The modified trace distance of coherence is constant on most pure states
aa r X i v : . [ qu a n t - ph ] A ug The modified trace distance of coherence is constant on most pure states
Nathaniel Johnston,
1, 2
Chi-Kwong Li, and Sarah Plosker
4, 2 Department of Mathematics and Computer Science, Mount Allison University, Sackville, NB, Canada E4L 1E4 Department of Mathematics and Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1 Department of Mathematics, College of William and Mary, Williamsburg, VA, USA 23187 Department of Mathematics & Computer Science, Brandon University, Brandon, MB, Canada R7A 6A9 (Dated: August 24, 2017)Recently, the much-used trace distance of coherence was shown to not be a proper measure of coherence, so amodification of it was proposed. We derive an explicit formula for this modified trace distance of coherence onpure states. Our formula shows that, despite satisfying the axioms of proper coherence measures, it is likely nota good measure to use, since it is maximal (equal to ) on all except for an exponentially-small (in the dimensionof the space) fraction of pure states. PACS numbers: 03.67.Ac, 03.65.Ta, 02.30.Mv, 03.67.Mn
I. INTRODUCTION
There are two key resources that set quantum informationtheory apart from classical information theory: coherence andentanglement. A necessary requirement for entanglement isthat the quantum system can be viewed as multiple quan-tum systems that interact. For over two decades, entangle-ment overshadowed coherence in the literature [1–8]. Now,much interest has percolated with respect to coherence, espe-cially as it pertains to quantum optics [9, 10], quantum biology[11–13], and thermodynamics [14–17], as well as other areas.Multiple quantum systems are not required for coherence, andit arises in any system that cannot be reduced to a classical one(essentially, whenever one deals with superpositions of quan-tum states).Formalizing the task of measuring coherence began in [18],which effectively gave a one-to-one correspondence betweencoherence measures and entanglement measures. Later, aframework [19] of four defining properties for a coherencemeasure to be proper was introduced, with these four proper-ties being seen as highly desirable, or even required, for validcoherence measures. Three of the most commonly used mea-sures of coherence, namely the ℓ -norm of coherence, the rel-ative entropy of coherence, and the robustness of coherence[20], have all been shown to be proper coherence measures.Another measure coherence, called the trace distance of co-herence [21], has also received quite a bit of attention, but wasrecently shown to not be a proper coherence measure in gen-eral [22] (although it is proper when restricted to qubit statesor X states). This led to a “modified” trace distance of coher-ence being proposed in [22], which was shown to indeed bea proper coherence measure, and has been further studied in[23, 24].Despite the modified trace distance of coherence being aproper measure of coherence, we demonstrate that it is notvery useful, since it is equal to its maximal value (i.e., ) forall but an exponentially-small proportion of pure states. Wealso provide several related results along the way, such as anexplicit formula for the modified trace distance of coherenceon pure states, and we show that the closest incoherent state to a pure state in this measure can always be chosen to havejust one non-zero entry. We also demonstrate numerically thatsimilar results likely hold for density matrices with a fixedrank larger than .In Section II, we review preliminary definitions and nota-tion needed for the remainder of the paper. In Section III, wepresent our main results on the modified version of the tracedistance of coherence: we show the non-uniqueness of theincoherent states ˜ δ closest to a given state ρ , we give a for-mula for computing the modified trace distance of coherencefor pure states and describe an optimal ˜ p and ˜ δ , and we showthat the modified trace distance of coherence is equal to —itsmaximum possible value—on all except for an exponentially-small (in the dimension of the space) fraction of pure states.We numerically extend our results to mixed states of higherrank in Section IV, and we provide concluding remarks in Sec-tion V. II. PRELIMINARIES AND THE TRACE DISTANCE OFCOHERENCE
Let I be the set of diagonal density matrices (incoherentstates). For any density matrix ρ , the trace distance of coher-ence is defined as the trace norm distance between ρ and theclosest incoherent state: C tr ( ρ ) def = min δ ∈I k ρ − δ k tr = min δ ∈I n X i =1 | λ i ( ρ − δ ) | , where λ i ( ρ − δ ) are the eigenvalues of the matrix ρ − δ .For a coherence measure C to be a proper measure of co-herence, it must satisfy the following four conditions [19]:(1) C ( ρ ) ≥ , with equality if and only if ρ ∈ I ;(2) C ( ρ ) ≥ C (Λ( ρ )) if Λ is an incoherent operation, i.e.a completely positive trace preserving linear (CPTP)map Λ( ρ ) = P i K i ρK † i whose Kraus operators satisfy K i I K † i ⊂ I ;(3) C ( ρ ) ≥ P j p j C ( ρ j ) where p j = tr ( K j ρK † j ) , ρ j =( K j ρK † j ) /p j , and { K j } is a set of incoherent Kraus op-erators (that is, Kraus operators that satisfy K i I K † i ⊂I ); and(4) P j p j C ( ρ j ) ≥ C ( P j p j ρ j ) for any set of states { ρ j } and any probability distribution { p j } .Items (3) and (4) above (monotonicity under selective mea-surements on average and non-increasing under mixing ofquantum states (convexity), respectively) are equivalent to C ( p ρ ⊕ p ρ ) = p C ( ρ ) + p C ( ρ ) for all block-diagonalstates ρ in the incoherent basis [22]; this equation is morereadily manipulated and was shown to be violated for the tracedistance of coherence (and in particular, condition (3) abovedoes not hold for the trace distance of coherence).In order to address this problem, the following modifiedtrace distance of coherence (originally called “modified tracenorm of coherence”) was proposed [22]: C ′ tr ( ρ ) def = min (cid:8) k ρ − pδ k tr : δ ∈ I , p ∈ [0 , ∞ ) (cid:9) . (1)The advantage of this measure is that it really is a proper co-herence monotone (i.e., it satisfies conditions (1)–(4) above),while still retaining the “spirit” of the trace distance of coher-ence. III. MAIN RESULTS
In the n = 2 case, the modified trace distance of coherencecoincides with the familiar ℓ -norm of coherence : C ℓ ( ρ ) def = X i = j | ρ ij | . While this fact is already known [23], we state it as a the-orem and include an alternate proof below that illustrates thenon-uniqueness of “the” state ˜ δ ∈ I attaining the minimumof C ′ tr for a given ρ . This non-uniqueness is important, as itplays a role in the pure state result that we will prove shortly. Theorem 1. If ρ ∈ M ( C ) then C ′ tr ( ρ ) = C ℓ ( ρ ) = 2 | ρ | .Proof. We compute k ρ − p · diag ( d , d ) k tr = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ρ − pd ρ ρ ρ − pd (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) tr = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ρ ρ − pd ρ − pd ρ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) tr ≥ | ρ | , where the inequality holds with equality if and only if ρ − pd = ρ − pd = µ for some µ with | µ | ≤ | ρ | . So, theoptimal solution set for diag ( dp , dp ) ∈ I equals (cid:8) diag ( ρ − µ, ρ − µ ) : −| ρ | ≤ µ ≤ min { ρ , ρ } (cid:9) . In particular, we can choose µ = 0 to get δ = diag ( ρ , ρ ) so that ρ − diag ( pd , pd ) = ρ E + ρ E , where E ij is an appropriately sized matrix with 1 in the ( i, j ) -th position and zeros elsewhere. The above expression hastrace norm equal to | ρ | , which equals C ℓ ( ρ ) . (cid:3) The above proof shows that the incoherent state attainingthe minimum in the modified trace distance of coherence isvery non-unique in general. For example, instead of choosing µ = 0 like we did in the proof, we could have chosen µ = −| ρ | to get the incoherent state diag ( ρ + | ρ | , ρ + | ρ | ) so that ρ − diag ( pd , pd ) = −| ρ | I + ρ E + ρ E , which also has trace norm equal to | ρ | .Next, we present an explicit method of computing C ′ tr ( | x ih x | ) (i.e., the modified trace distance of coherencewhen restricted to pure states) that is analogous to the methodthat was derived in [25] for the (standard) trace distance ofcoherence. It turns out that the formulas involved for thisversion of the trace distance are actually significantly simplerthan they were for the original version. In particular, we showthat C ′ tr ( | x ih x | ) is simply a function of the largest entry of | x i , which we denote by k| x ik ∞ := max j {| x j |} . Theorem 2.
Suppose | x i ∈ C n is a pure state.a) If k| x ik ∞ ≤ / √ then C ′ tr ( | x ih x | ) = 1 . Further-more, an optimal p in (1) is ˜ p = 0 .b) If k| x ik ∞ > / √ then C ′ tr ( | x ih x | ) = 2 k| x ik ∞ p − k| x ik ∞ . Furthermore, an optimal p and δ in (1) are ˜ p = 2 k| x ik ∞ − and ˜ δ = diag(1 , , , . . . , , respectively. Before proving Theorem 2, we note that C ′ tr ( ρ ) can becomputed numerically via the following semidefinite programfor arbitrary mixed states ρ :minimize: k ρ − D k tr subject to: D diagonal D (cid:23) O. (2)However, Theorem 2 provides a much more explicit way ofdealing with this quantity when ρ = | x ih x | is pure. We alsonote that we only describe “an” optimal p and δ in the state-ment of the theorem (as opposed to “the” optimal p and δ ),since the points attaining the minimum may not be unique, aswe noted after Theorem 1. Proof.
We prove the result by showing that in each of case (a)and (b), C ′ tr ( | x ih x | ) is bounded both above and below bythe indicated quantity. Also, we may assume without lossof generality that | x i = ( x , . . . , x n ) t ∈ R n with x ≥ · · · ≥ x n ≥ . This follows from the fact that the modi-fied trace distance of coherence is invariant under maps of theform | x i 7→ P U | x i , where P is a permutation matrix and U is a diagonal unitary matrix. In particular, this means that k| x ik ∞ = x .In case (a) of the theorem, C ′ tr ( | x ih x | ) ≤ trivially sincewe can choose p = 0 in definition (1). Similarly, in case (b)we can choose p = 2 x − and δ = diag(1 , , , . . . , , as suggested by the theorem, and then we have C ′ tr ( | x ih x | ) ≤k ρ − pδ k tr . To compute k ρ − pδ k tr we note that rank( ρ − pδ ) ≤ , and it is straightforward to verify that its non-zeroeigenvalues are λ ± = (1 − x ) ± x q − x , with corresponding (unnormalized) eigenvectors ~v ± = (cid:0) ± q − x , x , x , . . . , x n (cid:1) t , (3)respectively. Thus k ρ − pδ k tr = | λ + | + | λ − | = 2 x q − x , which establishes the desired C ′ tr ( | x ih x | ) ≤ x p − x in-equality.To obtain the opposite inequalities, we use weak duality ap-plied to the semidefinite program (2). In particular, a routinecalculation shows that the dual of that semidefinite programhas the form maximize: − h x | ( Y + Y ∗ ) | x i subject to: (cid:20) X YY ∗ Z (cid:21) (cid:23) O k X k , k Z k ≤ / Y ) = ~ . (4)Any feasible point that we can find for this dual prob-lem immediately (by weak duality) gives a lower bound on C ′ tr ( | x ih x | ) .In case (a) of the theorem, we can choose X = Z = I/ .To see how we choose Y , first note that there exists a partic-ular pure state | y i with the property that | y j | = | x j | for all j and h x | y i = 0 . To construct such a | y i , we just need to choosethe phases e iθ j of each entry y j of | y i , and we want them tosatisfy h x | y i = n X j =1 e iθ j | x j | = 0 . Well, since | x j | ≤ / for all j , such phases do indeed exist(this is basically just the triangle inequality—we can choose e iθ = 1 and then choose the other phases so as to work ourway back to the origin in the complex plane). Now that we have | y i , we choose Y = ( | y ih y | − | x ih x | ) / .Then (since | y j | = | x j | for all j ) we have diag( Y ) = ~ . Fur-thermore, | x i and | y i are orthogonal, so the eigenvalues of Y are ± / and some zeroes, so (cid:20) I/ YY ∗ I/ (cid:21) (cid:23) O . Thus allof the constraints of the SDP (4) are satisfied, and the corre-sponding objective value is −h x | ( Y + Y ∗ ) | x i = |h x | x i| − |h x | y i| = 1 − , which establishes the desired lower bound C ′ tr ( | x ih x | ) ≥ ,and completes the proof of part (a) of the theorem.On the other hand, in case (b) of the theorem, we can choose X = Z = I/ and Y = ( | v − ih v − |−| v + ih v + | ) / , where | v ± i is the normalization of the vectors ~v ± from Equation (3): | v ± i := 1 √ p − x ~v ± . By construction, Y has eigenvalues ± / and some zeroes,so (cid:20) I/ YY ∗ I/ (cid:21) is indeed positive semidefinite. The only otherconstraint to be checked is that diag( Y ) = ~ , and this followsfrom the fact that the entries of | v − i and | v + i have the sameabsolute values as each other.Thus X, Y, and Z define a feasible point of the SDP (4),and the corresponding objective value is easily computed tobe −h x | ( Y + Y ∗ ) | x i = |h x | v + i| − |h x | v − i| = 2 x q − x , which establishes the desired lower bound C ′ tr ( | x ih x | ) ≥ x p − x and completes the proof. (cid:3) In addition to providing an explicit formula for C ′ tr ( | x ih x | ) , Theorem 2 demonstrates that the modifiedtrace distance of coherence may have some limitations as ameasure of coherence (despite satisfying the requirements forit to be physically relevant presented in [19]), since it onlydepends on a the largest entry of | x i (see Figure 1).Furthermore, C ′ tr ( | x ih x | ) is constant and equal to its max-imal value for a very large proportion of the state space.For example, if | x i = (1 , , , , . . . , / √ and | y i =(1 , , , . . . , / √ n then C ′ tr ( | x ih x | ) = C ′ tr ( | y ih y | ) , whichseems very undesirable. Contrast this with the case of thetrace distance of coherence, the robustness of coherence, therelative entropy of coherence, the the ℓ -norm of coherence,all of which attain their maximum values only at the purestates | x i ∈ C n with | x j | = 1 / √ n for all j .This problem does not present itself in dimension n = 2 ,since every pure state | x i ∈ C has k| x ik ∞ ≥ / √ . How-ever, in higher dimensions, C ′ tr ( | x ih x | ) provides no informa-tion whatsoever on the vast majority of pure states, since con-centration of measure (see [26], for example) says that theproportion of pure states with k| x ik ∞ ≥ / √ decreases ex-ponentially in the dimension, and these are the only pure stateswith C ′ tr ( | x ih x | ) = 1 . The following theorem quantifies thisobservation explicitly. k| x ik ∞ C ′ tr ( | x ih x | ) FIG. 1: C ′ tr ( | x ih x | ) as a function of k| x ik ∞ . Theorem 3.
The proportion (with respect to uniform Haarmeasure) of pure states | x i ∈ C n for which C ′ tr ( | x ih x | ) = 1 is exactly − n/ n − . For example, this theorem says that already in dimension n = 19 , over 99.99% of pure states (chosen according to uni-form Haar measure) have C ′ tr ( | x ih x | ) = 1 (see Figure 2). n proportionFIG. 2: The proportion of pure states | x i ∈ C n with C ′ tr ( | x ih x | ) =1 , for values of n ranging from to . Notice the log scale on the y -axis. Proof of Theorem 3.
One way of generating Haar-uniformpure states | x i ∈ C n is to independently generate n N (0 , -distributed (i.e., normal-distributed with mean and vari-ance ) random variables y , y , . . . , y n , z , z , . . . , z n andthen set [27] | x i = ( y + iz , y + iz , . . . , y n + iz n ) k ( y + iz , y + iz , . . . , y n + iz n ) k . To start, we compute the probability that | x | := p | y | + | z | ≥ / √ , which is equivalent to | y | + | z | ≥ ( | y | + · · · + | y n | ) + ( | z | + · · · + | z n | ) . The sum of the squares of k independent N (0 , -distributedrandom variables follows a chi-squared distribution with k de-grees of freedom (facts like this one are contained in standardmathematical statistics textbooks like [28]), so we are askingexactly for P ( X ≥ Y ) , where X ∼ χ and Y ∼ χ n − .Then P ( X ≥ Y ) = P (cid:18) XY ≥ (cid:19) = P (cid:18) X/ Y / (2 n − ≥ n − (cid:19) . Well, X/ Y/ (2 n − follows an F-distribution with and n − de-grees of freedom, which is known to have probability densityfunction f , n − ( x ) = (cid:18) n − x + n − (cid:19) n . Thus we conclude that P ( | x | ≥ / √
2) = P (cid:18) X/ Y / (2 n − ≥ n − (cid:19) = Z ∞ n − f , n − ( x ) dx = − (cid:18) n − x + n − (cid:19) n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ n − = 1 / n − . Since the n events (cid:8) | x j | ≥ / √ (cid:9) have the same proba-bility regardless of j and are mutually exclusive, we concludethat P (max j {| x j |} ≥ / √
2) = nP ( | x | ≥ / √
2) = n/ n − , so the probability that | x j | ≤ / √ for all j (and thus C ′ tr ( | x ih x | ) = 1 ) is − n/ n − . (cid:3) IV. HIGHER-RANK DENSITY MATRICES
The modified trace distance of coherence behaves more likeone might naïvely expect on general mixed states, but a sim-ilar “almost-constant” phenomenon seems to occur if we fixthe rank of the density matrices (not necessarily equal to )and let the dimension grow. For example, Figure 3 illustrateswhat proportion of density matrices ρ (again, with respect touniform Haar measure [29]) of a given rank in a given dimen-sion have C ′ tr ( ρ ) = 1 .Numerically it seems that approximately 50% of states ρ ∈ M n ( C ) with rank( ρ ) ≤ n/ have C ′ tr ( ρ ) = 1 . Thesenumerics were computed using the YALMIP [30] optimiza-tion package for MATLAB, and our code for computing themodified trace distance of coherence can be downloaded from[31]. rank: n proportionFIG. 3: The proportion of mixed states ρ ∈ M n ( C ) with C ′ tr ( ρ ) =1 , for values of n ranging from to and various small ranks. Eachcurve corresponds to density matrices of a particular rank, which isindicated on the curve itself. The blue curve for rank- states is thesame as in Figure 2, but not on a log scale. V. CONCLUSIONS
We have shown that the modified trace distance of coher-ence is constant and equal to its maximum value of onall except for an exponentially-small ( n/ n − in dimension n ) fraction of pure states, and we have provided numericalevidence that suggests a similar phenomenon occurs fordensity matrices of any fixed rank. It would be interestingto pin down this numerical observation rigorously, but webelieve that the pure state result is enough to suggest thatother measures of coherence, which attain their maximalvalue at essentially a unique state, should be preferred overthe modified trace distance of coherence. Acknowledgements.
N.J. was supported by NSERC Discov-ery Grant number RGPIN-2016-04003. C.-K.L. is an affil-iate member of the Institute for Quantum Computing, Uni-versity of Waterloo. He is an honorary professor of the Uni-versity of Hong Kong and the Shanghai University. His re-search was supported by USA NSF grant DMS 1331021, Si-mons Foundation Grant 351047, and NNSF of China Grant11571220. S.P. was supported by NSERC Discovery Grantnumber 1174582, the Canada Foundation for Innovation, andthe Canada Research Chairs Program. [1] C.H. Bennett, D.P. Di Vincenzo, J. Smolin, and W.K. Wootters,Phys. Rev. A , 3824 (1996).[2] F.G.S.L. Brandão, Phys. Rev. A , 022310 (2005).[3] P. Hayden, M. Horodecki, and B. Terhal, J. Phys. A: Math. Gen. , 6891–6898 (2001).[4] M. Horodecki, Quantum Inf. Comput. , 3 (2001). [5] E.M. Rains, Phys. Rev. A , 173 (1999); 63, 173(E) (1999).[6] A. Shimony, Ann. N.Y. Acad. Sci., , 675–679 (1995).[7] M. Steiner, Phys. Rev. A , 054305 (2003).[8] G. Vidal and R. Tarrach, Phys. Rev. A , 141 (1999).[9] R. J. Glauber, Phys. Rev. , 2766 (1963).[10] E.C.G. Sudarshan, Phys. Rev. Lett. , 277 (1963).[11] S. Lloyd, J. Phys. Conf. Ser. , 012037 (2011).[12] S. Huelga and M. Plenio, Contemp. Phys. , 181 (2013).[13] C.-M. Li, N. Lambert, Y.-N. Chen, G.-Y. Chen, and F. Nori, Sci.Rep. , 885 (2012).[14] V. Narasimhachar and G. Gour, Nat. Commun. , 7689 (2015).[15] M. Lostaglio, D. Jennings, and T. Rudolph, Nat. Commun. ,6383 (2015).[16] P. ´Cwikli´nski, M. Studzi´nski, M. Horodecki, and J. Oppenheim,Phys. Rev. Lett. , 210403 (2015).[17] M. Lostaglio, K. Korzekwa, D. Jennings, and T. Rudolph, Phys.Rev. X , 021001 (2015).[18] J. Åberg, Quantifying superposition (2006), preprint availableat: http://arxiv.org/abs/quant-ph/0612146 [19] T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. , 140401 (2014).[20] C. Napoli, T. R. Bromley, M. Cianciaruso, M. Piani, N. John-ston, and G. Adesso. Phys. Rev. Lett. , 150502 (2016).[21] S. Rana, P. Parashar, and M. Lewenstein, Phys. Rev. A ,012110 (2016).[22] X.-D. Yu, D.-J. Zhang, G. F. Xu, and D. M. Tong, Phys. Rev. A , 060302(R) (2016).[23] B. Chen and S.-M. Fei, Notes on modified trace distance mea-sure of coherence (2017), preprint available at: https://arxiv.org/abs/1703.03265 [24] M.-L. Hu, X. Hu, J.-C. Wang, Y. Peng, Y.-R. Zhang, and H. Fan,
Quantum coherence and quantum correlations (2017), preprintavailable at: https://arxiv.org/abs/1703.01852 [25] J. Chen, S. Grogan, N. Johnston, C.-K. Li, and S. Plosker, Phys.Rev. A , 042313 (2016).[26] P. Hayden, Concentration of measure effects in quantum infor-mation, In Proceedings of Symposia in Applied Mathematics , 2010.[27] M. E. Muller, Comm. Assoc. Comput. Mach. , 19–20 (1959).[28] P. Sahoo, Probability and Mathematical Statistics (2013),available at: [29] More specifically, a rank- r ρ ∈ M n ( C ) is chosen by generat-ing a Haar-uniform | v i ∈ C r ⊗ C n and tracing out the firstsubsystem.[30] J. Löfberg, YALMIP: A Toolbox for Modeling and Optimiza-tion in MATLAB, In Proceedings of the CACSD Conference ,Taipei, Taiwan (2004).[31] Code is available at,Taipei, Taiwan (2004).[31] Code is available at