Investigating bound entangled two-qutrit states via the best separable approximation

TL;DR

This paper investigates the properties and distribution of bound entangled two-qutrit states using numerical and theoretical methods, revealing their negligible volume and sporadic occurrence on the separable state surface.

Contribution

It introduces a combined numerical and theoretical approach to analyze bound entangled states in two-qutrit systems, highlighting their rare and island-like distribution.

Findings

BE states have negligible volume in two-qutrit systems.

BE states form tiny islands on the separable state surface.

BE states are located under a layer of undistillable PPT states.

Abstract

We use the linear programming algorithm introduced by Akulin et al. [V. M. Akulin, G. A. Kabatiansky, and A. Mandilara, Phys. Rev. A 92, 042322 (2015)] to perform best separable approximation on two-qutrit random density matrices. We combine the numerical results with theoretical methods in order to generate random representative families of positive partial transposed bound entangled (BE) states and analyze their properties. Our results are disclosing that for the two-qutrit system the BE states have negligible volume and that these form tiny `islands' sporadically distributed over the surface of the polytope of separable states. %We devise a method for estimating numerically the average thickness of these formations and their frequency of occurrence. The detected families of BE states are found to be located under a layer of pseudo one-copy undistillable negative partial transposed states with the latter covering the vast majority of the surface of the separable polytope.

Tables

Table 1. Table 1: BE states detected.

	Weight	Rank of	Min Eigenvalue	Overlap
$R$	$B$	${\hat{\rho }}_{ent}^{\Gamma }$	of ${\hat{\rho }}_{sep}^{\Gamma }$	$⟨{\varphi }_0\left|{\hat{\rho }}_{\mathrm{ent}}^{\Gamma }\right|{\varphi }_0⟩$	$B_C$
$5.16$	$0.0151$	$1$	$0.0033$	$-0.091$	$0.0161$
$5.75$	$0.0014$	$1$	$0.0008$	$-0.235$	$0.0031$
$5.88$	$0.0036$	$1$	$0.0033$	$-0.325$	$0.0089$
$6.11$	$0.0046$	$1$	$0.0068$	$-0.275$	$0.0214$

Table 2. Table 2: Parameters in programs and accuracies achieved

System	two-qubit	two-qutrit
	$\left(N=4\right)$	$\left(N=9\right)$
Number
of input vectors:
$M=\lambda N^4$	$\lambda =15$	$\lambda =4$
Accuracy
of convergence
in the loop:
${ϵ}_C$	${10}^{-15}$	${10}^{-10}$
Accuracy
on the weight:
$\Delta B$	$5\times {10}^{-4}$	$5\times {10}^{-5}$
Accuracy
on eigevalues
of components of BSA:
$\Delta ϵ$	$3\times {10}^{-3}$	$5\times {10}^{-4}$
Secure threshold
for non-vanishing
eigenvalues $T_0$:	$>{10}^{-3}$	$>5\times {10}^{-4}$

Equations

$$D_{HS}\left(\hat{\rho }-{\hat{\rho }}^{\prime }\right)=\sqrt{\mathrm{tr}\left[{\left(\hat{\rho }-{\hat{\rho }}^{\prime }\right)}^2\right]}.$$

$$\hat{\rho }=(1-B){\hat{\rho }}_{\mathrm{sep}}+B{\hat{\rho }}_{\mathrm{ent}}.$$

$${\hat{\rho }}^{\Gamma }=(1-B){\hat{\rho }}_{\mathrm{sep}}^{\Gamma }+B{\hat{\rho }}_{\mathrm{ent}}^{\Gamma },$$

$$(1-B^{\prime }){\hat{\rho }}_{\mathrm{sep}}+B^{\prime }{\hat{\rho }}_{\mathrm{ent}}.$$

$$f\left(\hat{\rho },\beta \right)=(1-\beta ){\hat{\rho }}_{\mathrm{sep}}+\beta {\hat{\rho }}_{\mathrm{ent}},$$

$${\hat{\rho }}_C=(1-B_C){\hat{\rho }}_{\mathrm{sep}}+B_C{\hat{\rho }}_{\mathrm{ent}},$$

$$\hat{\rho }=(1-B){\hat{\rho }}_{\mathrm{sep}}+B|{\psi }_{ent}⟩⟨{\psi }_{ent}|.$$

$${\hat{\rho }}_c=(1-B_c){\hat{\rho }}_{\mathrm{sep}}+B_c{\hat{\rho }}_{\mathrm{ent}},$$

$$|a_0⟩=$$

$$|\psi ⟩=\sum _i^{N_K}\sqrt{{\alpha }_i}|b_i⟩|c_i⟩.$$

$${\hat{\rho }}_{\mathrm{ent}}=\sum _{ij}^{N_K}\sqrt{{\alpha }_i{\alpha }_j}|b_i⟩⟨b_j|\otimes |c_i⟩⟨c_j|.$$

$${\hat{\rho }}_{\mathrm{ent}}^{\Gamma }=\sum _{ij}^{N_K}\sqrt{{\alpha }_i{\alpha }_j}|b_i⟩⟨b_j|\otimes |c_j⟩⟨c_i|.$$

$$(1-B)⟨\varphi |{\hat{\rho }}_{\mathrm{sep}}^{\Gamma }|\varphi ⟩+B⟨\varphi |{\hat{\rho }}_{\mathrm{ent}}^{\Gamma }|\varphi ⟩<0.$$

$$(1-B)ϵ-B\gamma <0$$

$$⟨\varphi |{\hat{\rho }}_{\mathrm{sep}}^{\Gamma }|\varphi ⟩$$

$$⟨\varphi |{\hat{\rho }}_{\mathrm{ent}}^{\Gamma }|\varphi ⟩$$

Full text

††preprint: APS/123-QED

I Introduction

Best separable approximation (BSA) Sanpera provides a convex decomposition of a given mixed quantum state which fully unveils its entanglement properties Iha . More recently, an algorithm has been devised essent for numerically efficiently achieving this decomposition and in this paper we employ it in order to study the entanglement characteristics of random ensembles of two-qutrit states. The main focus of our studies is on states which are entangled but with positive partial transposed (PPT) counterparts Horodeckis98 ; Peres96 ; Horodeckis96 and in consequence undistillable/bound BennettBrassard96 ; Horodeckis97 ; Rains1999a ; Rains1999b ; DuerBruss2000 ; Rains2001 . In this work we signify the PPT bound entangled states simply as bound entangled (BE) states. Throughout the years, many important results have come to shed light on the intriguing class of BE states and by now many different classes of BE states Horodeckis98 ; BennettMor99 ; DiVincenzoMor2003 ; HuberLami18 ; Sentis2016 ; Oh ; Piani have been identified, the robustness of the classes has been proven Ghosh , tests for their detection Jens have been proposed, their non-local character Brunner , steerability properties Moroder and roles as activators activ have been revealed. In addition to the studies on BE states, there are efforts for identifying undistillable/bound entangled states with negative partial transposed (NPT) counterpart Horodecki0 ; DiVincenzo2000 ; Cirac ; Watrous ; Clarisse . For this case though undistillability not stemming from the PPT property, needs to be checked for an infinite number of copies of the state. The difficulty of the latter task is vast and yet there is no safe conclusion on the existence of such states. In this paper, we also approach this subject by studying numerically one-copy undistillable (POCU) NPT states.

The aim of this work is to add more knowledge on the BE class of states. Using BSA and some straightforward theoretical tools we first propose methods for creating random representative one-parametric families of BE. These methods permit us to generate a number of random families for two-qutrit system and draw conclusions on their characteristics. As expected, these families are found to be deposed on the surface of separable polytope and are parts of bigger ‘island’ formations. We provide some estimates on the average depth of these formations as well as on their frequency of appearance on the surface which are in agreement with previous results Jens . We undertake a similar procedure for generating families of POCU states and we conclude that these form a layer over the surface of separable polytope that covers up the ‘islands’ of BE states. From our studies we also conclude that the volume of BE states is finite but negligible for the two-qutrit systems and that the PPT criterion for entanglement detection is accurate in very high degree.

The structure of the paper is the following. In Secs. II and III our means for theoretical and numerical analysis are introduced: we revise BSA, derive some useful related lemmas for generating random families of BE states, and give details on the sampling methods for density matrices and on the algorithm for BSA analysis. In Sec. IV we present numerical results on two-qubits states which form a ground of comparison for higher dimensional systems. The main numerical outcomes concern two-qutrit states and these are found in Sec. V. In the last section, Sec. VI, we discuss questions emerging from the theoretical methods and numerical results of the current paper.

II Families of BE states emerging from BSA

BSA was introduced in Sanpera and its uniqueness was proven using the convex property of density matrices. In essent the representation was re-introduced underlying its geometric aspects which in turn lead to an efficient algorithm for its numerical realization. Here we follow the notation and notions introduced in essent – which are in consistency with those in Sanpera .

We start by providing some elements on the geometric aspects of mixed states which help in introducing BSA. Let us adopt the usual Bloch-sphere representation and consider pure states as unit vectors on its surface. For Hilbert space dimension $N>2$, physical states occupy only a sub-manifold of the surface of the hyper-Bloch sphere essent and any convex combination of these, lies inside this sub-manifold forming the convex hull body of mixed states. Separable states are states which can be written as convex combinations of pure product states and thus the separable convex set of states forms a separable polytope inside the convex hull body. States with sufficiently small length of Bloch vector are necessarily inside the volume1 ; Aubrun ; Gurvits and thus separable. The length of the Bloch vector of a density matrix is related to its purity $\mathrm{Tr}\rho^{2}$ but in this paper, we find it more convenient use the inverse quantity: the participation ratio $R=1/\mathrm{Tr}\rho^{2}$ volume1 with maximum value $N$ and minimum value $1$ for pure states. Also, for our analysis we are in need of a measure of distance between two density matrices $\hat{\rho}$ and $\hat{\rho}^{\prime}$ and for this we employ the measure of Hilbert-Schmidt distance:

[TABLE]

Given a density matrix $\widehat{\rho}$ describing a mixed multipartite quantum state, BSA is a unique convex decomposition over the separable polytope and set of entangled states ($=$ convex hull body $/$separable polytope):

[TABLE]

In (2) $\widehat{\rho}_{\mathrm{sep}}$ is, what we call in this work, the separable component, $\widehat{\rho}_{\mathrm{ent}}$ the essentially entangled part which cannot have any separable states as components, $B$ is a positive number in the range $\left[0,1\right]$ and $\mathrm{Tr}\left[\widehat{\rho}_{\mathrm{sep}}\right]=\mathrm{Tr}\left[\widehat{\rho}_{\mathrm{ent}}\right]=1$.

The uniqueness of BSA (2) is justified from the fact that among all possible convex decompositions of a state $\widehat{\rho}$ over the convex hull body of entangled states and convex polytope separable states, the positive number $B$ attains its minimum value. Since the decomposition is a result of extremization, both $\widehat{\rho}_{\mathrm{sep}}$ and $\widehat{\rho}_{\mathrm{ent}}$ are states on the boundaries of their representative sets. The essentially entangled component is on the surface of the convex hull body separating positive Hermitian matrices from non-positive ones and therefore of reduced rank. The rank theorem provides upper bounds on the rank of the essentially entangled component. This has been derived in Sanpera for the bipartite case and extended in essent for the general case: The maximum rank $d_{E\max}$ of an essentially entangled component $\widehat{\rho}_{\mathrm{ent}}$ for a system of dimension $N$ composed by $K$ subsystems each of them of dimension $N_{k}$, is $N-\sum_{k=1}^{K}N_{k}+K-1$. A direct consequence of this theorem is that for $N=4$ and $N=6$ the rank of $\widehat{\rho}_{\mathrm{ent}}$ is one.

Respectively, the separable component lies on the surface of the polytope of separable density matrices, separating separable from entangled states. Since both entangled and separable states are positive matrices, the rank of the separable component is not reduced in the general case. On the other hand under the PPT operation $\widehat{\rho}_{\mathrm{sep}}$ is often mapped on the borders of positive with non-positive Hermitian matrices and therefore its rank is reduced. As we explain further in this session this phenomenon always holds true when $N=4$ or $6$.

Let us list here some straightforward statements about the decomposition (2). For this purpose we apply the operation of partial transposition to left and right-hand sides of (2), obtaining the partial transposed version of it:

[TABLE]

where $\widehat{\rho}_{\mathrm{sep}}^{\Gamma}$ necessarily a separable state and $\widehat{\rho}_{\mathrm{ent}}^{\Gamma}$ a Hermitian operator that is non-positive $\widehat{\rho}_{\mathrm{ent}}^{\Gamma}\ngeq 0$ when $\widehat{\rho}_{\mathrm{ent}}$ is of rank one (see Appendix A). There we also conjecture that this property holds true in bipartite systems even if $\widehat{\rho}_{\mathrm{ent}}$ is of rank higher than one.

A state $\widehat{\rho}$ is separable iff $B=0$ in (2). 2. 2.

A state $\widehat{\rho}$ is entangled iff $B>0$ in (2). 3. 3.

A state $\widehat{\rho}$ is a PPT state iff $\widehat{\rho}^{\Gamma}\geq 0$ in (3). 4. 4.

A state $\widehat{\rho}$ is a NPT state iff $\widehat{\rho}^{\Gamma}\ngeq 0$ in (3). 5. 5.

A state $\widehat{\rho}$ is a BE state iff $\widehat{\rho}^{\Gamma}\geq 0$ in (3) and $B>0$ in (2). 6. 6.

Given the BSA (2) for a density matrix $\widehat{\rho}$ and with $B\neq 0$ then

[TABLE]

with $1\geq B^{\prime}>0$ and $B^{\prime}\neq B$ is the BSA for another density matrix $\widehat{\rho^{\prime}}$, the one equal to (4).

The last statement, a simple consequence of convexity of the sets involved, permits us to start with an entangled state $\widehat{\rho}$ with known BSA (2) as generator/seed, and create a one-parametric family of entangled states

[TABLE]

with the parameter $\beta$ taking values in the range $\left[0,1\right]$.

If in (5) one employs as seed a BE state then a family of BE states can be created for $\beta:\left[0,B_{C}\right]$ where $B_{C}$ is a critical value of the weight which can be identified by partially transposing (5) and setting this to zero. This critical weight defines a state $\widehat{\rho}_{C}$

[TABLE]

on the boundary of the BE family with NPT states. Then simply $\delta=D_{HS}\left(\widehat{\rho}_{C}-\widehat{\rho}_{\mathrm{sep}}\right)$ quantifies the ‘depth’ of this BE family. We proceed with two Lemmas which provide an alternative way for constructing families of BE states using as seeds NPT states with specific properties.

• •

Lemma 1. An NPT entangled state (2) with $\widehat{\rho}_{\mathrm{sep}}^{\Gamma}$ in (3) being of full rank, gives rise to a family of BE states $f\left(\widehat{\rho},B\rightarrow 0\right)$.

• •

Lemma 2. An NPT entangled state (2) with $\widehat{\rho}_{\mathrm{sep}}^{\Gamma}$ in (3) being of reduced rank, gives rise to a family of BE states $f\left(\widehat{\rho},B\rightarrow 0\right)$ if there is no eigenvector $\left|\phi\right\rangle$ in the null eigenvalues subspace of $\widehat{\rho}_{\mathrm{sep}}^{\Gamma}$ such that $\left\langle\phi\right|\widehat{\rho}_{\mathrm{ent}}^{\Gamma}\left|\phi\right\rangle\ngeq 0$.

The proofs of the Lemmas can be found in Appendix B. A direct consequence of Peres-Hordecki Peres96 ; Horodeckis96 criterion and Lemma 1 is that for $N=4$ and $6$ the $\widehat{\rho}_{\mathrm{sep}}^{\Gamma}$ in (3) is of reduced rank and therefore this lies on the borders between separable states and non-positive Hermitian operators. In addition, for such dimensions there is no NPT states satisfying Lemma 2. We numerically confirm these statements in Section IV.

In Sec. V we employ the Lemmas $1$ and $2$ to identify NPT states which can serve as seeds of one-parametric families of BE states $f\left(\widehat{\rho},\beta\right)$ for $\beta:\left[0,B_{C}\right]$. The families of BE states created this way can be considered as representative ones and analysis on them provides estimates on the average characteristics of BE formations.

III On the algorithm performing BSA and sampling methods

In this paper we perform numerical analysis and it is important to give information on the algorithm performing BSA, as well as, on the sampling methods for the density matrices.

Concerning the algorithm, we closely follow the procedure prescribed in essent achieving an accuracy $\Delta B$ of the order $10^{-4}$ on the estimation of the extremum weight $B$ in (2) for two-qutrit states. This accuracy signifies that if a state is analyzed by the algorithm more than once, the deviation of the results is of this order or less. In essent is prescribed that the number of random vectors sampling the convex space at each step of the method should be of the order $N^{4}$ where $N$ the dimension of the Hilbert space of the combined system. We have found out that this number should be increased by a factor $\lambda$ for an accuracy consistent with the desired accuracy. The BSA algorithm in addition to $B$ provides the two density matrices components (2) and the accuracy $\Delta\epsilon$ in their spectrum is critical for the analysis performed in this paper. By analyzing many times several states and also with the use of the rank theorem, we draw conclusions on $\Delta\epsilon$ and most importantly we provide sufficient thresholds $T_{0}$ for non-vanishing eigenvalues. Our conclusions are summarized in the Table 2 of Appendix C where we also provide more details on the programs. Finally in our analysis we have excluded states with one or more eigenvalue less than $10^{-4}$ since the algorithm is applicable only to full rank states. For these cases the algorithm suggested in Love appears as good alternative to our method –especially when $\widehat{\rho}_{\mathrm{ent}}$ is of rank one.

While for generating random pure states the generally admitted Haar measure exists, sampling mixed states is an intriguing subject with different options Karol1 ; Karol2 and ongoing research. In this paper we have employed two independent methods for sampling our space since the question on the dependence of the volume of BE states on the measure is also a question of potential interest (see Slater1 and references therein for more intense studies on this subject). Our first choice is the flat measure –as we call it in this paper, which has been used in the very first works on the volume of separable volume1 and BE states volume2 . More specifically, the eigenvalues and eigenvectors of a density matrix are treated as two different sets; the first are sampled uniformly over its geometric space (simplex) and the second use unitary projectors uniformly sampled over the Haar Measure. The first step is straightforward using the instructions in the Appendix of volume1 , for the second step we have used random unitary matrices produced according to the procedure Karol3 and relevant programs Poland . The second sampling method that we use is the induced measure Karol1 which has gained a lot of attention the last years. This measure follows the more natural procedure using ancillary systems and tracing out degrees of freedom of pure random states of higher dimensions. The dimension of the ancillary system defines the measure and metric of the produced random ensemble. If the dimension is the same as for the system one obtains a Hilbert-Schmidt ensemble, while if the dimension is higher one obtains induced measure ensembles (with Bures ensemble as a subcase). We have found out that each of these ensembles is circumscribed normally distributed around a certain degree of participation ratio $R$, so in order to obtain a distribution of states which covers the full range of $R$ we use different dimensions for the ancillary system and also a method suggested in Karol1 that includes projection onto maximally entangled states for the ancillary system (instead of tracing out). For our programs we have used Ginibre matrices generated by the Mathematica package Poland . In Appendix D we present graphically distributions of two-qubit and three-qubit density matrices according to flat and (combined) induced measure.

IV BSA on two-qubit states

According to the Peres-Horodecki criterion Peres96 ; Horodeckis96 , two-qubit systems cannot accommodate BE states. We think though that it is worth analyzing this case since the results provide information on the BSA properties for this system which has not been reported elsewhere. We perform BSA on $1200$ randomly sampled NPT density matrices, $600$ according to flat measure and $600$ according to induced measure and in Fig. 1 we summarize the joint outcomes.

The rank theorem Sanpera applied on a two-qubit system, dictates that $\widehat{\rho}_{\mathrm{ent}}$ is necessarily of rank one and therefore (2) attains the simpler form:

[TABLE]

Our numerical results confirm Eq. (7) and as it would be expected, the weight $B$ in Fig 1 (a) in average drops with the decrease of purity. For $R>3$ confirming previous results volume1 ; Gurvits , no entangled states are detected. Interestingly in Fig 1(b) we observe that the concurrence concu of $\left|\psi_{ent}\right\rangle$ in (7), on average is increasing with $R$, and that this is a maximally entangled state for a big fraction of density matrices ($\approx 72\%$ for our data). In the Appendix E we show how the latter observation can be employed for enhancing the distillation procedure of two-qubit NPT states.

Confirming Peres-Horodecki criterion we see in Fig. 1 (c)-(d), that none of the conditions of Lemmas $1-2$, which could lead to families of bound states from NPT states are fulfilled. More specifically, $\rho_{sep}^{\Gamma}$ is of reduced rank (below the threshold $T_{0}$, see Appendix C) and the overlap of the null eigenvector $\left|\phi_{0}\right\rangle$ of $\rho_{sep}^{\Gamma}$ with the $\rho_{ent}^{\Gamma}=\left|\psi_{ent}\right\rangle\left\langle\psi_{ent}\right|$ always negative. Finally, we have not observed any differentiation on the results between flat and induced measure sampling.

V BSA on two-qutrit states

We start by performing BSA on PPT states in order to obtain an estimate on the BE states’ volume as compared to the NPT’s and separable ones’. We have tested $30000$ random states (among which $3392$ PPT) sampled according to flat measure and identified only $4$ BE states. The results are summarized in Fig. 2 (a), where it becomes obvious that the volume of BE states is negligible (but finite in accordance with Aubrun ), below the $0.1$% of total volume. In Fig. 2 (b) we present the corresponding graph for the three-qubit system testing $5000$ random states sampled over flat measure. Fig. 2 (b) reproduces Fig. $7$ in volume2 with more accurate methods and gives a volume of BE states approximately $2$% of the total volume. This comparative study on composite systems of similar dimensions confirms the known fact Aubrun that the tensor product structure of the Hilbert space does matter for bound entanglement.

In Table $I$ we list the properties of the few identified BE states: $R$, $B$ weight, the rank of $\widehat{\rho}_{\mathrm{ent}}$, the minimum eigenvalue of the partial transposed separable component $\widehat{\rho}_{sep}^{\Gamma}$ (see Lemma 1), and the overlap of the minimum eigenvalue eigenvector $\left|\phi_{0}\right\rangle$ of $\widehat{\rho}_{sep}^{\Gamma}$ with $\widehat{\rho}_{ent}^{\Gamma}$ (see Lemma 2). We also employ BE states as seeds for BE families and calculate the critical weight $B_{C}$ (6) of the latter.

We proceed by performing BSA on NPT states. This a time-consuming numerical task and we have analyzed via BSA $600$ random NPT states –$300$ for flat and $300$ for the induced measure. We present the results in Fig. 3 and we observe that as for the two-qubit systems (see Fig. 1), the weight $B$ in average drops with participation ratio. In Fig. 3 (b) the distribution on rank of the essentially entangled component, $\widehat{\rho}_{ent}$ of the random states is presented and as predicted by the rank theorem this does not exceed the value $5$.

In Fig. 3 (c), we now see that NPT states appear above the secure threshold $T_{0}$ for non-vanishing eigenvalues (see Appendix C) thus fulfilling the criteria of Lemma $1$ with certainty. These states can serve as seed states able to generate BE families of states. Similarly we detect some states fulfilling the criteria of Lemma $2$. Finally, few of the states fulfill the criteria of both Lemmas. In overall, we have detected $40$ NPT states which can generate BE families and since the threshold $T_{0}$ is sufficient but not necessary for their detection, we may conclude that the surface of separable polytope is covered by BE states at a considerable rate $>10$%. On the other hand, from our results we have not been able to draw safe conclusions on the differentiation of results due to different samplings of the states under analysis.

In the next step, we use the detected seed NPT states to generate BE families and estimate the corresponding critical weight $B_{C}$ (6) and depth $\delta$. The results are summarized in Fig. 4 (a) and (c). Combining the rate of occurrence of NPT seed states with the average depth of the families we confirm the results of analysis on PPT states; the volume of BE states does not exceed $1/1000$ of the total volume of states.

The identified BE families $f$ are not isolated and these are part of bigger formations, ‘islands’ of BE states which are deposed on the surface of the separable polytope, see Fig. 4 (d). This phenomenon is the subject of Ghosh and more recently jagged islands of BE states have been constructed in Slater2 . From the point of view of BSA (2) it is evident that infinitesimal deviations on a seed NPT state $\widehat{\rho}$ will give infinitesimal deviations to its components and in turn BE families in the neighborhood of the initial one.

V.1 Generating families of POCU states

We proceed with a closely related matter to PPT bound entanglement, the one of undistillability of NPT states, and we examine whether random families of POCU states can be constructed in a similar fashion as for BE states. The answer is positive and the numerical analysis gives evidence that in their vast majority ($100\%$ of tested cases) NPT states generate one-parametric families of POCU states as $f\left(\widehat{\rho},B\rightarrow 0\right)$. To estimate the depth $\sigma$ of these POCU families we follow similar steps as for the BE families: we start with the BSA of a random NPT state $\widehat{\rho}$ (2) and we decrease $B$ in (5) until we identify numerically the critical weight $B_{c}$ such that $\widehat{\rho}_{c}$

[TABLE]

is a POCU state. The states $f\left(\widehat{\rho},B\right)$, with $0<B<B_{c}$ comprise a family of POCU states. Then $\sigma=D_{HS}\left(\widehat{\rho}_{c}-\widehat{\rho}_{\mathrm{sep}}\right)$ provides the depth of the POCU family. In the Fig. 4 (b) we present the distribution of $\sigma$ for $200$ POCU families which have been created using as generators random $NPT$ states sampled according to flat measure.

BE states are undistillable for any number of copies and further analysis on the BE families constructed in the previous section shows the expected phenomenon: BE families are subsets of POCU families. To numerically prove this, we use the detected NPT seed states for BE families, as generators for POCU families and we compare the critical weights $B_{C}$ and $B_{c}$. The (sorted) results are presented graphically in Fig. 4 (c) where one may observe that $B_{C}<B_{c}$ holds in all cases under test. In a similar way, $\delta<\sigma$ for all tested NPT seed states.

Finally a few words about the algorithm that we employ to check numerically the one-copy undistillability of states. This relies on the Lemma 2 of DiVincenzo2000 and according to it, it is sufficient to check whether projections $\widehat{\rho}_{2\otimes 3}=P_{A}\otimes 1_{B}\widehat{\rho}P_{A}^{\dagger}\otimes 1_{B}$ on the density matrix result in $\widehat{\rho}_{2\otimes 3}^{\Gamma}\ngeq 0$. If the latter holds true we conclude that the state is distillable while if all random ($50.000$) projections that we try out give $\widehat{\rho}_{2\otimes 3}^{\Gamma}>0$ we characterize the state as POCU state. To parametrize the component $\left|a_{0}\right\rangle$ of the projector $P_{A}=\left|a_{0}\right\rangle\left\langle a_{0}\right|+\left|a_{1}\right\rangle\left\langle a_{1}\right|$ with $\left\langle a_{0}\right|\left.a_{1}\right\rangle=0$ we use the convenient representation from Klimov for a qutrit state in terms of angles,

[TABLE]

which permits us to have also an analytic expression for $\left|a_{1}\right\rangle$. The random search on the projectors is performed as random search on the angles of $\left|a_{0}\right\rangle$ (9) and on the free angle parameters of $\left|a_{1}\right\rangle$.

VI Discussion

Summarizing the main analytical and numerical findings of this paper, one first useful conclusion is that random one-parametric BE families can be generated at an efficient rate by performing BSA on NPT states. Applying the aforementioned technique we have been able to create a number of such families and draw some preliminary conclusions on their properties and their placement in the convex space of density matrices; the families lie on the surface of the separable polytope without much penetrating the NPT volume. Then we have proposed a similar procedure for generating one-parametric families of POCU NPT states and we have observed that the latter form a layer over the surface of the separable polytope and covering up the BE formations. Finally by applying BSA on a considerable number of PPT states we have seen that BE states for two-qutrit systems are so rare that PPT criterion can be applied with confidence of more than $99,9\%$. For our numerical studies we have employed two independent sampling methods and our generic conclusions seem to hold for both methods. However in order to draw decisive conclusions on the sampling matter as well as for providing precise quantitative estimations on quantities under examination in this paper (such as $\delta$, $\sigma$), a much higher number of BSA tests is necessary.

A straightforward extension of this paper would be to perform BSA analysis on higher-dimensional bipartite systems and track the rate of growth of the BE states’ volume. The linear programming algorithm essent used in this work scales as $N^{12}$ where $N$ the dimensional of the total Hilbert space, and dimensions up to $N=16$ are still tractable if high computational power is available. Another possibility for reaching higher dimensions is to use the current programs to conclude on the BE detection ability of more computationally accessible methods, such as covariance matrix criteria Eisert , and proceed with the latter.

On the theoretical level a finding of this paper deserving further investigation, is that both (PPT) BE and (NPT) POCU states are correlated with a BSA weight $B$ of the same (low) order of magnitude. This phenomenon gives the suggestion that undistillability of a state might be related not only with its character under partial transposition but also with its BSA geometric properties. Along the same line, the example in the Appendix E gives evidence that BSA can provide useful knowledge to a distillation process and we believe that more results can be discovered on this subject. Finally, in this work we use the Peres-Horodecki criterion to explain the fact that for $N=4$ and $6$ partial transposition always maps the separable component of a state to the borders of separable polytope and non-positive operators. An independent proof which could be helpful for a better understanding of the geometric space of entangled states and related maps, is still pending.

Acknowledgement

The authors are grateful to Andreas Osterloh, Jens Siewert, Karol ̵̇Życzkowski for essential feedback and exchange. The authors acknowledge financial support from the Nazarbayev University ORAU grant “Dissecting the collective dynamics of arrays of superconducting circuits and quantum metamaterials” (no. SST2017031) and MES RK state-targeted program BR$05236454$. A.M. is also thankful to ICTP for warm hospitality and financial support.

Appendix A On the non-positivity of the partially transposed essentially entangled component

Let us consider bipartite symmetric systems of total dimension $N=N_{K}\times N_{K}$ with the assumption that $\widehat{\rho}_{\mathrm{ent}}$ in (2) is of rank 1, or else $\widehat{\rho}_{\mathrm{ent}}=\left|\psi\right\rangle\left\langle\psi\right|$. Let the Schmidt decomposition for $\left|\psi\right\rangle$ be

[TABLE]

The essentially entangled component in the aforementioned basis is

[TABLE]

and after partial transposition on the second system, 10 becomes

[TABLE]

One can check that any vector $\frac{1}{\sqrt{2}}\left(\left|b_{k}\right\rangle\left|c_{l}\right\rangle-\left|b_{l}\right\rangle\left|c_{k}\right\rangle\right)$ with $k\neq l$ is an eigenvector with negative eigenvalue $-\sqrt{\alpha_{k}\alpha_{l}}$. Therefore the minimum dimension of the negative subspace of $\widehat{\rho}_{\mathrm{ent}}^{\Gamma}$ is $\frac{{N_{K}}!}{\left({N_{K}}-2\right)!2!}$.

Now let us consider a higher rank $M<d_{Emax}$ (see rank theorem in Sec. II) for $\widehat{\rho}_{\mathrm{ent}}=\sum_{m=1}^{M}\lambda_{m}\left|\psi_{m}\right\rangle\left\langle\psi_{m}\right|$. Every eigenvector $\left|\psi_{m}\right\rangle$ of $\widehat{\rho}_{\mathrm{ent}}$ after partial transposition transforms into a negative operator $\hat{V}_{m}$ negative subspace of dimension $\frac{{N_{K}}!}{\left({N_{K}}-2\right)!2!}$. However this does not necessarily imply that $\widehat{\rho}_{\mathrm{ent}}^{\Gamma}=\sum_{m=1}^{M}\lambda_{m}\hat{V}_{m}$ is negative; in the general case the probability that the different negative subspaces have some overlap (and in consequence $\widehat{\rho}_{\mathrm{ent}}^{\Gamma}\ngeq 0$) is obviously very small especially as $N$ is increasing. On the other hand what we have observed in the two-qutrit system (but also in previous studies essent ) that among the $M$ eigenvectors there is a principal one $\left|\psi_{p}\right\rangle$ with $\lambda_{p}>0.5$. This can be understood from the fact that the essentially entangled component is located towards the outer surface of the convex body and of relatively high purity. In Fig. 5 we exhibit the distribution of $\widehat{\rho}_{\mathrm{ent}}$ and $\widehat{\rho}_{\mathrm{sep}}$ over the participation ratio for two-qutrit random states that confirms this statement. We conjecture that this phenomenon persists in higher dimensions of bipartite systems, i.e., there is principal eigenvector whose non-positivity under partial transposition dominates, and in consequence $\widehat{\rho}_{\mathrm{ent}}^{\Gamma}\ngeq 0$.

Appendix B Proofs of Lemmas

Let us consider a NPT state $\widehat{\rho}$ with BSA (2) and also their partially transposed counterparts $\widehat{\rho}^{\Gamma}$, (3). According to statement $6$ in Sec. II the state $\widehat{\rho}$ generates a family of states $f(\widehat{\rho},B)$ as in (5) and let us denote by $f(\widehat{\rho},B)^{\Gamma}$ the partially transposed counterpart of the family.

By the definition of NPT states, at least a vector $\left|\phi\right\rangle$ exists such that $\left\langle\phi\right|\widehat{\rho}^{\Gamma}\left|\phi\right\rangle<0$, or employing (3) such that

[TABLE]

Since $\widehat{\rho}_{\mathrm{sep}}^{\Gamma}\geq 0$ and $\widehat{\rho}_{\mathrm{ent}}^{\Gamma}\ngeq 0$, it is convenient to introduce here parameters $\epsilon\geq 0$, $\gamma>0$ and re-express (12) as

[TABLE]

where

[TABLE]

Taking cases on the rank of $\widehat{\rho}_{\mathrm{sep}}^{\Gamma}$:

• •

If $\widehat{\rho}_{\mathrm{sep}}^{\Gamma}$ has full rank, then this implies that is a strictly positive operator and then for any vector $\left|\phi\right\rangle$, $\epsilon>0$. In consequence there is a positive parameter $B_{\ast}=\frac{\epsilon}{\epsilon+\gamma}$ (derived by (13)) such that if $B<B_{\ast}$, $\left\langle\phi\right|\widehat{\rho}^{\Gamma}\left|\phi\right\rangle>0$. It is easy to check that under these conditions $\left\langle\phi\right|f(\widehat{\rho},B<B_{\ast})^{\Gamma}\left|\phi\right\rangle>0$ and therefore searching among all $\left|\phi\right\rangle$ one can identify a global minimum (positive) value for $B_{\ast}$ that we denote as $B_{C}$. This implies that a family of BE states exists for $B<B_{C}$.

• •

If $\widehat{\rho}_{\mathrm{sep}}^{\Gamma}$ has at least one [math] eigenvalue but the corresponding to this eigenvalue, eigenvector $\left|\phi_{0}\right\rangle$ gives $\left\langle\phi_{0}\right|\widehat{\rho}_{\mathrm{ent}}^{\Gamma}\left|\phi_{0}\right\rangle>0$. For this vector obviously $\left\langle\phi_{0}\right|f(\widehat{\rho},B\rightarrow 0)^{\Gamma}\left|\phi_{0}\right\rangle>0$. For any other vector $\epsilon>0$ holds and there is a critical $B_{\ast}$ that $\left\langle\phi\right|f(\widehat{\rho},B<B_{\ast})^{\Gamma}\left|\phi\right\rangle>0$. Similarly to previous case, one can identify a global minimum (positive) value for $B_{\ast}$ and identify a family of BE states.

Appendix C On the accuracy limits of the programs

The algorithmic procedure that we use essent for our programs consists of a random sampling of the convex hull body and convex polytope of separable states, a linear programming routine (simplex method) that selects the vectors among the sampled ones which satisfy the constrains of BSA and a ‘loop’ to gradually converge to the unique solution. The simplex method is applied exactly and thus the final accuracy depends on the number $M=\lambda N^{4}$ of vectors/states which are used to sample the space at each step and on the prescribed accuracy of convergence $\epsilon_{C}$ in the loop. In turn the running time has a dependence $M^{3}$ on the number of vectors and has approximately inverse dependence on $\epsilon_{C}$. In the programs constructed for this work we try to balance between accuracy and running time since we aim to perform some low-level statistics. Taking into account all the aforementioned factors we summarize in the Table II the parameters of the programs used in this work together with their accuracy limits. The latters have been estimated by testing many different states, multiple times each one of these, and also with the use of the rank theorem. The program used for the numerical analysis of two-qutrit states can be found on the page www.qubit.kz

Appendix D Distributions of Density Matrices

In Figs. 6-7 the distributions of two-qubit and three-qubit density matrices over the participation ratio $R$ are presented for sampling according to flat measure and to (combined) induced measure respectively.

Appendix E Using BSA to skip the local filtering protocol for two-qubit states

All entangled two-qubit states are distillable Horodeckis97 , meaning that if a sufficient number of copies of the state are provided and local operations and classical communication are allowed, then at least an EPR pair, $\left|\psi_{EPR}\right\rangle=1/\sqrt{2}\left(\left|00\right\rangle+\left|11\right\rangle\right)$ can be produced. If the fidelity $F$ of the state $\hat{\rho}$ with the EPR pair, $F_{EPR}\left(\hat{\rho}\right)=\mathrm{Tr}\left[\hat{\rho}\left|\psi_{EPR}\right\rangle\left\langle\psi_{EPR}\right|\right]$, is greater than $1/2$, distillation can be achieved via the recurrence protocol hash , followed up by the hashing protocol hash . In the habitual case, where $F_{EPR}<1/2$ the additional initial step of local filtering should be taken Horodeckis97 in order to achieve a density matrix Gisin ; HORO with $F_{EPR-}>1/2$, where $\left|\psi_{EPR-}\right\rangle=1/\sqrt{2}\left(\left|01\right\rangle-\left|10\right\rangle\right)$. Then simple local unitary operations are applied to convert the locally filtered state to a state with $F_{EPR}>1/2$. In what follows we show how the knowledge about the BSA of a mixed state with $F_{EPR}<1/2$ may help to skip the local filtering step.

As we have observed in Fig 1 (b), the essentially entangled component $\left|\psi_{ent}\right\rangle$ in (7) is a maximally entangled state for a big part ($\approx 72\%$) of tested density matrices. For the states $\widehat{\rho}$ where the latter holds true, i.e. the concurrence of $\left|\psi_{ent}\right\rangle$ is greater than $0.99$, we calculate the fidelity $F_{ent}$ of the random density matrices with $\left|\psi_{ent}\right\rangle$, and we plot it versus $F_{EPR}$ in Fig  8. One can observe that in many cases (for the states on the upper-left quartile), $F_{ent}>1/2$ while $F_{EPR}<1/2$. According to our statistics on $1200$ random states, no knowledge of $BSA$ requires the application of local-filtering protocol in $\approx 99\%$ of cases, while a knowledge of the BSA states requires the application of local-filtering protocol only in $\approx 50\%$ of cases. For the rest of $50\%$ it is sufficient to apply local unitary operations converting $\left|\psi_{ent}\right\rangle$ to $\left|\psi_{EPR}\right\rangle$.