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The conditional version of sandwiched Tsallis relative entropy (CSTRE) is employed to study the bipartite separability of one parameter family of N -qudit Werner-Popescu states in their 1:N-1 partition. For all N, the strongest limitation on bipartite separability is realized in the limit and is found to match exactly with the separability range obtained using an algebraic method which is both necessary and sufficient. The theoretical superiority of using CSTRE criterion to find the bipartite separability range over the one using Abe-Rajagopal (AR) q-conditional entropy is illustrated by comparing the convergence of the parameter x with respect to q, in the implicit plots of AR q-conditional entropy and CSTRE.

Entropic characterization of separability [

Quite similar to the definition of sandwiched Rényi relative entropy [

D ˜ q T ( ρ | | σ ) = Tr { ( σ 1 − q 2 q ρ σ 1 − q 2 q ) q } − 1 q − 1 (1)

Equation (1) reduces to traditional relative Tsallis entropy D q T ( ρ | | σ )

D q T ( ρ | | σ ) = Tr [ ρ q σ 1 − q ] − 1 q − 1 (2)

when r and s commute with each other.

The conditional forms of D ˜ q T ( ρ | | σ ) are defined as [

D ˜ q T ( ρ A B | | ρ B ) = Q ˜ q ( ρ A B | | ρ B ) − 1 1 − q (3)

and

D ˜ q T ( ρ A B | | ρ A ) = Q ˜ q ( ρ A B | | ρ A ) − 1 1 − q (4)

with Q ˜ q ( ρ A B | | ρ B ) , Q ˜ q ( ρ A B | | ρ A ) being respectively given by

Q ˜ q ( ρ A B | | ρ B ) = Tr { [ ( I A ⊗ ρ B ) 1 − q 2 q ρ A B ( I A ⊗ ρ B ) 1 − q 2 q ] q } (5)

Q ˜ q ( ρ A B | | ρ A ) = Tr { [ ( ρ A ⊗ I B ) 1 − q 2 q ρ A B ( ρ A ⊗ I B ) 1 − q 2 q ] q } (6)

In Ref. [

S q T ( A | B ) = 1 q − 1 [ 1 − Tr ρ A B q Tr ρ B q ] , (7)

S q T ( B | A ) = 1 q − 1 [ 1 − Tr ρ A B q Tr ρ A q ] . (8)

Quite like the AR q-conditional entropies S q T ( A | B ) , S q T ( B | A ) , both the conditional versions of sandwiched Tsallis relative entropy D ˜ q T ( ρ A B | | ρ B ) , D ˜ q T ( ρ A B | | ρ A ) reduce to the respective von-Neumann entropies S ( A | B ) , S ( B | A ) in the limit q → 1 .

Both AR- and CSTRE-criteria have been employed in Refs. [

The investigation of separability range in one parameter families of mixed states through AR- and CSTRE-criteria has revealed that whenever the marginal is not maximally mixed and hence does not commute with the global density matrix, the CSTRE criterion yields stricter separability range than its commuting version, the AR-criterion [

This article is organized in four sections including the introductory section (Section 1) in which we recall the non-additive entropic separability criteria such as AR-, CSTRE-criteria and discuss the motivation behind this work. Section 2 introduces the N-qudit Werner-Popescu state as a generalization of noisy one-parmeter family of N-qubit GHZ states to its qudit counterpart. Section 3 examines the 1 : N − 1 separability range of one parameter family of N-qudit Werner-Popescu states using different separability criteria. A comparison of the results obtained through AR-, CSTRE criteria is carried out and the superiority of CSTRE criterion is illustrated through the implicit plots of x versus q in both AR-, CSTRE methods (Section 3). Finally Section 4 provides a summary of the results.

The Werner-Popescu state with N-qudits [

ρ N d ( x ) = ρ ( A 1 , A 2 , ⋯ , A N ) = 1 − x d N [ I d ( A 1 ) ⊗ I d ( A 2 ) ⊗ ⋯ ⊗ I d ( A N ) ] + x | Φ d N 〉 〈 Φ d N |

Here 0 ≤ x ≤ 1 and I d ( A i ) , i = 1 , 2 , ⋯ , N are d × d unit matrices belonging to the subsystem space of each qudit A i , i = 1 , 2 , ⋯ , N . The pure state | Φ d N 〉 is given by

| Φ d N 〉 = 1 d ∑ k = 0 d − 1 | k 〉 A 1 ⊗ | k 〉 A 2 ⊗ ⋯ ⊗ | k 〉 A N . (9)

and it is an analogue of GHZ state to d-level systems. Notice that when d = 2 , i.e., for qubits, k = 0 , 1 and Equation (9) reduces to the N-qubit GHZ state

| GHZ N 〉 = 1 2 ( | 0 1 0 2 ⋯ 0 N 〉 + | 1 1 1 2 ⋯ 1 N 〉 )

The eigenvalues of ρ N d ( x ) are given by

λ 1 = 1 − x d N [ ( d N − 1 ) fold degenerate ] , λ 2 = 1 + ( d N − 1 ) x d N non-degenerate (10)

The focus here is to find the 1 : N − 1 separability range of ρ N d ( x ) using CSTRE criterion.

Denoting the first qubit as subsystem A and the remaining N − 1 qubits as subsystem B, the density matrix of the N − 1 qubit marginal is given by

ρ B = = T r A 1 ρ ( A 1 , A 2 , … , A N ) = T r A 1 ρ N d (x)

It can be seen that the eigenvalues η i of the N − 1 qubit marginal ρ B of ρ N d ( x ) , obtained by reducing over the first qubit, are given by

η 1 = 1 − x d N − 1 [ ( d N − 1 − d ) - fold degenerate ] , η 2 = 1 + ( d N − 2 − 1 ) x d N − 1 [ d - fold degenerate ] (11)

Also, the subsystem ρ A , the single qudit marginal of ρ N d ( x ) , corresponds to the maximally mixed state I d / d , I d being d × d unit matrix.

In order to find the separability range of the state ρ N d in its 1 : N − 1 partition using CSTRE criterion, one needs to evaluate the eigenvalues γ i of the sandwiched matrix

Γ = ( I A ⊗ ρ B ) 1 − q 2 q ρ N d ( x ) ( I A ⊗ ρ B ) 1 − q 2 q (12)

so that

D ˜ q T ( ρ N d ( x ) | | ρ B ) = ∑ i γ i q − 1 1 − q (13)

can be evaluated. Thus, in the evaluation of D ˜ q T ( ρ N d ( x ) | | ρ B ) , the non-negative eigenvalues γ i play a crucial role. In order to obtain the form of the eigenvalues γ i for arbitrary N, an analysis of their form for different N ( N = 2,3,4,5 ) and d ( d = 3 , 4 , 5 , 6 ) is carried out to arrive at a generalization for any N, d.

It can be readily seen from

γ 1 = ( 1 − x d N ) ( 1 − x d N − 1 ) 1 − q q , ( d N − d 2 ) -fold degenerate γ 2 = ( 1 − x d N ) ( 1 + ( d N − 2 − 1 ) x d N − 1 ) 1 − q q , ( d 2 − 1 ) -fold degenerate γ 3 = ( 1 + ( d N − 1 ) x d N ) ( 1 + ( d N − 2 − 1 ) x d N − 1 ) 1 − q q , non-degenerate . (14)

The 1 : N − 1 separability range of ρ N d ( x ) , for each combination of N = 2 , 3 , 4 , 5 and d = 3 , 4 , 5 , 6 obtained using CSTRE approach allows us to generalize this range to any N and d.

Using

0 ≤ x ≤ 1 1 + d N − 1 (15)

One can note that the 1 : N − 1 separability range given in Eq.(15) is the same as that obtained in Ref. [

Number of levels (d) | Number of parties (N) | γ 1 ( d N − d 2 ) fold degenerate | γ 2 ( d 2 − 1 ) fold degenerate | γ 3 |
---|---|---|---|---|

3 | 2 | - | ( 1 − x 9 ) ( 1 3 ) 1 − q q | ( 1 + 8 x 9 ) ( 1 3 ) 1 − q q |

3 | ( 1 − x 27 ) ( 1 − x 9 ) 1 − q q | ( 1 − x 27 ) ( 1 + 2 x 9 ) 1 − q q | ( 1 + 26 x 27 ) ( 1 + 2 x 9 ) 1 − q q | |

4 | ( 1 − x 81 ) ( 1 − x 27 ) 1 − q q | ( 1 − x 81 ) ( 1 + 8 x 27 ) 1 − q q | ( 1 + 80 x 81 ) ( 1 + 8 x 27 ) 1 − q q | |

5 | ( 1 − x 243 ) ( 1 − x 81 ) 1 − q q | ( 1 − x 243 ) ( 1 + 26 x 81 ) 1 − q q | ( 1 + 242 x 243 ) ( 1 + 26 x 81 ) 1 − q q | |

4 | 2 | - | ( 1 − x 16 ) ( 1 4 ) 1 − q q | ( 1 + 15 x 16 ) ( 1 4 ) 1 − q q |

3 | ( 1 − x 64 ) ( 1 − x 16 ) 1 − q q | ( 1 − x 64 ) ( 1 + 3 x 16 ) 1 − q q | ( 1 + 63 x 64 ) ( 1 + 3 x 16 ) 1 − q q | |

4 | ( 1 − x 256 ) ( 1 − x 64 ) 1 − q q | ( 1 − x 256 ) ( 1 + 15 x 64 ) 1 − q q | ( 1 + 255 x 256 ) ( 1 + 15 x 64 ) 1 − q q | |

5 | ( 1 − x 1024 ) ( 1 − x 256 ) 1 − q q | ( 1 − x 1024 ) ( 1 + 63 x 256 ) 1 − q q | ( 1 + 1023 x 1024 ) ( 1 + 63 x 256 ) 1 − q q | |

5 | 2 | - | ( 1 − x 25 ) ( 1 5 ) 1 − q q | ( 1 + 24 x 25 ) ( 1 5 ) 1 − q q |

3 | ( 1 − x 125 ) ( 1 − x 25 ) 1 − q q | ( 1 − x 125 ) ( 1 + 4 x 25 ) 1 − q q | ( 1 + 124 x 125 ) ( 1 + 4 x 25 ) 1 − q q | |

4 | ( 1 − x 625 ) ( 1 − x 125 ) 1 − q q | ( 1 − x 625 ) ( 1 + 24 x 125 ) 1 − q q | ( 1 + 624 x 625 ) ( 1 + 24 x 125 ) 1 − q q | |

5 | ( 1 − x 3125 ) ( 1 − x 625 ) 1 − q q | ( 1 − x 3125 ) ( 1 + 124 x 625 ) 1 − q q | ( 1 + 3124 x 3125 ) ( 1 + 124 x 625 ) 1 − q q | |

6 | 2 | - | ( 1 − x 36 ) ( 1 6 ) 1 − q q | ( 1 + 35 x 36 ) ( 1 6 ) 1 − q q |

3 | ( 1 − x 216 ) ( 1 − x 36 ) 1 − q q | ( 1 − x 216 ) ( 1 + 5 x 36 ) 1 − q q | ( 1 + 215 x 216 ) ( 1 + 5 x 36 ) 1 − q q | |

4 | ( 1 − x 1296 ) ( 1 − x 216 ) 1 − q q | ( 1 − x 1296 ) ( 1 + 35 x 216 ) 1 − q q | ( 1 + 1295 x 1296 ) ( 1 + 35 x 216 ) 1 − q q | |

5 | ( 1 − x 7776 ) ( 1 − x 1296 ) 1 − q q | ( 1 − x 7776 ) ( 1 + 215 x 1296 ) 1 − q q | ( 1 + 7775 x 7776 ) ( 1 + 215 x 1296 ) 1 − q q |

Number of levels (d) | Number of parties (N) | CSTRE separability range |
---|---|---|

3 | 2 | ( 0,0.25 ) |

3 | ( 0,0.1 ) | |

4 | ( 0,0.0357 ) | |

5 | ( 0,0.0121 ) | |

4 | 2 | ( 0,0.2 ) |

3 | ( 0,0.0588 ) | |

4 | ( 0,0.0153 ) | |

5 | ( 0,0.0039 ) | |

5 | 2 | ( 0,0.1666 ) |

3 | ( 0,0.0384 ) | |

4 | ( 0,0.0079 ) | |

5 | ( 0,0.0016 ) | |

6 | 2 | ( 0,0.1428 ) |

3 | ( 0,0.0270 ) | |

4 | ( 0,0.0046 ) | |

5 | ( 0,0.0007 ) |

In all these states, the single qudit density matrix turns out to be I d / d thus commuting with the corresponding density matrix implying that the in general non-commutative CSTRE approach yields the results equivalent to commutative AR-approach [

It can be seen that D ˜ q T ( ρ 4 ( 3 ) ( x ) | | ρ B ) is negative for x > 0.5633 when q = 1 implying that ( 0, 0.5633 ) is the separability range through Von-Neumann conditional entropy, whereas it is negative for x > 0.0357 in the limit q → ∞ leading to ( 0, 0.0357 ) as the separability range through CSTRE-criterion.

Even though the separability range of ρ N d ( x ) , obtained using both CSTRE and AR-conditional entropy are same, there is a difference in the way the parameter x converges to the value x ∞ , the value of x for which lim q → ∞ S q ( A | B ) = 0 , lim q → ∞ D ˜ q T ( ρ N ( d ) ( x ) | | ρ B ) = 0 .

Criterion | 3-level | 4-level | 5-level | ||||||
---|---|---|---|---|---|---|---|---|---|

3-party | 4-party | 5-party | 3-party | 4-party | 5-party | 3-party | 4-party | 5-party | |

CSTRE | 0.3837 | 0.3114 | 0.2744 | 0.3108 | 0.2396 | 0.2116 | 0.2610 | 0.1943 | 0.1730 |

AR | 0.3162 | 0.1889 | 0.1104 | 0.2425 | 0.1240 | 0.0623 | 0.1961 | 0.0890 | 0.0399 |

parameter x is rapidly decreasing in AR method even for q = 2 thus confirming its relatively rapid convergence in comparison with that of CSTRE in the limit q → ∞ .

The rapid convergence of the parameter x with increasing values of q in the case of AR q-conditional entropy is illustrated in

It is also evident from

Similarly a comparison of

In this article, the CSTRE criterion is employed to find out the 1 : N − 1 separability range of N-qudit Werner-Popescu states. It is observed that the 1 : N − 1 separability range obtained through both CSTRE and AR q-conditional entropy criteria match with each other for these states. The maximally mixed and hence commuting nature of the single qudit density matrix with the

Werner-Popescu state is found to be the reason behind the matching of the 1 : N − 1 separability ranges due to commutative AR-criterion and non-commutative CSTRE criterion. The relatively smoother convergence of the parameter x with respect to increasing q is observed in the case of implicit plots of CSTRE in comparison with the convergence in the case of AR q-conditional entropy thus establishing the supremacy of CSTRE criterion over the AR-criterion. The 1 : N − 1 separability range obtained for N-qudit Werner Popescu states using entropic criteria is seen to match with that obtained using an algebraic necessary and sufficient condition for separability.

Anantha S. Nayak acknowledges the support of Department of Science and Technology (DST), Govt. of India through the award of INSPIRE fellowship; A. R. Usha Devi is supported under the University Grants Commission (UGC), India (Grant No. MRP-MAJOR-PHYS-2013-29318).

Nayak, A.S., Sudha, Usha Devi, A.R. and Rajagopal, A.K. (2018) One Parameter Family of N-Qudit Werner-Popescu States: Bipartite Separability Using Conditional Quantum Relative Tsallis Entropy. Journal of Quantum Information Science, 8, 12-23. https://doi.org/10.4236/jqis.2018.81002