arXiv:quant-ph/0307192v2 29 Dec 2003 ρ minimum utdt uniytemxdeso h iodro tt:t measures state: a two of are disorder entropy the There Neumann or Von mixedness to the state. quantify decoherence mixed to environmental suited generally by a induced because in is property evolve 9]. fundamental state 8, a pure 7, is any [6, mixedness question of degree open and The fascinating entanglement a between bip remains relation nontrivial mixedness the es- smallest system, well the quantum is qubits, tite two states the for pure even and of tablished, qualification entanglement the the [5] of for teleportation quantification theory and comprehensive [4] compu- a cryptography quantum While quantum [2], in [3], communication entanglement tation quantum by played of role fields key the the highlighted have [1] es htteodrn fsae nue yec esr is measure each state by a induced For states [9]. of different ordering (maxi- in the 1 inequivalent general that to in sense states) are they (pure but 0 states), mixed from mally range (1)-(2) Eqs. entropies ttsia hsc n unu rbblt hoy n th and theory, probability quantum and physics statistical where usse r bandb ata tracing partial by obtained are subsystem space a entropy ear sacne obnto fpoutstates product of combination convex a as tt.Fraqatmstate quantum a For state. 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Fisica di Dipartimento  log h agnldniymtie feach of matrices density marginal the i S ≡ 1 oiiewihssc that such weights positive V ρ ρ E − D S hc a ls oncin with connections close has which separable na in fabpriesse ihHilbert with system bipartite a of ( ( NNSz iNpl,Gup ol iSlro 48 Baroniss 84081 Salerno, di Coll. Gruppo Napoli, di Sez. INFN Tr ρ ρ ρ ] p p ρ 1 ) , D For . 2 ) spoel unie yany by quantified properly is  − ead dso arzoIlmnt,adSli eSiena De Silvio and Illuminati, Fabrizio Adesso, Gerardo ≡ ≡ iesoa ibr space Hilbert dimensional S < µ pure 1]i tcnb written be can it if [10] D ( ρ D p − ρ 2 1 ttso bipartite a of states ) 1 n taqie its acquires it and , ρ 1 ρ , . 2 o uestates pure For . S [1 ρ = = = − purity Tr I P µ P D 2 Dtd coe 2 2003) 22, October (Dated: ( , ρ /D 1 i i )] p p ρ µ i i . The . e ρ , 1 = fa of 1 i lin- the (2) (1) (3) ar- he ⊗ . . retois hsrsl atal eeaie omixed to generalizes partially result This entropies. ar PT 1] ttn that stating transpo [14], partial (PPT) positive of criterion Peres-Horodecki the where ihtesm lbladmria pcr u ihdifferent with st but spectra only marginal i.e. properties, the and entanglement [12], to global states due same isospectral the hand, with so-called other the the On sta of mixed existence [6]. maximally separable the to necessarily close are sense sufficiently the states in all separability, that and mixedness between spondence Concerning eto n state any of ment h atr hr xs eaal n nage ttsthat states entangled to and purities respect marginal separable with given exist that, there Von finding latter, the the entropies, compare linear then We and formatio entropies. Neumann of linear marginal entanglement the the mixed- and relating an- bound marginal an upper provides fixed states alytical these for of Knowledge identify states (MEMMS). nesses and entangled mixednesses entanglement maximally marginal of the and behavior global the varying study with We marginal and measures. global entropic by spanned in manifold states entangled three-dimensional and separable discriminate that bounds entropi [12]. criterion the majorization as the and such [13], provid criterion entanglement, to for quantified be conditions can sufficient property this for s states, happen for quantum and not ble theory does probability this classical Because in states whole. correlated a as disordere system more be the f may than unique components the individual share their states that can entangled ture mixednesses, particular, marginal In and given. global be the on based glement, epc otefis qubit, first the to respect the ma- the satisfy never can criterion. they jorization that showing e the LPTPS, of tangled characterization analytical an provide We (LPTPS). nageetncsayt raea nage state, entangled an the create to consider necessary we entanglement states, formation mixed of for tanglement entanglement of sure r ie hnpoutsae hnmaue ythe by measured when states product than mixed ore rs o ytm ftoqbt edsrmnt the discriminate we qubits two of systems For ures. nse odn o uesae.W dniyaclass a identify We states. pure for holding dnesses jrzto criterion. ajorization nti okw rsn ueia tde n analytical and studies numerical present we work this In ob pcfi,ltu osdratoqbtsse endin defined system two-qubit a consider us let specific, be To nse,addtriea nltcluprbound upper analytical an determine and dnesses, 4 rieqatmssesb oprn lbland global comparing by systems quantum artite dmninlHletspace Hilbert -dimensional ρ T 1 lro NMURd Salerno, di UdR INFM alerno, stepriltasoeo h est matrix density the of transpose partial the is E mixed F ( ρ ρ ) fsc ytmi opeeyqaie by qualified completely is system a such of tts hr sawa ulttv corre- qualitative weak a is there states, ≡ S) Italy (SA), i { p min ρ µ i 1] hc unie h mutof amount the quantifies which [15], ( ,ψ 1 ssprbei n nyif only and if separable is ρ , pcmeasures opic 2 i T } 1 r espr hnpoutstates product than pure less are X ) mµ,nν sufficient i H p i = E ( = ( C | ψ 2 odtosfrentan- for conditions i ⊗ ρ ih ) ψ C nµ,mν i 2 | h entangle- The . ) , samea- a As . ρ T 1 ρ epara- sition ≥ with ates en- the ea- for (4) n- 0 te n d c e , 2 where the minimization is taken over those probabilities pi while for SV = 0 there cannot be states for SV 1 = SV 2. { } 6 and pure states ψi that realize the density matrix ρ = This qualitative behavior is reflected in some analytical { } i pi ψi ψi . For two qubits, the entanglement of formation properties. Firstly, the Von Neumann entropy satisfies the can be| easilyih | computed [16], and reads triangle inequality [18] P

EF (ρ) = ( (ρ)) , (5) SV 1 SV 2 2SV SV 1 + SV 2 . (8) F C | − |≤ ≤ 1 (x) H 1+ 1 x2 , (6) The leftmost inequality is saturated for pure states, while F ≡ 2 −   the rightmost one for product states ρ⊗ = ρ1 ρ2. This  p  (7) ⊗ H(x) = x log2 x (1 x) log2(1 x) . means that, for any state ρ with reduced density matrices ρ1,2, − − − − ⊗ SV (ρ) SV (ρ ), so that product states are indeed maxi- The quantity (ρ) is called the concurrence of the state ρ and ≤ C mally mixed states with fixed given marginals. The lower is defined as (ρ) max 0, √λ1 √λ2 √λ3 √λ4 , C ≡ { − − − } boundary to the region of coexistence is determined by the where the λi ’s are the eigenvalues of the matrix ρ(σy ∗ { } ⊗ entropic criterion, stating that for separable states 2SV σy)ρ (σy σy) in decreasing order, σy is the Pauli spin ma- ≥ ⊗ max SV 1, SV 2 . As soon as this inequality is violated, only trix and the star denotes complex conjugation in the compu- entangled{ states} can be found. The structure of the bundles tational basis ij i j , i,j = 0, 1 . Because (x) identified in the space V and depicted in Fig. 1(a) shows is a monotonic{| convexi ≡ |functioni ⊗ | i of x [0, 1]}, the concurrenceF E ∈ 2 that the most entangled states fall in the region of largest (ρ) and its square, the tangle τ(ρ) (ρ), can be used to marginals, and yields the following numerical upper bound: defineC a proper measure of entanglement.≡ C All the three quan- tities , , and take values ranging from zero EF (ρ) (ρ) τ(ρ) EF (ρ) min SV 1, SV 2 . (9) (separable states)C to one (maximally entangled states). ≤ { } This bound is obviously very loose since it can be saturated only for SV 1 = SV 2, and then only by pure states. Pure II. CHARACTERIZING ENTANGLEMENT IN THE SPACE states can thus be viewed as maximally entangled states with OF VON NEUMANN ENTROPIES equally distributed marginals. This naturally leads to the ques- tion of identifying the maximally entangled mixed states with To unveil the connection between entangle- fixed, arbitrarily distributed marginal mixednesses. We will ment, global, and marginal mixednesses, let us name these states “maximally entangled marginally mixed first consider the three-dimensional space V states” (MEMMS). MEMMS should not be confused with E ≡ SV 1 SV (ρ1), SV 2 SV (ρ2), SV SV (ρ) spanned the “maximally entangled mixed states” (MEMS) introduced by{ the≡ global and marginal≡ Von Neumann≡ entropies.} We in Ref. [8], which are maximally entangled states with fixed randomly generate several thousands density matrices [17], global mixedness. and plot them as points in the space V as shown in Fig. 1(a). We assign to each state a differentE color according to the value of its entanglement of formation EF . Red points denote III. MEMMS: MAXIMALLY ENTANGLED STATES WITH separable states (EF = 0). Entangled states fall in four FIXED MARGINAL MIXEDNESSES bundles with increasing EF : green points denote states with 0 < EF 1/4; cyan points are states with 1/4 < EF 1/2; In order to determine the MEMMS, we begin by reminding ≤ ≤ blue points are states with 1/2 < EF 3/4; and magenta that both the entanglement and the mixednesses are invariant ≤ points denote states with 3/4 < EF 1. We find a under local unitary transformations. Without loss of general- ≤ qualitative behavior, according to which the entanglement ity, we can then consider in the computational basis density tends to increase with decreasing global mixedness and with matrices of the following form: increasing marginal mixednesses. Qualitatively, more global mixedness means more randomness and then less correlations x 0 0 e between the subsystems. One then expects that by keeping the 0 y f 0 ρ = , marginal mixednesses fixed, the maximally mixed states must  0 f w 0  be those with the least correlation between subsystems, i.e.  e 0 0 z  they must be product states. This is in fact the case: product   states are “maximal” states in the space V . In Fig. 1(a) they with e √xz, f √yw for the positivity of ρ. The E | | ≤ | | ≤ lie on the yellow plane SV = (SV 1 + SV 2)/2. Proceeding PPT criterion entails ρ entangled if and only if e > √yw downward through Fig. 1(a), we first find a small region or f > √xz. We can always choose e and f| |to be real containing only separable states lying above the horizontal and| positive| from local invariance. The concurrence of such red-orange plane SV = (log4 12)/2 [9]. Below this plane a state reads (ρ) = 2max f √xz,e √wy, 0 . The there is a “region of coexistence” in which both separable role of the firstC two terms can{ be− interchanged− by local} uni- and entangled states can be found. Going further down tary operations, so, in order to obtain maximal concurrence, one identifies an area of lowest global and largest marginal we can equivalently annihilate one of the four parameters mixednesses that contains only entangled states. Pure states x, y, w, z . We choose to set y = 0, so that entanglement { } are obviously located at SV = 0 on the line SV 1 = SV 2, arises as soon as e > 0. This implies f = 0; exploiting then 3 the constraint of normalization Tr ρ = 1, we arrive at the fol- lowing form of the state

x1 0 0 c/2 00 0 0 , (10)  0 01 x1 x2 0  − −  c/20 0 x2    with x1 +x2 1, c 2√x1x2. This form is particularly use- ful because every≤ parameter≤ has a definite meaning: c (ρ) is the concurrence and it regulates the entanglement≡C of for- mation Eq. (5), while the reduced density matrices are simply ρ1 = diag x1, 1 x1 , ρ2 = diag 1 x2, x2 . The problem { − } { − } of maximizing c (or EF ) keeping x1, x2 fixed is then trivial { } and leads to c =2√x1x2. Let us mention that, to gain maxi- mal concurrence while assuring ρ 0, in our parametrization ≥ x1 must be identified with the greatest eigenvalue of the less pure marginal density matrix, and x2 with the lowest of the more pure one (or vice versa). The MEMMS, up to local unitary operations, have thus the simple form

x1 0 0 √x1x2 (a) 00 0 0 ρm = . (11)  0 01 x1 x2 0  − −  √x1x2 0 0 x2    Let us remark that these states are maximally entangled with respect to any entropic measure of marginal mixedness (ei- ther Von Neumann, or linear, or generalized entropies) be- cause the eigenvalues of ρ1,2 are kept fixed. Their global Von Neumann entropy is limited by 1/2, and they reduce to pure states for x1 = 1 x2. Notice that density matrices of the form Eq. (11) have− at most rank two. The entanglement of formation of MEMMS is EF (ρm) = (2√x1x2). Unfortu- nately, it cannot be expressed analyticallyF as a function of the marginal entropies SV (ρmi ) = H(xi) due to transcendence of the binary entropy function H(x) Eq. (7).

IV. CHARACTERIZING ENTANGLEMENT IN THE SPACE OF LINEAR ENTROPIES

We now consider the linear entropy as a measure of mixed- ness. The advantage of using the linear entropy is that it is directly related to the purity of the state by Eq. (2), and its definition does not involve any transcendental function. In this case the entanglement is properly quantified by the (b) tangle τ(ρ) 2(ρ) because the linear entropy of entan- ≡ C glement for pure states is related to the Von Neumann en- FIG. 1: Plots of 60,000 randomly generated physical states of two tropy of entanglement Eq. (3) by the same relationship that qubits in the space of global and marginal mixednesses, as quanti- connects the tangle to the entanglement of formation. We fied by Von Neumann entropy (a), or linear entropy (b). Red dots again randomly generate some thousands density matrices ρ indicate separable states; green, cyan, blue, and magenta dots denote of two qubits and plot them as points in the three-dimensional nonseparable states. The entanglement of formation (a), or tangle (b), grows going from the green to the magenta zone (see text for space L SL1 SL(ρ1), SL2 SL(ρ2), SL SL(ρ) , as shownE in≡ Fig. { 1(b).≡ The distribution≡ of colored≡ bundles} is details). In both plots, the yellow surfaces represent product states, while above the red-orange planes no entangled states can be found. analogous to that of the previous case Fig. 1(a), but with the All quantities plotted are dimensionless. linear entropies and the tangle replacing, respectively, the Von Neumann entropies and the entanglement of formation. 4

A comparison with the previous case shows some prima facie analogies and some remarkable differences. We find again a general trend of increasing entanglement with de- creasing global and increasing marginal mixednesses. Prod- uct states lie on the yellow surface of equation SL = 2(SL1 + SL2)/3 (SL1SL2)/3. With respect to the linear entropy, taken as− a measure of mixedness, they are not maximally mixed states with fixed marginals. This fact has deep con- sequences that we will explore in Sec. VI. Going downward through Fig. 1(b) we find a small region of only separable states above the horizontal red-orange plane SL = 8/9 [9]. Below this plane there is again the region of coexistence of separable and entangled states, while in the region of lowest global and largest marginal mixednesses we find only entan- gled states. Pure states are always located at SL = 0 on the line SL1 = SL2, and again, there are no states for SL = 0 and SL1 = SL2. Qualitatively, the structure is very similar 2 6 FIG. 2: Plot of the tangle C (ρ) as a function of the marginal lin- to that obtained in the space V of Von Neumann entropies. E ear entropies for 30,000 randomly generated entangled states of two Some numerics changes due to different definitions, in partic- qubits. The yellow shaded surface represents the maximally entan- ular the lower boundary of the region of coexistence is now gled states for fixed marginal mixednesses (MEMMS) defined by determined by the entropic criterion for the linear entropies: Eq. (11). They saturate the inequality Eq. (13). All quantities plotted if a state is separable, then SL 2(max SL1, SL2 )/3, or, are dimensionless. ≥ { } equivalently, µ min µ1, µ2 . A state violating these in- equalities must necessarily≤ { be entangled.} The tangle satisfies a numerical upper bound analogous to Eq. (9) for the entan- marginals should be large as well: in particular, if the mixed- glement of formation, ness of one of the subsystemsis zero, then the state of the total 2 system is not entangled. Finally, from Eqs. (5) and (13) it im- (ρ) min SL1, SL2 . (12) C ≤ { } mediately follows that the entanglement of formation satisfies the following bound: V. MEMMS IN THE SPACE OF LINEAR ENTROPIES: ANALYTICAL BOUNDS EF (ρ) 1 1 SL1 1 1 SL2 . ≤F r ∓ − ± − ! Obviously, states of the form Eq. (11) are MEMMS in the  p  p  (14) space L as well. The relation between the eigenvalues xi of This bound establishes a relation between the entanglement E 2 the reduced density matrices ρm and the tangle is (ρm)= of a mixed state and its marginal mixednesses. Although i C 4x1x2, while for the marginal linear entropies we have SLi = the entanglement of formation for systems of two qubits is 2 2 2[x + (1 xi) ]. This time we can straightforwardly invert known [16], our inequality Eq. (14) provides a generalization i − these relations to obtain the following analytical upper bound to mixed states of the equality EF (ρp) = SL(ρp1,2 ) on the tangle of any mixed state of two qubits (see Fig. 2) holding for pure states. F p
2 (ρ) 1 1 SL1 1 1 SL2 , (13) C ≤ ∓ − ± −    where the minus signp must be associatedp with the lowest VI. LPTPS: ENTANGLED STATES THAT ARE LESS PURE marginal linear entropy. For equal marginals this bound re- THAN PRODUCT STATES 2 duces simply to (ρ) SL1 = SL2 and the equality is reached for pureC states.≤ The bound Eq. (13) bears some re- Unlike the Von Neumann entropy, the linear entropy is not markable consequences. In particular, it entails that maximal additive on product states, and does not satisfy the triangle entanglement decreases at a very fast rate with increasing dif- inequality. As mentioned above, with respect to the linear ference of marginal linear entropies, as can be seen in Fig. 2. entropy, product states are not maximally mixed states with In fact, it introduces the following general rule: in order to fixed marginal mixednesses (see Fig. 1(b)), and this entails obtain maximally entangled states one needs to have the low- the existence of states (separable or entangled) that are less est possible global mixedness, the largest possible marginal pure than product states (LPTPS). We wish to characterize the mixednesses, and the smallest possible difference between the entangled LPTPS and to study their tangle as a function of latter. That the marginals should be as close as possible is their “distance in purity” from product states. Because for intuitively clear, because if the two subsystems have strong the latter µ = µ1µ2, then for any state ρ to be a LPTP state quantum correlations between them, they must carry about (separable or entangled) the following condition must hold: the same amount of information. For instance, the MEMS de- fined in Ref. [8] have always the same marginal spectra. The ∆µ µ1µ2 µ 0 , (15) ≡ − ≥ 5 where ∆µ defines the natural distance from the purity of prod- the class of maximally entangled states with fixed marginal uct states. The LPTPS that saturate inequality Eq. (15) are mixednesses (MEMMS). They allow to obtain an analytical isospectral to product states. We are looking for entangled upper bound for the entanglement of formation of generic LPTPS, and we can then exploit again the form Eq. (10) of mixed states in terms of the marginals. This provides a partial the density matrix, and impose the condition Eq. (15) to ob- generalization to mixed states of the exact equalities holding tain the following constraints: for pure states. We may reasonably expect that similar bounds and inequalities can be determined for more complex systems 1 2x1 2x2 +2x1x2 0 that do not allow for a direct computation of entanglement 2− − ≥ (16) c 4x1x2(1 2x1 2x2 +2x1x2) . measures.  ≤ − − The difference between linear and Von Neumann entropies For all entangled LPTPS this implies that allows to detect a class of states that are less pure than product states with fixed marginal mixednesses (LPTPS). We charac- 4 1 SL2,1 SL1,2 − , (17) terized the entangled LPTPS and showed that their maximal ≤ 2 1+p 1 SL2,1 entanglement decreases with decreasing purity. Finally, we − singled out in both spaces V and L a large region of coex- which simply means that nop entangled states
can exist too istence of separable and entangledE statesE with the same global close to the maximally mixed state, as shown in Fig. 3(a). In and marginal mixednesses. In this region a complete char- the range of parameters c, x1, x2 constrained by Eq. (16), it acterization of entanglement can be achieved only once the is immediately verified that{ the largest} eigenvalue of the den- amount of classical correlations in a quantum state has been sity matrix Eq. (10) is always smaller than the largest eigenval- properly quantified. This problem awaits further study, as well ues of the marginals ρ1,2 so that entangled LPTPS are never as the question of characterizing entanglementwith global and detected by the majorization criterion. We now wish to de- marginal mixednesses for higher–dimensional and continuous termine the maximally entangled LPTPS with fixed ∆µ. This variable systems. problem can be easily recast as a constrained maximization of 2 ∆µ = x1x2(1 2x1 2x2 +2x1x2)/2 c with fixed c. Its − − − solution yields a class of states with x1 = x2 = (3 √5)/4, and a linear relation between the distance ∆µ and− the tangle Acknowledgments c2: Financial support from INFM, INFN, and MURST under 2 c = 2(∆µmax ∆µ) , (18) projects PRIN–COFIN (2002) is acknowledged. − where ∆µmax = (5√5 11)/8 0.0225. The tangle of maximally entangled LPTPS− decreases≃ linearly with increas- 2 ing ∆µ from the maximumvalue cmax = 2∆µmax at ∆µ =0, and vanishes beyond the critical value ∆µmax (See Fig. 3(b)). When the mixednesses are measured using the Von Neumann entropy, one finds that all LPTPS lie below the plane of prod- uct states in the space V (See Fig. 1(a)). Therefore, in this case, the lack of precisionE with which the linear entropy char- acterizes the Von Neumann entropy turns out to be an useful resource to detect a class of entangled states that could not be otherwise discriminated by the entropic and majorization criteria.

VII. CONCLUDING REMARKS AND OUTLOOK (a) (b)

In conclusion, we explored and characterized qualitatively FIG. 3: Entangled LPTPS. (a) Plot of 10,000 numerically generated and quantitatively the entanglement of physical states of two entangled LPTPS in the space EL of global and marginal linear en- qubits by comparing their global and marginal mixednesses, tropies. As in Fig. 1(b), the yellow surface represents product states. as measured either by the Von Neumann or the linear entropy. (b) Plot of the tangle C2(ρ) of LPTPS as a function of the distance We found that entanglement generally increases with decreas- ∆µ from the purity of product states. The red line represents the ing global mixedness and with increasing local mixednesses, maximally entangled LPTPS given by Eq. (18). All quantities plot- and we provided several numerical bounds. We determined ted are dimensionless.

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