 5. yellow: N = 10. orange N = 15. red: N = 20. Right: Typical purity and nonG in the plane ( µ, δ ) varying the dimension N = 2 , . . . ,
20. The blue line correspond to the approximate formula reported in the text.
Using Eqs. (3) we may write the relation between the typical nonG and the typicalpurity as δ f ( µ ) = c − (cid:18) µ (cid:19) c . Comparison with numerical findings is reported in the right panel of Fig. 4.
5. Conclusion
In conclusion, we have analyzed the properties of random quantum states generatedin finite dimensional Hilbert spaces in terms of their nonG character and purity. Wehave found that both quantities distribute according to a Gaussian-like distribu-tion with variance that decreases by increasing the dimension, i.e. they concentrateovember 16, 2018 11:18 WSPC/INSTRUCTION FILE nonGNum
Purity and non-Gaussianity in truncated Hilbert spaces around typical values. We also found that the typical nonG and the typical purityare monotone functions of the dimension d . In particular, the average purity de-creases to zero whereas the average nonG increases to an asymptotic value. Besides,we have found that, at fixed dimension, the points corresponding to the randomstates in the plane ( µ, δ ) are confined in well-defined regions whose area decreaseswith the dimension. For increasing dimension higher nonG correspond to higherpurities. Acknowledgments
We thank Konrad Banaszek for several discussions about non-Gaussianity. Thiswork has been partially supported by the CNR-CNISM convention.
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