Estimating the concurrence for quantum states via symmetric measurements

The gist The authors derive improved lower bounds of concurrence induced by symmetric measurements, which retains experimental feasibility without state tomography, and resolve a related inequality conjecture that implies numerous previous results based on symmetric measurements are strictly stronger than the one based on realignment.

Introduction to Concurrence Bounds

A pivotal issue in the theory of quantum entanglement is the quantification and estimation of entanglement for composite systems One of the most well-known entanglement measures is concurrence[2], which has the analytic formula for two-qubit quantum states[3] and is closely related to the entanglement of formation[3]. However, in highdimensional systems, due to the infimum involved in the calculation, analytical formulas for concurrence are obtained only for some very specific quantum states[4, 5]. Instead of the analytical formula for concurrence, much progress has been made toward the lower bounds of concurrence for general bipartite states. In 2005, a lower bound of concurrence[6] was proposed based on positive partial transpose (PPT) criterion[7, 8] and realignment criterion[9, 10]. Subsequently, the lower bound of concurrence based on local uncertainty relations and correlation matrix was introduced [11]. Since then various lower bounds have been derived using various methods [12–17], including the ones based on the extended realigned matrix[18, 19]. Recently, the lower bound of concurrence based on symmetric measurements was proposed[20] by employing the conical 2-design property, and this bound has been generalized to versions of generalized symmetric measurements[21] and generalized equiangular measurements[22] by using a similar method.

Symmetric Measurements and Conical 2-Designs

Symmetric informationally complete positive operator-valued measure (SIC POVM)[23] and complete set of mutually unbiased bases(MUBs)[24] are the two most famous examples of complex projective 2-designs, which are further generalized to general SIC POVM (GSIC POVM)[25] and mutually unbiased measurements (MUMs)[26]. In fact, Graydon and Appleby proposed the notion of conical 2-designs[27] as a generalization of complex projective 2-designs. Both GSIC POVM and informationally complete MUMs are the examples of this notion[28, 29]. The informationally complete (N, M)-POVM, also called symmetric measurements[30], is the important example of conical 2-designs[31, 32]. It has been shown in Refs.[20, 21, 33] that any conical 2-design can be used to give a lower bound of concurrence and a Schmidt number criterion.

Improved Lower Bound Theorem

Theorem 1 states that for a bipartite state ρ acting on HA ⊗ HB, the concurrence C(ρ) satisfies C(ρ) ≥ K s2 d(d − 1)∥Mµ,ν(ρ)∥tr − sµ2 + (dA − 1)(M2 AxA + d2 A)dAMA(MA − 1)sν2 + (dB − 1)(M2 BxB + d2 B)dBMB(MB − 1)! The proof involves relating the matrix Mµ,ν(ρ) to the state's Schmidt decomposition and showing that∥P(ρ) − P(σ)∥tr ≤ sµ2 + (dA − 1)(M2 AxA + d2 A)dAMA(MA − 1)sν2 + (dB − 1)(M2 BxB + d2 B)dBMB(MB − 1)! This theorem generalizes the lower bounds of concurrence based on symmetric measurements proposed in Refs.[20, 33].

Comparison with Realignment Criterion

Theorem 2 demonstrates that the lower bound of concurrence based on symmetric measurements [20] is always tighter than the one based on realignment [6]. The derivation shows that M(M − 1)xM2 − d∥Eˆ(XF)Eˆ†∥tr ≥∥R(ρ)∥tr − 1. This result implies that the lower bounds of q-concurrence and α-concurrence based on symmetric measurements [39] are always tighter than the one based on realignment [40–42] too.

Genuine Tripartite Entanglement

Theorem 3 provides a result for genuine multipartite entanglement (GME) concurrence, stating that CGME(ρ) ≥ 3 p d(d − 1)K (fµ,ν(ρ) − L) − 2 r d − 1 d. This bound is derived using an informationally complete (NA, MA)-POVM with the free parameter xA on C d and an informationally complete (NB, MB)-POVM with the free parameter xB on C d ⊗ C d.

Conclusions and Open Problems

The work has derived an extended version of lower bounds of concurrence induced by symmetric measurements, which can tighter than the original one for some states. Furthermore, it has proven that the Conjecture in Ref.[20] is true, explaining that numerous previous results based on symmetric measurements are found to be strictly stronger than the ones based on realignment. The authors conjecture that the essential reason behind this may be that the lower bounds of concurrence induced by arbitrary two distinct conical 2-designs are comparable. Another interesting problem is for which states the lower bounds of concurrence induced by conical 2-designs attain the exact value of concurrence. It is also worth studying whether these lower bounds can detect all PPT entangled states.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (NSFC) under Grants No.12471427 and No.12171044. The Hermitian basis operators used to construct (8, 2)-POVM in Example 1 are given by the Gell-Mann matrices Gα,k.

Appendix A

In Example 1, we used (8, 2)-POVM with the Hermitian basis operators Gα,k given by the following Gell-Mann matrices. G11 = 1/√20 1 0; 1 0 0; 0 0 0, G21 = √20-i i; i i i; i i i, G31 = √20 1-1; i-1-1; 0-1 0, G41 = √20 0-i; 0 0 i; i i i, G51 = √20 0 1; 0 i-1; i-1-1, G61 = √20 0-i; 0 i i; 0-i i i, G71 = √21-1 0; 0-1 0; 0-2 0, and G81 = √61 0 0; 0 1 i; i i-2. G8 is given in Appendix A.

References

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[3] S. A. Hill and W. K. Wootters, Entanglement of a pair of quantum bits, Phys. Rev. Lett. 78, 5022 (1997).

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[7] A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).

1. Improved entanglement detection for high-dimensional systems: The system can now derive improved lower bounds of concurrence induced by symmetric measurements, which can also be experimentally identified without state tomography, enabling experimental verification of entanglement in complex quantum states using only symmetric measurements rather than full state tomography.

2. Conjecture resolution and comparison: The AI system can rigorously test the conjecture that numerous previous results based on symmetric measurements are strictly stronger than the one based on realignment by comparing the derived lower bounds (Equation 18) with those from realignment (Equation 19), proving that although lower bound (18) is intended for unknown states, it is always tighter than lower bound (19) for known states.

3. Tripartite entanglement quantification: The system can establish a lower bound of genuine tripartite entanglement concurrence based on symmetric measurements using Theorem 3, allowing it to quantify the GME concurrence of a tripartite state by calculating the expression in Equation (43).

4. Design comparison analysis: The AI system can analyze and compare lower bounds derived from different measurement schemes, specifically showing that the lower bounds of q-concurrence and α-concurrence based on symmetric measurements [39] are always tighter than the one based on realignment [40–42] too.

5. Conical 2-design optimization: The system can address the conjecture regarding arbitrary designs by concluding that the lower bounds of concurrence induced by arbitrary two distinct conical 2-designs are comparable, guiding future research toward identifying which designs can achieve the tightest lower bound of concurrence.