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Characterizing Entanglement with Global and Marginal Entropic Characterizing entanglement with global and marginal entropic measures Gerardo Adesso, Fabrizio Illuminati, and Silvio De Siena Dipartimento di Fisica “E. R. Caianiello”, Universit`adi Salerno, INFM UdR di Salerno, INFN Sez. di Napoli, Gruppo Coll. di Salerno, 84081 Baronissi (SA), Italy (Dated: October 22, 2003) We qualify the entanglement of arbitrary mixed states of bipartite quantum systems by comparing global and marginal mixednesses quantified by different entropic measures. For systems of two qubits we discriminate the class of maximally entangled states with fixed marginal mixednesses, and determine an analytical upper bound relating the entanglement of formation to the marginal linear entropies. This result partially generalizes to mixed states the quantification of entaglement with marginal mixednesses holding for pure states. We identify a class of entangled states that, for fixed marginals, are globally more mixed than product states when measured by the linear entropy. Such states cannot be discriminated by the majorization criterion. PACS numbers: 03.67.Mn, 03.65.Ud, 03.67.-a, 03.65.Yz I. INTRODUCTION AND BASIC NOTATIONS Concerning mixed states, there is a weak qualitative corre- spondence between mixedness and separability, in the sense The modern developments in quantum information theory that all states sufficiently close to the maximally mixed state [1] have highlighted the key role played by entanglement in are necessarily separable [6]. On the other hand, due to the the fields of quantum communication [2], quantum compu- existence of the so-called isospectral states [12], i.e. states tation [3], quantum cryptography [4] and teleportation [5]. with the same global and marginal spectra but with different While a comprehensive theory for the qualification and the entanglement properties, only sufficient conditions for entan- quantification of the entanglement of pure states is well es- glement, based on the global and marginal mixednesses, can tablished, even for two qubits, the smallest nontrivial bipar- be given. In particular, entangled states share the unique fea- tite quantum system, the relation between entanglement and ture that their individual components may be more disordered mixedness remains a fascinating open question [6, 7, 8, 9]. than the system as a whole. Because this does not happen for The degree of mixedness is a fundamental property because correlated states in classical probability theory and for separa- any pure state is induced by environmental decoherence to ble quantum states, this property can be quantified to provide evolve in a generally mixed state. There are two measures sufficient conditions for entanglement, such as the entropic suited to quantify the mixedness or the disorder of a state: the criterion [13], and the majorization criterion [12]. In this work we present numerical studies and analytical Von Neumann entropy SV which has close connections with statistical physics and quantum probability theory, and the lin- bounds that discriminate separable and entangled states in the three-dimensional manifold spanned by global and marginal ear entropy SL which is directly related to the purity µ of a state. For a quantum state ρ in a D dimensional Hilbert space entropic measures. We study the behavior of entanglement they are defined as follows − with varying global and marginal mixednesses and identify H the maximally entangled states for fixed marginal mixed- SV (ρ) Tr [ρ log ρ] , (1) nesses (MEMMS). Knowledge of these states provides an an- ≡ − D D 2 D alytical upper bound relating the entanglement of formation SL(ρ) 1 Tr ρ [1 µ(ρ)] , (2) ≡ D 1 − ≡ D 1 − and the marginal linear entropies. We then compare the Von − − Neumann and linear entropies, finding that, with respect to arXiv:quant-ph/0307192v2 29 Dec 2003 2 where µ Tr ρ is the purity of the state ρ. For pure states ≡ the latter, there exist separable and entangled states that for ρp = ψ ψ , µ =1; for mixed states µ< 1, and it acquires its given marginal purities µ1 2 are less pure than product states | ih | I , minimum 1/D on the maximally mixed state ρ = D/D. The (LPTPS). We provide an analytical characterization of the en- entropies Eqs. (1)-(2) range from 0 (pure states) to 1 (maxi- tangled LPTPS, showing that they can never satisfy the ma- mally mixed states), but they are in general inequivalent in the jorization criterion. sense that the ordering of states induced by each measure is To be specific, let us consider a two-qubit system defined in different [9]. For a state ρ of a bipartite system with Hilbert the 4-dimensional Hilbert space = C2 C2. The entangle- space 1 2 the marginal density matrices of each H ⊗ H ≡ H ⊗ H ment of any state ρ of such a system is completely qualified by subsystem are obtained by partial tracing ρ1,2 = Tr2,1 ρ. the Peres-Horodecki criterion of positive partial transposition Any state ρ is said to be separable [10] if it can be written (PPT) [14], stating that ρ is separable if and only if ρT1 0, i T1 ≥ as a convex combination of product states ρS = i piρ1 where ρ is the partial transpose of the density matrix ρ with i i ⊗ T1 ρ2, ρ1,2 1,2, with pi positive weights such that i pi =1. respect to the first qubit, (ρ ) = (ρ) . As a mea- ∈ H P mµ,nν nµ,mν Otherwise the state is entangled. For pure states of a bipartite sure of entanglement for mixed states, we consider the en- P system, the entanglement E(ρp) is properly quantified by any tanglement of formation [15], which quantifies the amount of of the marginal mixednesses as measured by the Von Neu- entanglement necessary to create an entangled state, mann entropy (entropy of entanglement) [11]: EF (ρ) min piE( ψi ψi ) , (4) ≡ {pi,ψi} | ih | E(ρp) S(ρp1 ) S(ρp2 ) . (3) i ≡ ≡ X 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.
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