Susceptibility of qudit-qudit entanglement to quantum noise: insights from the negativity rank

The gist The negativity rank provides greater (reduced) robustness to targeted (depolarizing) noise in qudit-qudit entanglement, offering insights into how structural properties of entanglement impact its relationship with various forms of quantum noise.

Robustness to Depolarization

The global depolarizing noise channel introduces an unpolarized source into an ensemble, updating a two-qudit density matrix as Ep(ρ) = (1 − p)ρ + pI/d2, where d2 is the dimension of the Hilbert space and ρ is an element in the set of d2 × d2 trace-1 Hermitian positive semidefinite matrices. In terms of the Bloch vector r of (d4 − 1) scalar coefficients governing the generalized Pauli (σ) decomposition of a density matrix ρ(r) = I+r·σd2, the depolarizing channel simply shrinks this vectorization of the ensemble to Ep (ρ (r)) = ρ (r(1 − p)). The evolution of negativity upon global depolarization is more naturally expressed recursively because the partial transpose (PT) is a linear map, and Ep (ρ)Γ = Ep ρΓ. Since the negativity is the sum of negative eigenvalues in the spectrum of ρΓ, Ni = (1 − p)Ni−1 − pn−/d2. The rate of negativity reduction under depolarizing noise is governed by its rank n−, with higher rank associated with more rapid decay.

Qudit-Qudit APPT Boundary

For pure states, any separable state can be transformed to be maximally entangled by application of an appropriate entangling unitary. Mixed ensembles, however, exhibit constrained entanglement generation. The absolutely positive partial transpose (APPT 1) subspace is equivalent to the absolutely separable (ASEP) subspace for qubit-qudit. With the understanding of negativity decay under depolarizing noise established in Section II A, the APPT boundary can be straightforwardly calculated for mixtures expressible as depolarized pure states. This set of ensembles includes the wellstudied isotropic states, and r0 agrees with the standard APPT boundary calculated for them. Furthermore, the spectrum of the APPT ensemble resulting from the decay of the Bell state under depolarizing noise turns out to be an extremal point of the APPT set.

Maximum Purity of Qutrit-Qutrit APPT Ensemble

For qutrit-qutrit states, there are two matrix inequalities that determine APPT classification. The goal of this section is to relate the purity of a state to the value of its L1 and L2 negative eigenvalues. The maximum purity of an APPT ensemble is P APPT max = 17/121. This maximum APPT purity is achieved when the initial state has P0 = 1 and µmin = −1.

Maximal Rank NPT Subspaces

The upper bound on max n− for a dA × dB dimensional Hilbert space has been shown to be (dA − 1)(dB − 1). This bound on n− is governed by the maximum dimension of an NPT subspace.

Robustness to Statewise Mixture

The negativity rank of an entangled pure state is shown to be exactly the maximum number of arbitrary separable states that can be included in arbitrary mixture while the ensemble remains NPT. This result achieves a tight bound that determines the maximum by proving that the negativity rank of a pure state is invariant upon arbitrary mixture with any separable state. A mixed ensemble in the form of Eq. (31) composed of an entangled pure state ψ0⟩ ∈ NPT with Schmidt rank s and N ≤ s(s−1)2 - 1 additional arbitrary pure separable states is NPT.

Entangled Mixtures

The minimum number of entangled states capable of eliminating the entanglement through classical mixture scales linearly with the Schmidt rank. A constructive strategy of mixing s − 1 entangled states to produce a PPT mixture proves that the boundary in Theorem IV.5 is tight for an arbitrary ψ0⟩ entangled state. This constructive strategy of mixing s − 1 entangled states to produce a PPT mixture proves that the boundary in Theorem IV.5 is tight for an arbitrary ψ0⟩ entangled state.

Discussion

The negativity rank of an entangled pure state is shown to be exactly the maximum number of arbitrary separable states that can be included in arbitrary mixture while the ensemble remains NPT. The negativity rank is found to set the rate of negativity decay under global depolarizing noise. The present depolarizing noise analysis also generates an analytic derivation for the maximum purity of a qutrit-qutrit APPT ensemble. This work provides guidance for defining specifications of strategically tailored entanglement.

Conclusion

The negativity rank provides guidance for defining specifications of strategically tailored entanglement. The structural incompatibility that produces these invariances of negativity rank or NPT classification may yield further opportunities for embedding quantum information into subspaces that retain entanglement amid logical noise channels.

1. textbf Engagement Robustness Quantification through Negativity Rank Monitoring (Section II A): This enables AI systems to dynamically assess how quantum correlations degrade under global depolarizing noise by tracking the 'negativity rank n−', which is stated to be governed by its rank n−, with higher rank associated with more rapid decay.

2. textbf APPT Boundary Calculation for Mixed Ensembles (Section II B): AI systems can analytically derive the boundary for generating entanglement, using the result that the Bloch vector norm at which the Bell state becomes PPT under the effect of depolarizing noise is found to be r0 = p0 rpure = 2/d squared + 2p / (p/d squared - 1).

3. textbf Maximum Purity Constraint for Qutrit-Qutrit States (Section II C): AI systems can calculate the maximum purity of a qutrit-qutrit APPT state to provide a computationally amenable softening of constraints for a semidefinite program, which can complement the search for APPT entangled states.

4. textbf Noise Boundary Identification (Section II A/B): The framework allows for direct calculation of (1) certain noise boundaries for the Absolutely Positive Partial Transpose (APPT) subspace beyond which no unitary operator is capable of generating negativity and (2) the maximum qutritqutrit APPT purity.

5. textbf Separable Mixture Capacity Bound (Section IV A): AI systems can determine tight bounds on the number of arbitrary separable states that can be mixed with an NPT pure state while retaining the NPT classification, establishing that N is inclusively upperbounded by the negativity rank of ψ0⟩.

6. textbf Entangled Mixture Capacity Bound (Section IV B): The system can establish bounds on mixing entangled states, showing that a mixed ensemble remains NPT when composed of an entangled pure state and up to s − 2 additional arbitrary entangled pure states, where 's' is the Schmidt rank.

7. textbf Invariance of Negativity Rank under Mixture (Section IV A/B): The analysis demonstrates that for a mixture of an entangled pure state and an arbitrary separable state, the negativity rank is invariant upon arbitrary mixture with any separable state, meaning N can be as large as s(s−1)2.

8. textbf Noise-Resilient Entanglement Witness Filtering (Section V Discussion): The tight purity upper bound of 17/121 allows for the evaluation of entanglement witnesses, which can guide numerical searches for APPT entangled ensembles by identifying states that might be sensitive to PPT entanglement.

One aspect of complexity for quantum many-body systems resides in their ability to exhibit entanglement, uniquely quantum correlations central to quantum computation. To quantitatively explore the diminishing effect that quantum noise has on entanglement resources, we focus on the negativity witness, an entanglement measure based on local time reversal transformations that provides a perfect witness in simple physical contexts. Beyond calculation of the negativity, we analyze the associated eigenvalue spectra of partially transposed density matrices for qudit-qudit bipartitions, finding that higher negativity rank provides greater(reduced) robustness to targeted(depolarizing) noise. In particular, we extend the maximum number of separable states that can be arbitrarily mixed into an entangled pure ensemble while retaining negativity to be exactly equal to the original negativity rank. Furthermore, we report how understanding the role of the negativity rank with respect to the completely depolarizing channel allows direct calculation of (1) certain noise boundaries for the Absolutely Positive Partial Transpose (APPT) subspace beyond which no unitary operator is capable of generating negativity and (2) the maximum qutrit-qutrit APPT purity. Beyond proving tight bounds and contributing to the ongoing search for APPT entanglement, these results expand our understanding of the nuanced relationship between computation with entanglement and practical environments for operating quantum devices.