Negativity vs Entropy in Entanglement Witnessing
James Schneeloch, H Shelton Jacinto, Christopher C. Tison, Paul M. Alsing
NNegativity vs entropy in entanglement witnessing
James Schneeloch, ∗ H Shelton Jacinto, Christopher C. Tison, and Paul M. Alsing Air Force Research Laboratory, Information Directorate, Rome, New York, 13441, USA (Dated: September 16, 2020)In this work, we prove that while all measures of mixedness can be used to witness entanglement,no measure of mixedness is more sensitive than the negativity of the partial transpose. However,computing either the negativity or differences between von Neumann entropies to witness entangle-ment requires complete knowledge of the joint density matrix (and is therefore not practical at highdimension). In light of this, we examine joint vs marginal purities as a witness of entanglement,(which can be obtained directly through interference measurements) and find that comparing puri-ties is actually more sensitive at witnessing entanglement than using von Neumann entropies whilealso providing tight upper and lower bounds to it in the high-entanglement limit.
I. INTRODUCTION
Quantum entanglement is the principal resource con-sumed in many applications of quantum information suchas quantum computing, communication, and enhancedquantum metrology. Understanding its fundamental na-ture goes hand in hand with developing adequate tech-niques to fully characterize it in the exceptionally high-dimensional systems being employed today, such as quan-tum computations on 53-qubit states [1], or in pairs ofparticles entangled in high-dimensional degrees of free-dom [2].In this article, we use the quantum Renyi entropy oforder α to look at measures of mixedness as a hallmarkof quantum entanglement. In doing this, we find thatthe von Neumann entropy (i.e., Renyi for α = 1) is bothless sensitive and requires more resources to witness en-tanglement than measuring the state purity (a functionof the Renyi entropy for α = 2). Moreover, the state pu-rity can be used to bound the value of the von Neumannentropy, which is more valuable in quantum information.Along the way, we discover that when there are no re-source limitations to determining the full quantum state,the negativity of the partial transpose N supersedes allmeasures of mixedness at witnessing entanglement. II. FOUNDATION: ENTANGLEMENT FROMMIXEDNESS AND MAJORIZATION
In classical probability, joint distributions are neverless mixed than the marginal distributions obtained fromthem [3]. In the language of Shannon entropy, the jointentropy is never less than the marginal entropy; two ran-dom variables never take less information to communi-cate than one. However, this need not be the case whencomparing the mixedness of joint and marginal quantumstates.To quantify the mixedness of quantum states, we mea-sure the mixedness of the probability distribution gen- ∗ [email protected] erated by the eigenvalues of the density matrix. This isthe least mixed ensemble of pure states that can consti-tute the state being measured. What makes quantumstates special is that it is possible for the joint state oftwo parties AB to be less mixed than the marginal stateof either A or B . For example, AB can be in a purequantum state such as a Bell state, while the reducedstates of A and B are both maximally mixed. This canonly happen however, if the joint state is entangled [4].Indeed, it was proven in [4] that given a separable state,the joint density matrix of AB cannot be less mixed thaneither that of A or B because the probability eigenvaluesof AB are majorized by those of both A and B . This isknown as the majorization criterion of separability.Majorization is a relation between two probability dis-tributions (or density matrices) in which one can be ob-tained from the other through a series of mixing opera-tions. Given two density matrices ˆ ρ and ˆ σ , we say thatˆ ρ majorizes ˆ σ , (denoted ˆ ρ (cid:31) ˆ σ ) when the sum of the n largest eigenvalues of ˆ ρ is greater than or equal to thesum of the n largest eigenvalues of ˆ σ for all n . Whenˆ ρ (cid:31) ˆ σ , there exists a series of Robin-Hood [5] mixingoperations that will convert ˆ ρ into ˆ σ , but not the otherway around. Although measures of mixedness are well-defined functions over all density matrices, it is possible(and common) for two density matrices to be incompa-rable with respect to each other (i.e., where neither den-sity matrix majorizes the other). This begs the questionwhether there are states whose entanglement cannot bewitnessed by comparing one measure of mixedness, butcan by another, motivating this study. III. QUANTUM STATE PURITY AS ASUPERIOR ENTANGLEMENT WITNESS TOVON NEUMANN ENTROPY
In this section, we examine measures of mixednessbased on the second-order moment of the density ma-trix (i.e., Tr[ˆ ρ ]), in comparison to the von Neumannentropy given as − Tr[ˆ ρ log(ˆ ρ )]. In particular, we showhow comparing the joint and marginal state purities isgenerally more sensitive at witnessing entanglement thancomparing joint and marginal von Neumann entropies, a r X i v : . [ qu a n t - ph ] S e p FIG. 1: (Left to right) Scatter plots and respective purity histograms for 10 D = (2 , , , , , , S ( A | B ) isplotted against the first-order conditional entropy S ( A | B ). The red dotted line is where S ( A | B ) = S ( A | B ), andall points in the lower right quadrant are states whose entanglement is witnessed by S ( A | B ) <
0, but not by S ( A | B ) <
0. (Bottom) This table gives the percentages of the total number of generated states whose entanglementwas witnessed with the entropy function (and the negativity N ) in the first column. Here, we measured thepercentages both by sampling the eigenvalues uniformly on the probability simplex, and on the redistributed set ofeigenvalues obtained by raising them by the normalizing exponent and renormalizing.even though fewer resources are required to experimen-tally determine the state purity. While the von Neumannentropy requires full knowledge of the density matrix (re-quiring state tomography to be performed), state puri-ties can be measured directly by interfering two identicalcopies of the system in an experiment [6, 7]. To facili-tate a side-by side comparison of von Neumann entropyand state purity at witnessing entanglement, we considercomparing the Renyi entropies of order α (given by S α )without loss of generality: S α ( A ) = S α (ˆ ρ A ) ≡ − α log (cid:16) Tr[ˆ ρ αA ] (cid:17) , (1)lim α → S α ( A ) = S ( A ) ≡ − Tr[ˆ ρ A log(ˆ ρ A )] , (2)lim α →∞ S α ( A ) = S ∞ ( A ) = − log (cid:0) max i { λ Ai } (cid:1) , (3) S α ( A | B ) = S α ( AB ) − S α ( B ) . (4)Here we see that S ( A ) is the von Neumann entropy ofsystem A , and S ( A ) is a monotonically decreasing func-tion of Tr[ˆ ρ A ] known as the quantum collision entropy.Ironically, S ∞ ( A ) is known as the min-entropy. We definethe Renyi conditional entropy S α ( A | B ) for convenience.Whenever S α ( A | B ) is negative, the joint state of AB isless mixed than the marginal state of B , witnessing en-tanglement. A. Monte Carlo simulations of random densitymatrices
In order to compare the effectiveness of comparing vonNeumann entropies to comparing state purities as wit-nesses of entanglement, we performed Monte-Carlo simu-lations on 1 million 2-quDit systems, for D = { , , , } .The fair sampling of random density matrices is accom-plished by the algorithm in [8, 9] to generate an ensembleof eigenvalues uniform on the probability simplex (givenby the set of vectors (cid:126)λ such that { λ i } ≥ (cid:80) i λ i = 1).Although uniform on the simplex, this ensemble is bi-ased against pure states at high dimension simply be-cause the fraction of the total hypervolume occupied bystates near the corner vertices (representing nearly purestates) decreases exponentially with dimension [10]. Inorder to better cover the full range of values that thequantum entropy can take, we created a second ensem-ble by raising the generated probability eigenvalues to afixed power (dependent on dimension) and renormaliz-ing to produce a different set of eigenvalues less biasedagainst pure states. Once we have both ensembles of ran-domly sampled diagonal density matrices, we rotate themby taking randomly selected unitary transformations uni-form according to the Haar ensemble. This distributionof unitary transformations is uniform in that the ensem-ble remains invariant under any additional unitary trans-formation. In the table at the bottom of Fig. 1 we showthat the percentages of randomly generated states whoseentanglement is witnessed, either by different measuresof mixedness or by the negativity of the partial trans-pose N , increase when renormalized and dramatically soat higher dimension.In Fig. 1, we show scatter plots of the von Neu-mann conditional entropy S ( A | B ) vs the second-orderRenyi conditional entropy S ( A | B ). In all situations,we find that there are substantially more states where S ( A | B ) < S ( A | B ) > B. Side note: Increased sensitivity when usinghigher-order moments
Using higher-order moments of the density matrix mayyield more sensitive entanglement witnesses than the pu-rity, but at the expense of becoming progressively moredifficult to obtain directly from experiment. In particu-lar, the direct measurement of Tr[ˆ ρ n ] requires interfering n copies of the state ˆ ρ , which becomes impractical as n grows large.That said, it is straightforward to show that for allstates with maximally mixed marginal systems, everystate whose entanglement is witnessed by S α ( A | B ) < α (cid:48) > α . This comes from the fact that theRenyi entropy of order α is a monotonically decreasingfunction of α .As a particularly striking example of how sensitivethese higher-order entropies can be, we consider the caseof the N ⊗ N Werner state, which is a mixture of the Bellstate | Φ (cid:105)(cid:104) Φ | and the maximally mixed state: ρ ( W erner ) AB = p | Φ (cid:105)(cid:104) Φ | + (1 − p ) I N , (5) | Φ (cid:105) ≡ √ N N (cid:88) i =1 | i (cid:105)| i (cid:105) . (6)The probability eigenvalue vectors for the Werner state are: (cid:126)λ ( AB ) = (cid:16) p + 1 − pN , − pN , ..., − pN (cid:17) , (7) (cid:126)λ ( A ) = (cid:126)λ ( B ) = (cid:16) N , ..., N (cid:17) . (8)The entanglement of the Werner state is witnessedwhenever S α ( A | B ) <
0. For constant p , S α ( A | B ) de-creases as α increases; and for constant α , S α ( A | B ) de-creases as p increases. To keep the value of S α ( A | B )constant at increasing α , there must be a correspondingdecrease in p . The threshold Bell state fraction p forwhich S α ( A | B ) = 0 must also decrease as α increases.Clearly for Werner states, higher-order Renyi entropiesmake for more sensitive witnesses of entanglement thanlower-order. Indeed, if one uses S ( A | B ), one findsthat the threshold value of p , ( p c ), does not scale fa-vorably at high dimension N . Instead, p c asymptoti-cally approaches 1 / N → ∞ . On the other hand,using S ( A | B ) scales more favorably, and has an ana-lytic value of p c = 1 / √ N + 1, decreasing toward zerofor large dimension. Going beyond second order, using S ∞ ( A | B ) scales better still, with an analytic value of p c = 1 / ( N + 1), a quadratic improvement over the colli-sion entropy. Even here, the favorability of the scaling isunderstated. Recall that the 53-qubit state has dimen-sion of 2 ≈ . × , and a Werner state of sucha dimension can still have its entanglement witnessedby comparing purities for any Bell state fraction greaterthan 1 . × − . IV. THE SUPREMACY OF THE NEGATIVITY
In 1998, the Horodeckis [11] showed that all states witha positive partial transpose are undistillable, being ei-ther separable or bound-entangled [12]. In 2003, TohyaHiroshima proved [13] that if a joint state ˆ ρ AB is undistil-lable, then it must satisfy the majorization criterion. To-gether, this proves that all states with a positive partialtranspose satisfy the majorization criterion. Therefore,any state violating the majorization criterion must havea negative partial transpose (NPT). In this way, we seethat by computing the negativity (a measure sensitive toNPT states), the entanglement present will be witnessedin at least all states that might have otherwise been wit-nessed by comparing measures of mixedness. That doesnot mean however, that comparing measures of mixed-ness is obsolete.Although the negativity is a tractable measure of en-tanglement (not suffering the NP-hardness [14] of faith-ful entanglement measures) the difficulty in reconstruct-ing a density matrix from experimental data becomes in-tractable at high dimension due to the sheer number of el-ements that a density matrix may contain. Although to-mography is not too challenging for a state made of one ortwo qubits, the number of elements to be determined in-creases exponentially with the number of qubits. Indeed,a 53-qubit state (realizable on state-of-the-art quantumcomputing experiments [1]) has a total of 4 . × density matrix elements, a number so large that it is in-tractable to store, let alone compute with. In particular,4 . × bits is over ten billion zettabytes, exceedingthe world’s estimated data storage capacity by approxi-mately nine orders of magnitude [15]. V. CONCLUSION: MERITS OF DIFFERENTENTANGLEMENT WITNESSES
In our investigations, we examined how well comparingthe mixedness of a joint quantum state to the mixednessof its subsystems witnesses entanglement. While the vonNeumann entropy is a popular measure of mixedness, wefind that even comparing the joint and marginal puri-ties (i.e., Tr[ˆ ρ ] as measured by second-order Renyi en-tropy) witnesses entanglement in more quantum statesthan when using the von Neumann entropy. This is goodnews, as there exist direct measurements of Tr[ˆ ρ ] by in-terfering two copies of a quantum state [6, 16], so that full state tomography is unnecessary. In addition, straight-forward upper and lower bounds exist for the von Neu-mann entropies given a constant state purity. The maxi-mum entropy distribution for constant purity is uniformexcept for one outcome, while the corresponding mini-mum entropy distribution is a discrete top-hat distribu-tion with one non-uniform nonzero outcome (see [17] fordetails). When full state tomography is possible, how-ever, computing the negativity of the partial transposeis more sensitive than comparing any measure of mixed-ness. ACKNOWLEDGMENTS
We gratefully acknowledge support from the Air ForceOffice of Scientific Research LRIR 18RICOR028 andLRIR 18RICOR079, as well as insightful discussions withDr. Michael Fanto, Dr. Ashley Prater-Bennette, and Dr.Richard Birrittella. Any opinions, findings and conclu-sions or recommendations expressed in this material arethose of the author(s) and do not necessarily reflect theviews of the Air Force Research Laboratory. [1] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin,R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A.Buell, et al. , Nature , 505 (2019).[2] J. Schneeloch, C. C. Tison, M. L. Fanto, P. M. Alsing,and G. A. Howland, Nat. Commun , 042338 (2019).[3] In the limit that the joint and marginal probability dis-tributions are equally mixed, the joint distribution isa diagonal distribution with diagonal elements equal tothe marginal probabilities. No joint distribution is lessmixed because already single elements correspond to themarginal probabilities. All other joint distributions whosesums give the marginal distributions are necessarily moremixed because they can be obtained by mixing differentpermutations of the joint distribution.[4] M. A. Nielsen and J. Kempe, Phys. Rev. Lett. , 5184(2001).[5] Robin Hood mixing operations take two unequal ele-ments of a probability distibution and bring them closerto their arithmetic mean, by adding to the smaller ele-ment and subtracting from the larger element an equalamount.[6] A. K. Ekert, C. M. Alves, D. K. L. Oi, M. Horodecki,P. Horodecki, and L. C. Kwek, Phys. Rev. Lett. ,217901 (2002).[7] T. A. Bruni, Quantum Information & Computation ,401 (2004). [8] K. ˙Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewen-stein, Phys. Rev. A , 883 (1998).[9] F. Mezzadri, Notices of the AMS , 592 (2007).[10] The probability of a “nearly pure” state (defined as hav-ing a maximum probability eigenvalue of at least 1 / N -dimensional simplex is N × (1 − N ) , which decreases exponentially toward zerofor large N .[11] M. Horodecki, P. Horodecki, and R. Horodecki, Phys.Rev. Lett. , 5239 (1998).[12] A state ˆ ρ AB is undistillable if there does not exist anyseries of local operations and classical communication(LOCC) that can create pure maximally entangled two-qubit states (known as ebits) from a sufficiently largenumber of copies of ˆ ρ AB .[13] T. Hiroshima, Phys. Rev. Lett. , 057902 (2003).[14] S. Gharibian, Quantum Information & Computation ,343 (2010).[15] M. Shirer and J. Rydning, “IDC’s Global StorageSphereForecast Shows Continued Strong Growth in the World’sInstalled Base of Storage Capacity,” , accessed:2020-08-11.[16] F. A. Bovino, G. Castagnoli, A. Ekert, P. Horodecki,C. M. Alves, and A. V. Sergienko, Phys. Rev. Lett. ,240407 (2005).[17] D. W. Berry and B. C. Sanders, Journal of Physics A:Mathematical and General36