Generalized entropic criterion for separability

We discuss the entropic criterion for separability of compound quantum systems for general non-additive entropic forms based on arbitrary concave functions $f$. For any separable state, the generalized entropy of the whole system is shown to be not smaller than that of the subsystems, for any choice of $f$, providing thus a necessary criterion for separability. Nevertheless, the criterion is not sufficient and examples of entangled states with the same property are provided. This entails, in particular, that the conjecture about the positivity of the conditional Tsallis entropy for all $q$, a more stringent requirement than the positivity of the conditional von Neumann entropy, is actually a necessary but not sufficient condition for separability in general. The direct relation between the entropic criterion and the largest eigenvalues of the full and reduced density operators of the system is also discussed.

We discuss the entropic criterion for separability of compound quantum systems for general nonadditive entropic forms based on arbitrary concave functions f . For any separable state, the generalized entropy of the whole system is shown to be not smaller than that of the subsystems, for any choice of f , providing thus a necessary criterion for separability. Nevertheless, the criterion is not sufficient and examples of entangled states with the same property are provided. This entails, in particular, that the conjecture about the positivity of the conditional Tsallis entropy for all q, a more stringent requirement than the positivity of the conditional von Neumann entropy, is actually a necessary but not sufficient condition for separability in general. The direct relation between the entropic criterion and the largest eigenvalues of the full and reduced density operators of the system is also discussed. The concept of quantum entanglement [1] has aroused great interest in recent years, due to its deep implications for quantum computation [2], quantum cryptography [3] and quantum teleportation [4]. The relation between entropy and quantum entanglement has also attracted the attention from several authors [5][6][7][8][9][10][11][12][13][14][15]. It is well known, for instance, that the von Neumann entropy of a compound quantum system may be larger or smaller than that of a subsystem [16,17]. However, if the system is in a separable (i.e., unentangled) state, the von-Neumann entropy of the whole system is not smaller than that of a subsystem [5,6]. Unfortunately, the converse is not true, i.e., the same may occur when the system is in an inseparable (i.e., entangled) state, so that this entropy provides only a necessary test for separability. The von Neumann based criterion is actually rather weak, being less stringent than other equally simple necessary conditions [5,18,19]. As discussed in [7,14,15], the von Neumann entropy is in fact not a good entanglement indicator even in those cases where entanglement is fully determined by the eigenvalues of the density operator ρ.
These facts suggest consideration of other information measures which could capture more effectively the effects associated with the separability or inseparability of a compound quantum system. In particular, it has been shown that non-additive information measures like that of Tsallis [20] do provide more stringent conditions for separability [11,12]. Moreover, this entropy depends on a parameter q which can be optimized. In fact, for q → ∞, necessary and sufficient conditions for separability were obtained with this entropy for some important classes of states, like Werner states for n qubits and also n qudits [11,12]. In other situations [13], entanglement was detected however at finite values of q, rather than in the q → ∞ limit. Hence, the questions arise of whether this entropy could provide a necessary and/or sufficient test in general and whether other information measures could lead to the same result.
In this article we will examine more general entropic forms based on arbitrary concave functions, which include as particular cases the von Neumann and Tsallis entropies. We will show that any of these forms provide necessary conditions for separability, which are not sufficient in general. It will also become clear why the Tsallis form provides necessary and sufficient conditions for Werner states in the q → ∞ limit, and why it is not so in other situations. Finally, other entropic forms providing similar results are given.
Let us consider a quantum system described by a density operator ρ. We will examine the general entropic forms [15] where f is a smooth concave function (f ′ (p) decreasing for p ∈ (0, 1)) satisfying f (0) = f (1) = 0, and p i , i = 1, . . . , n, are the eigenvalues of ρ ( i p i = 1). We assume a finite dimension n. The von Neumann entropy is recovered for with k > 0, while the Tsallis entropy corresponds to [20] which approaches −p ln p for q → 1. The generalized entropies (1) satisfy most basic properties of the conventional entropy, except those related with additivity. In particular, S f (ρ) ≥ 0, with S f (ρ) = 0 iff the system is in a pure state (ρ 2 = ρ), while its maximum is attained for the fully mixed state ρ = I/n [21]. Concavity of f ensures concavity of S f (ρ) [17] (S f ( j q j ρ j ) ≥ j q j S f (ρ j ) for 0 ≤ q j ≤ 1, j q j = 1). It can be shown [15,21] . The condition [pf ′′ (p)] ′ = 0 determines in fact Eq. (2). The Tsallis entropy is, accordingly, sub(super)-additive for q > 1 (q < 1). A fundamental property of the forms (1) which will be employed in this work, and which justifies their use as information measures, is that if ρ is more mixed than a density operator ρ ′ , then for any f of the previous form [17]. Labeling the eigenvalues of ρ and ρ ′ in decreasing order, i.e. p 1 ≥ p 2 ≥ . . . ≥ p n , ρ is said to be more mixed (or disordered) than i.e., if and f is concave (Eq. (4) follows then from the mean value theorem; note that the allowed values of S i form a convex set defined by Moreover, it can be shown [17] that ρ is more mixed than ρ ′ if and only if Tr f (ρ) ≥ Tr f (ρ ′ ) for any concave f , i.e., iff Eq. (4) holds ∀ f of the previous form (the conditions f (0) = f (1) = 0 fix just an arbitrary linear term ap + b that can be added to f without affecting concavity or Eq. (4)). If the dimensions of ρ and ρ ′ differ, we may apply the same definition of more mixed by adding zero eigenvalues to the density with the smallest dimension, which leaves S f unchanged.
Let us consider now a system composed of two subsystems A and B. The quantity where ρ A = Tr B ρ is the reduced density matrix of system A and Tr = Tr A Tr B , plays the role of a conditional entropy. In the von Neumann case, Eq. (6) becomes the usual conditional entropy [17], whereas in the Tsallis case, it is proportional to the q-conditional entropy defined in [11,12], S q (B|A) = S A f (ρ)/Tr ρ q A . For a discrete classical system described by a joint probability distribution p ij , Eq. (6) is always nonnegative, i.e., since for any concave f satisfying f (0) = 0, we have f (p + q) ≤ f (p) + f (q) if p ≥ 0, q ≥ 0 (it may be also seen that the set of probabilities {p ij } is more mixed well as for any density diagonal in a basis of product states (ρ = i,j p ij |i A j B i A j B |). Nevertheless, in the general quantum case, S A f (ρ) may of course be negative. In particular, for a pure state ρ = |Ψ Ψ|, S f (ρ) = 0 and the positive eigenvalues of ρ A and ρ B are identical [17], whence For f (p) = −p log 2 p, this is just the usual definition of the entanglement of a pure state |Ψ [22,23]. Negative values of S A f (ρ) are then indicative of distinctive quantum correlations. In particular, for the case (3) it has been conjectured [11][12][13] that the sign of the difference (6) may provide a criterion for determining the separability of ρ [13]. Let us recall that a mixed state ρ is separable (or clasically correlated) iff it can be written as a convex combination of uncorrelated densities [24], with α ω α = 1. Otherwise it is called entangled or inseparable. For the Tsallis case, it has been shown [11,12] that the criterion S A f (ρ) ≥ 0 leads, for q → ∞, to the necessary and sufficient condition for separability for some important classes of states, like Werner states. Nevertheless, we will show here that this does not hold in general. In particular, for an entangled state S A f (ρ) and S B f (ρ) may in fact be both positive for any concave f (including the q → ∞ limit in the Tsallis case), indicating that entanglement cannot be always detected by such entropic criteria (or, in general, by information based on the eigenvalues of ρ and ρ A,B alone). This may occur already for a two qubit system, where the Peres necessary criterion for separability [18] is known to be sufficient [19], so that the entropic criterion is here weaker than the Peres criterion.
Let us first show that Eq. (6) is indeed positive for any separable ρ. A fundamental theorem demonstrated in [25] states that if ρ is separable, then ρ is more mixed than ρ A and ρ B (disorder criterion for separability). Hence, Eq. (4) implies that if ρ is separable, then and similarly, S B f (ρ) ≥ 0, for any concave f (satisfying f (0) = 0). This is in fact an equivalent entropic formulation of the disorder criterion. For a separable state, Eq. (10) will therefore hold ∀q > 0 in the case (3), implying Tr ρ q − Tr A ρ q A ≤ 0 (≥ 0) if q > 1 (0 < q < 1). Note that this entails S α (ρ) ≥ S α (ρ A ) ∀ α > 0, where S α (ρ) = 1 1−α ln Tr ρ α is the Rényi entropy [5,26] (which is additive but not of the form (1), and approaches the von Neumann entropy for α → 1). The disorder criterion is, however, not sufficient [25], so that Eq. (10) provides in general only a necessary test for separability, as will be explicitly seen below.
For a system of two qubits, Eq. (10) is actually an immediate consequence of the more obvious fact that for any separable state, where p 1 (p A 1 ) denotes the largest eigenvalue of ρ (ρ A ). This is so because the difference is a non-negative operator if all ω α ≥ 0 [27]. Hence, denoting with |i any eigenstate of ρ, we have For a two qubit system, (11) already implies that ρ is more mixed than ρ A : i j=1 p j ≤ p A 1 + p A 2 = 1 for i = 2, 3, 4. There are two important remarks to make here. First, if p 1 > p A 1 , the state is certainly entangled, but ρ A is not necessarily more mixed than ρ, entailing that S A f (ρ) is not necessarily negative for any f . Nevertheless, in the Tsallis case, as well as for any set of entropic functions where k > 0 and g q (p) is a convex increasing function satisfying g q (0) = 0, g q (1) = 1 and S f (ρ) will be a decreasing function of the largest eigenvalue p 1 for sufficiently large q and finite dimension (S f (ρ) ≈ k(1 − d 1 g q (p 1 )) in this limit, with d 1 the multiplicity of p 1 ). Hence, if p 1 > p A 1 , S A f (ρ) will become negative for sufficiently large q, and the entropic criterion will be able to detect entanglement. In other words, for q → ∞, S A f (ρ) < 0 iff p 1 > p A 1 , which is a sufficient condition for inseparability. Note that Eq. (3) is of the form (14) for q > 1 and satisfies (15). Another example is [15] which is concave ∀ q, approaches 1 2 p(1 − p) for q → 0 (q = 2 case in (3)) and is of the form (14) for q > 0.
Nonetheless, and this is the second important remark, there are entangled states for which p 1 ≤ p A 1 and p B 1 , i.e., for which the greatest eigenvalue of ρ remains smaller than that of ρ A and ρ B . This may occur already for a system of two qubits, in which case ρ will remain more mixed than ρ A and ρ B , and S A f (ρ), S B f (ρ) will both be non-negative for any concave f . This type of entanglement will therefore not be detected by the previous entropic criterion.
An example is the state considered in [18], where |Ψ 0 = (|↑↓ −|↓↑ )/ √ 2 is the singlet (a maximally entangled state) and |↑↑ a maximally polarized separable state. As shown in [18], Peres criterion determines that this state is entangled ∀ x > 0: the partial transpose of ρ (defined as the transposition with respect to the indexes of system A), which is still a density operator if ρ is separable, has always a negative eigenvalue for x > 0, namely However, as the eigenvalues of ρ are (x, 1 − x, 0, 0), and those of ρ A and ρ B are (1 − x/2, x/2), the greatest eigenvalue of ρ (p 1 = x for x > 1 2 ) is greater than that of ρ A (p A 1 = 1 − x/2) only for x > x c = 2/3 [ Fig. 1]. Hence, for 0 < x < 2/3, entanglement will not be detected by S A,B f (ρ), for any f . This can also be directly seen from the explicit expression Since for a two state system, the entropy f (p) + f (1 − p) is a decreasing function of the largest eigenvalue (f ′ (p) − f ′ (1 − p) < 0 for p > 1/2 and f concave), in this case , for any f . The sign of S A f (x) is independent of the choice of entropic function f in this example, i.e. independent of q in the Tsallis case or in Eq. (16), as shown in Fig. 1. For normalization purposes, we have plotted the quantitȳ where g q (p) = p q in the Tsallis case (3) (so thatS A f (ρ) = S q (B|A)) and g q (p) = (e qp −1)/(e q −1) for Eq. (16).
This situation is actually not very special. Consider for instance the more general state where |uv = |u A |v B is an arbitrary separable pure state of the two qubits. This state is again entangled ∀ x > 0, since the partial transpose of ρ has a negative eigenvalue On the other hand, the eigenvalues of ρ are while those of ρ A , ρ B are again (1 − x/2, x/2). Hence, p 1 = (1 + z)/2, p A 1 = (1 − x/2), and p 1 > p A 1 only for x > x c = 2r/(1 + 2r) .
Thus, S A f (ρ) < 0 iff x c < x < 1, for any concave f . Again, the entropic criterion fails to detect entanglement for 0 < x < x c . For r = 1, we recover the results of the previous example, whereas for r = 0, i.e. |uv = |↑↓ , σ 1 = −x/2 and x c = 0, so that S A f (ρ) < 0 ∀ x > 0. This is the only case where the entropic criterion predicts the full interval of inseparability.
Let us still consider the example of refs. [18,28], with |Ψ = a|↑↓ + b|↓↑ , |a| 2 + |b| 2 = 1. As shown in [18], this state is entangled just for x > x e = (1 + 2|ab|) −1 , since the lowest eigenvalue of the partial transpose is However, the eigenvalues of ρ are (x, (1−x)/2, (1−x)/2, 0) while those of ρ A , ρ B are (1±x(|b| 2 −|a| 2 ))/2. The largest eigenvalue of ρ (p 1 = x for x > 1/3) is greater than that of ρ A (p A 1 = (1 + x||a| 2 −|b| 2 |)/2) only for x > x c = (2−||a| 2 − |b| 2 |) −1 . But x c ≥ x e , with x c = x e just for |a| = |b| or in the trivial separable cases b = 0 or a = 0. Hence, if |a| = |b| and ab = 0, S A f (ρ) will not detect entanglement for x e < x < x c . Note also that for x > x c , S A f (ρ) is in this case not necessarily negative for any f , but will become negative for sufficiently large q in the Tsallis case or in Eqs. (14) or (16), as shown in Fig. 2. The value of x where S A f (ρ) = 0 converges actually exponentially fast to x c for q → ∞ in (3) or (16). This will occur whenever the degeneracies of p 1 and p A 1 coincide. The entropic criterion will provide, however, necessary and sufficient conditions for separability for any density ρ diagonal in the Bell basis [7], i.e. the basis of maximally entangled states |Ψ 0 , |Ψ 1 = (|↑↓ + |↓↑ )/ √ 2, |Ψ 2,3 = (|↑↑ ± |↓↓ )/ √ 2. In such a case, is known to be entangled iff p 1 > 1/2 [5], where p 1 = Max[{q i }] is the largest eigenvalue of ρ. This may be obtained directly with Peres criterion, as the partial transpose of ρ has eigenvalues 1 2 − q i . Now, for any pure Bell state |Ψ i Ψ i |, the reduced density matrices are fully mixed, with eigenvalues ( 1 2 , 1 2 ), so that the same will oc-entropic function f . However, the condition p 1 ≤ p A 1 becomes sufficient in some important cases, which include any density diagonal in the Bell basis in a two qubit system, and also Werner-like states in n qubit (or qudit) systems. In these cases the inequality S A f (ρ) ≥ 0 will lead, for q → ∞ in Eq. (3) or (14), to the necessary and sufficient condition for separability.
The condition S A f (ρ) ≥ 0 for any concave entropic function f is equivalent to the requirement that ρ be more mixed than ρ A , a general necessary condition for separability [25]. Let us remark that this requirement is stronger than the condition S A f (ρ) ≥ 0 ∀ q > 0 in (3) (or ∀q in (16)). Other families of concave entropic functions are required in general to detect that ρ is not more mixed than ρ A when the first violation of Eqs. (5) occurs for i > 1, although in many cases this can also be seen with the entropies (3) or (16). In such situations S A f (ρ) will remain positive for q → ∞ but may become negative at finite values of q.
RR and NC acknowledge support from CIC and CON-ICET, respectively, of Argentina.