Phase transitions for random states and a semi-circle law for the partial transpose
aa r X i v : . [ qu a n t - ph ] A p r Phase transitions for random states and a semicircle law for the partial transpose
Guillaume Aubrun, ∗ Stanis law J. Szarek,
2, 3, † and Deping Ye ‡ Institut Camille Jordan, Universit´e Claude Bernard Lyon 1, 69622 Villeurbanne CEDEX, France. Case Western Reserve University, Cleveland, Ohio 44106-7058, USA. Institut de Math´ematiques de Jussieu, Universit´e Pierre et Marie Curie, 75005 Paris, France Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, Canada A1C 5S7
For a system of N identical particles in a random pure state, there is a threshold k = k ( N ) ∼ N/ k particles each typically share entanglement if k > k , and typicallydo not share entanglement if k < k . By “random” we mean here “uniformly distributed on thesphere of the corresponding Hilbert space.” The analogous phase transition for the positive partialtranspose (PPT) property can be described even more precisely. For example, for N qubits the twosubsystems of size k are typically in a PPT state if k < k := N/ − / k > k . Since, for a given state of the entire system, the induced state of a subsystem isgiven by the partial trace, the above facts can be rephrased as properties of random induced states.An important step in the analysis depends on identifying the asymptotic spectral density of thepartial transposes of such random induced states, a result which is interesting in its own right. PACS numbers: 02.40.Ft, 03.65.Db, 03.65.Ud, 03.67.MnKeywords: Quantum states, entanglement, partial transpose, random induced states, Wishart ensemble
INTRODUCTION
If all that we know about a quantum system is its di-mension n (the number of levels) and that it is well iso-lated from the environment, a reasonable model – or atleast a reasonable first guess – for the state of the systemis a unit vector selected at random from the sphere of an n -dimensional complex Hilbert space H . If the systeminteracts with some part of the environment, representedby an ancilla space H a , the quantum formalism suggestsas a model the so-called (random) induced state , obtainedafter partial tracing, over H a , a random pure state on thespace H ⊗ H a . The same description applies if we are pri-marily interested in a subsystem of an isolated system, thesetup that is addressed in the abstract.The above is just one example of how a randomparadigm arises naturally in the quantum context. Inthe last few years probabilistic considerations have be-come a very fruitful approach in quantum informationtheory, the highlights being the fundamental paper [1]by Hayden, Leung and Winter and, more recently, Hast-ings’s proof that suitably chosen random channels providea counterexample to the additivity conjecture for classicalcapacity of quantum channels [2].Although random states have been considered for manyyears, their properties (e.g., are they typically entangled? )remained elusive. In this note we describe a reasonablygeneral way to handle such questions. Of course, the in-duced state ρ being random, we can not expect to be ableto tell what ρ is. However, we may be able to infer someproperties of ρ if they are generic (that is, occur withprobability close to 1). As it turns out, for many naturalproperties a phenomenon of phase transition takes place(at least when dim H is sufficiently large): the generic be- havior of ρ “flips” to the opposite one when s := dim H a changes from being a little smaller than certain thresholddimension s to being larger than s .For simplicity, we will focus on the random inducedstates mentioned at the beginning of the Introduction.This leads (see [3, 4]) to a natural family of probabilitymeasures on D ( H ), the set of states on H , where s , thedimension of the ancilla space, is a parameter. For speci-ficity, consider H = C d ⊗ C d and let us concentrate ontwo properties: entanglement and positive partial trans-pose (PPT). This choice is based, first, on the importanceof these concepts and, second, on the differences in theirrespective mathematical features, which allow to presentthe diverse techniques needed to handle the problems.Concerning the importance aspect, we note that de-tecting and exploiting entanglement – originally discov-ered in the 1930’s [5] – is a central problem in quantum in-formation and quantum computation at least since Shor’swork [6] on integer factoring. Next, the positive partialtranspose is the simplest test for entanglement (Peres–Horodecki PPT criterion, see [7, 8]) and is at the centerof the important distillability conjecture [9], a positive an-swer to which would give a physical/operational meaningto the PPT property. On the other hand, from the com-putational complexity point of view, verifying the PPTproperty is easy (just check whether the partial transposeof the state ρ is positive semi-definite), while decidingwhether ρ is entangled is a computationally intractable(NP-hard) problem [10].In the special case when n := dim H equals s = dim H a ,partial tracing over the ancilla space H a leads to the uni-form distribution on D ( C n ) (i.e., uniform with respect tothe Lebesgue measure, or Hilbert–Schmidt volume, de-noted by vol). More generally, when s > n , the cor-responding probability measure µ n,s has a density withrespect to the Lebesgue measure on D ( C n ) which has asimple form [3] dµ n,s d vol ( ρ ) = 1 Z n,s (det ρ ) s − n , (1)where Z n,s is a normalization factor. Note that (1) de-fines the measure µ n,s (in particular) for every real s > n ,while the partial trace construction makes sense only forinteger values of s . If s < n , the measure µ n,s is con-centrated on the boundary of D ( C n ), but still can bedescribed analytically. Another way to implement thesemeasures is to start from the complex Wishart–Laguerrematrices W n,s ( n × n , with s degrees of freedom) [11], aclassical ensemble in statistics and mathematical physics,and to normalize them to have trace 1.In spite of the explicitness of the formula (1), it isnot easy to find – even approximately, and even for s = n = d – the probability that a random induced statehas PPT or is entangled. This is because these traitsare not encoded in a simple way in the spectral proper-ties of ρ . It was shown in [12] – via methods of high-dimensional probability – that the proportion of states(measured in the sense of µ n,n , i.e., the Legesgue mea-sure) that are un-entangled, or separable , is extremelysmall in large dimensions. This was extended to the casewhen s = dim H a is slightly larger than n = d in [13],while, on the other hand, it was proved in [1] that ran-dom induced states on C d ⊗ C d are typically separablewhen s is proportional to n = d . The paper [12] alsoestablished that un-entangled states are extremely rareeven among PPT states (again, when s = n = d ). How-ever, even such simple question as “ Does the proportionof the PPT states among all states go to as the dimen-sion increases? ,” originally asked in [14], has not beenrigorously addressed prior to the work that we describein this note. The results we summarize go a long wayin filling the gaps in understanding of the phenomena inquestion (see [15, 16] for details and references). We showthat the threshold between entanglement and separabil-ity occurs when s is roughly of order n / = d , and thatthe threshold between NPT (i.e., non-PPT) and PPT iswhen s ∼ n = 4 d .The heuristics behind the consequences stated in theabstract is now as follows. If we have a system of N particles (with D levels each) which is in a random purestate, and two subsystems of k particles each, then the“joint state” of the subsystems is modeled by a randominduced state on C d ⊗ C d with d = D k and s = D N − k .In particular, the relation k = N/
5, or N = 5 k , corre-sponds exactly to s = d . A similar argument applies tothe PPT property. Another consequence of the results is that, for a largerange of parameters, when the ancilla dimension s isroughly between 4 d and d , a generic random state isboth PPT and entangled. Such states are bound entan-gled , or undistillable [9] and, in spite of being entangled,are useless for purposes such as teleportation or super-dense coding (cf. [17]). However, since for small systems(2 ⊗ ⊗
3, see [18, 19]) PPT is equivalent to sep-arability, one is tempted to think that bound entangledstates are an anomaly, and that the PPT property re-mains a good proxy for separability in higher dimensions.Our results imply that this heuristic becomes mislead-ing for large systems and that PPT and separability arequantitatively very different properties.While, as we postulated, random induced states formthe most natural family of probability measures on D ( C n ), our methods are fairly robust and allow handlingof other random models. For example, another popularway to construct random states is to consider mixturesof pure states. Our analysis applies to this model as well:if ν n,s is the distribution of s P si =1 | ψ i ih ψ i | , where ( ψ i )are independent uniform pure states, then all the resultspresented for the measures µ n,s remain valid mutatis mu-tandis for ν n,s . THE RESULTS
We recapitulate the setting: n = dim H , ψ is a (ran-dom) unit vector uniformly distributed on the sphere of H⊗ C s , and ρ = tr C s | ψ ih ψ | is a random state on H whosedistribution is denoted by µ n,s . Further, we assume that n = d > H = C d ⊗ C d ; states on H will be consid-ered entangled, PPT etc., with respect to this particularsplitting. For definiteness, the partial transpose Γ will bethe transposition in the second factor, i.e., defined (by itsaction on product states) via ( τ ⊗ τ ) Γ = τ ⊗ τ T .The first result describes the phase transition betweengeneric entanglement and generic separability. Theorem 1 [16]
There exist effectively computable con-stants
C, c > and a threshold function s = s ( d ) satis-fying cd s ( d ) Cd log d, (2) such that if ρ is a random state on C d ⊗ C d distributedaccording to the measure µ d ,s and if ε > , then (i) for s (1 − ε ) s ( d ) we have P ( ρ is separable) − c ( ε ) d ) and (ii) for s > (1 + ε ) s ( d ) we have P ( ρ is entangled) − c ( ε ) s ) , where c ( ε ) > depends only on ε . Let us mention that our methods extend also to themultipartite setting and to “unbalanced” systems suchas C d ⊗ C d , d = d – see [16] for precise statements.The idea behind the proof of Theorem 1, based on toolsfrom high-dimensional convexity, is quite general and canbe used to estimate thresholds for other properties ofrandom induced states (beyond separability), providedthe set of states with given property is a convex subset K ⊂ D ( H ) and has some minimal invariance properties.However, in the special case of PPT we have a more pre-cise result. Theorem 2 [15]
Let ρ be a random state on C d ⊗ C d distributed according to µ d ,s . Set s ( d ) = 4 d and let ε > . Then (i) for s (1 − ε ) s ( d ) we have P ( ρ is PPT) − c ( ε ) d ) and (ii) for s > (1 + ε ) s ( d ) we have P ( ρ is non-PPT) − c ( ε ) s ) , where c ( ε ) > depends only on ε . As noted in [15], it is likely that the sharp estimatein (i) is of order exp ( − c ( ε ) d ); this conjecture leads tointeresting large deviation problems for matrices ρ Γ .The proof of Theorem 2 (except for the exponential es-timates on the probabilities, which require a unified ap-proach common to both Theorems) depends on methodsof random matrix theory and, specifically, on the follow-ing result that identifies asymptotic spectral density of thepartial transpose of random induced states, and which isof independent interest.If A is a Hermitian matrix, we will denote by λ max ( A )and λ min ( A ) the largest and the smallest eigenvalues of A . If a ∈ R and σ >
0, the semicircular distribution µ SC( a,σ ) is the probability measure with support [ a − σ, a + 2 σ ] and density (2 πσ ) − p σ − ( x − a ) . Wethen have Theorem 3 [15]
Given α > , let ρ d be a random mixedstate on C d ⊗ C d distributed according to µ d , ⌊ αd ⌋ . Then,as d tends to + ∞ , the eigenvalue distributions of ρ Γ d ap-proaches the deterministic measure µ SC(1 , /α ) in the fol-lowing sense: for any interval I ⊂ R , the proportion ofeigenvalues of ρ Γ d inside the rescaled interval d I con-verges (in probability) towards µ SC(1 , /α ) ( I ) .Moreover, we also have convergence of the extremeeigenvalues λ max ( d ρ Γ d ) and λ min ( d ρ Γ d ) to respectively / √ α and − / √ α , the endpoints of the supportof µ SC(1 , /α ) . It is easy to numerically “check” the conclusion of The-orem 3 (this was first noticed in [20]). For example, Fig-ure 1 shows sample distributions of eigenvalues of a par- tially transposed random state on C ⊗ C , when α = 1and α = 4 (sample size 1 in each case). Eigenvalues of r G (x10 −4 )−5 0 5 10 15 Eigenvalues of r G (x10 −4 )−5 0 5 10 15 FIG. 1. Histogram showing distribution of the eigenvalues of ρ Γ , where ρ is a random state on C ⊗ C chosen accordingto the distribution µ , ( α = 1, top) or µ , ( α =4, bottom). In both cases the median eigenvalue is about = 4 × − . Because of the link between random induced states andthe Wishart ensemble W n,s , Theorem 3 holds also for thatensemble (real or complex, although it is the complex set-ting that is most relevant to the quantum theory); in thatcase the rescaling factor d is not needed. We emphasizethat this is rather unexpected since the asymptotic spec-tral density of the Wishart ensemble itself is given by theMarchenko–Pastur distribution [21]. THE MATHEMATICS BEHIND THE RESULTS
Although Theorems 1 and 2 have similar statements,the tools used in their proofs are very different, whichparallels the differences in the computational complexityof PPT vs. that of entanglement. However, combiningall the tools is often necessary to obtain the full strengthof the results.We first describe the proof of Theorem 1, which is ofgeometric nature and where the concept of mean width plays a central role. To present it, let us introduce basicconcepts associated to a convex body K ⊂ R m containingthe origin in the interior (see [22] for more background).The gauge of K is the function k · k K defined for x ∈ R m by k x k K = inf { t > x ∈ tK } . The polar (or dual) body of K is defined as K ◦ = { y ∈ R m : h x, y i ∀ x ∈ K } . If u is a vector from the unit sphere S m − , the support function of K in the direction u is h K ( u ) :=max x ∈ K h x, u i = k u k K ◦ . Note that h K ( u ) + h K ( − u ) isthe distance between the two hyperplanes tangent to K and normal to u . The mean width of K is then defined as w ( K ) := Z S m − h K ( u ) dσ ( u ) = Z S m − k u k K ◦ dσ ( u ) , where dσ is the normalized spherical measure on S m − .In our setting, the relevant convex body is K = S ◦ ,where S is the set of mixed separable states on H = C d ⊗ C d . The ambient space R m is the space of self-adjoint trace 1 operators on H (hence m = d − K ◦ = ( S ◦ ) ◦ = S (the bipolar theorem) and sinceseparability of ρ is equivalent to k ρ k S
1, the crucialquestion is whether w ( S ◦ ) is smaller or larger than 1. Ananalysis of this question leads to the following value ofthe threshold function appearing in Theorem 1 s ( d ) = w ( S ◦ ) . Assertions (i) and (ii) can then be derived from concen-tration of measure, and the heart of the proof is showing(2), especially the upper bound.Determining the threshold s ( d ) requires finding thetypical value of the gauge associated to S , computingwhich – as we mentioned – is a hard problem. We takean indirect route and find the order of magnitude of thethreshold using the machinery of high-dimensional geom-etry, especially the so-called M M ∗ -estimate.The M M ∗ -estimate (see [22, 23]) is a general theoremwhich relates the mean width of a convex body and themean width of its polar. While the abstract formula-tion may require an affine change of coordinates, in thepresent situation, because of the symmetries of S (invari-ance under local unitary conjugations), we can deducevia simple representation theory the inequalities1 w ( S ) w ( S ◦ ) C log d, where C > w ( S ) canbe estimated by standard techniques of high-dimensionalprobability [12], this allows to establish the order of mag-nitude of w ( S ◦ ) (and hence of s ( d )) up to polylog factors. As indicated earlier, the same scheme yields estimatesfor the thresholds corresponding to other properties in-cluding the PPT, but it does not allow to recover theprecise order 4 d appearing in Theorem 2. However,the latter result (except for quantitative estimates on theprobabilities, which require further work, again based onthe concentration of measure) follows readily from The-orem 3, which describes very precisely the spectrum ofthe partial transpose of a random induced state: thePPT condition is equivalent to λ min ( ρ Γ d ) >
0, which isgeneric if 1 − / √ α >
0; similarly, λ min ( ρ Γ d ) < − / √ α < α = sd = 4 is the criticalvalue. In turn, to show Theorem 3 we use the momentmethod, a standard technique from random matrix the-ory. The idea is to identify the asymptotic spectral den-sity by computing its moments. This leads to problems inasymptotic combinatorics: the moments of semicirculardistributions are given by the Catalan numbers, corre-sponding to the dominant combinatorial terms, while thestatements about convergence of extreme eigenvalues areproved by refining the calculations and carefully estimat-ing contributions of lower order combinatorial terms. CONCLUSIONS
We established that random induced states on H = C d ⊗ C d exhibit a phase transition phenomenon with re-spect to the dimension s of the ancilla space. We exempli-fied the phenomenon on two properties: positive partialtranspose, for which the threshold value of s is 4 d , andentanglement, for which the threshold is d (up to a poly-log factor). This allows to determine whether two sub-systems of an isolated system typically share (or typicallydo not share) entanglement when knowing only the sizesof those subsystems, and similarly for the PPT property.In fact, we provide a “black box” approach which appliesto many natural properties of quantum states. Our re-sults motivate further study of the geometry of sets ofquantum states, and that of large deviation behavior ofsome random matrix ensembles related to quantum in-formation theory.We expect the probabilistic methods to continue toplay a major role in quantum theory. Indeed, the latterfield usually involves high-dimensional objetcs; for exam-ple, the quantum analogue of a byte (a state on ( C ) ⊗ – a qubyte, one may say) “lives” in a space of dimension2 − curse of dimension-ality ), randomness is boosted by the presence of manyfree parameters (one may call this phenomenon the bless-ing of dimensionality ). The current level of understand-ing of these aspects of the theory is arguably comparableto that of combinatorics in the 1950’s, when the powerof the probabilistic method [24] began to be appreciatedand, subsequently, the study of random graphs becamean intensive area of research. Acknowledgements : Part of this research was performed dur-ing the fall of 2010 while SJS and DY visited the Fields In-stitute and while GA and SJS visited Institut Mittag-Leffler.The research of GA is supported in part by the
Agence Na-tionale de la Recherche grants ANR-08-BLAN-0311-03 andANR 2011-BS01-008-02. The research of SJS is supportedin part by grants from the
National Science Foundation(U.S.A.) , from the
U.S.-Israel Binational Science Foundation ,and by the second ANR grant listed under GA. The researchof DY has been initiated with support from the Fields In-stitute, the NSERC Discovery Accelerator Supplement Grant ∗ [email protected] † [email protected] ‡ [email protected][1] P. Hayden, D. Leung and A. Winter, Comm. Math. Phys. (2006) 95-117.[2] M. B. Hastings, Nature Physics, (2009) 255-257.[3] K. ˙Zyczkowski and H.-J. Sommers, J. Phys. A (2001)7111-7125.[4] I. Bengtsson and K. ˙Zyczkowski, Geometry of QuantumStates. Cambridge University Press, Cambridge, 2006.[5] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. (1935) 777-780.[6] P. W. Shor, Proceedings of the 35th Annual Symposiumon Foundations of Computer Science, 124-134, IEEEComput. Soc. Press, Los Alamitos, CA, 1994. [7] M. Horodecki, P. Horodecki and R. Horodecki, Phys.Lett. A (1996) 1-8.[8] A. Peres, Phys. Rev. Lett. (1996) 1413-1415.[9] M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev.Lett. (1-2) (1928) 32-52.[12] G. Aubrun and S. J. Szarek, Phys. Rev. A (2006)022109.[13] D. Ye, J. Phys. A: Math. Theor. (2010) 315301 (17pp).[14] K. ˙Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewen-stein, Phys. Rev. A (1998) 883-892.[15] G. Aubrun, Partial transposition of random statesand non-centered semicircular distributions.
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