Identifying families of multipartite states with non-trivial local entanglement transformations (2302.03139v2)
Abstract: The study of state transformations by spatially separated parties with local operations assisted by classical communication (LOCC) plays a crucial role in entanglement theory and its applications in quantum information processing. Transformations of this type among pure bipartite states were characterized long ago and have a revealing theoretical structure. However, it turns out that generic fully entangled pure multipartite states cannot be obtained from nor transformed to any inequivalent fully entangled state under LOCC. States with this property are referred to as isolated. Nevertheless, the above result does not forbid the existence of particular SLOCC classes that are free of isolation, and therefore, display a rich structure regarding LOCC convertibility. In fact, it is known that the celebrated $n$-qubit GHZ and W states give particular examples of such classes and in this work, we investigate this question in general. One of our main results is to show that the SLOCC class of the 3-qutrit totally antisymmetric state is isolation-free as well. Actually, all states in this class can be converted to inequivalent states by LOCC protocols with just one round of classical communication (as in the GHZ and W cases). Thus, we consider next whether there are other classes with this property and we find a large set of negative answers. Indeed, we prove weak isolation (i.e., states that cannot be obtained with finite-round LOCC nor transformed by one-round LOCC) for very general classes, including all SLOCC families with compact stabilizers and many with non-compact stabilizers, such as the classes corresponding to the $n$-qunit totally antisymmetric states for $n\geq4$. Finally, given the pleasant feature found in the family corresponding to the 3-qutrit totally antisymmetric state, we explore in more detail the structure induced by LOCC and the entanglement properties within this class.
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- Notice that the examples of conversions that cannot be achieved by concatenating one-round protocols do not prove this. This is because the output state is automatically not weakly isolated (it must be finite-round reachable) and the input state can be one-round convertible to a different state.
- We say a matrix X𝑋Xitalic_X quasi-commutes with another matrix A𝐴Aitalic_A if and only if X†AX=kA∝Asuperscript𝑋†𝐴𝑋𝑘𝐴proportional-to𝐴X^{\dagger}AX=kA\propto Aitalic_X start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_A italic_X = italic_k italic_A ∝ italic_A for some k∈ℂ𝑘ℂk\in\mathbb{C}italic_k ∈ blackboard_C.
- Note that we exchange the order of α2subscript𝛼2\alpha_{2}italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and β1subscript𝛽1\beta_{1}italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT here as opposed to the notation that we use in Observation 11 to denote the states in MA3subscript𝑀subscript𝐴3M_{A_{3}}italic_M start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT.
- The argument we use here to show that B⊗B−1⊗𝟙⊗n−2∈𝒮|An⟩tensor-product𝐵superscript𝐵1superscript1tensor-productabsent𝑛2subscript𝒮ketsubscript𝐴𝑛B\otimes B^{-1}\otimes\mathbbm{1}^{\otimes n-2}\in\mathcal{S}_{|A_{n}\rangle}italic_B ⊗ italic_B start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ⊗ blackboard_1 start_POSTSUPERSCRIPT ⊗ italic_n - 2 end_POSTSUPERSCRIPT ∈ caligraphic_S start_POSTSUBSCRIPT | italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ end_POSTSUBSCRIPT is the same argument used in Ref. [56] (Sec. II) to prove that permutation-symmetric states have symmetries of the form B⊗B−1⊗𝟙⊗n−2tensor-product𝐵superscript𝐵1superscript1tensor-productabsent𝑛2B\otimes B^{-1}\otimes\mathbbm{1}^{\otimes n-2}italic_B ⊗ italic_B start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ⊗ blackboard_1 start_POSTSUPERSCRIPT ⊗ italic_n - 2 end_POSTSUPERSCRIPT.
- See p.8 of Ref. [58] for the fact that Zariski closure on ℂdsuperscriptℂ𝑑\mathbb{C}^{d}blackboard_C start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT implies Euclidean closure on ℂdsuperscriptℂ𝑑\mathbb{C}^{d}blackboard_C start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT.
- It is easy to see that the Bolzano-Weierstrass theorem also applies to bounded sequences in ℂdsuperscriptℂ𝑑\mathbb{C}^{d}blackboard_C start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT by viewing them as sequences in ℝ2dsuperscriptℝ2𝑑\mathbb{R}^{2d}blackboard_R start_POSTSUPERSCRIPT 2 italic_d end_POSTSUPERSCRIPT.
- The considered state might then be LU-equivalent to the initial state.
- Note that if there exists a consistency condition with x1(λ)=0superscriptsubscript𝑥1𝜆0x_{1}^{(\lambda)}=0italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_λ ) end_POSTSUPERSCRIPT = 0 and x2(λ)≠0superscriptsubscript𝑥2𝜆0x_{2}^{(\lambda)}\neq 0italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_λ ) end_POSTSUPERSCRIPT ≠ 0 while θ𝜃\thetaitalic_θ is an irrational multiple of π𝜋\piitalic_π, then the system of equations is inconsistent.
- We obtain Eq. (20) by first multiplying each equation in 𝐁α′→=φ′→+θ→𝐁→superscript𝛼′→superscript𝜑′→𝜃\mathbf{B}\vec{\alpha^{\prime}}=\vec{\varphi^{\prime}}+\vec{\theta}bold_B over→ start_ARG italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG = over→ start_ARG italic_φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG + over→ start_ARG italic_θ end_ARG by a factor z∈ℂ𝑧ℂz\in\mathbb{C}italic_z ∈ blackboard_C on both sides, and then exponentiating both sides of each equation.
- J. J. Sakurai. Modern Quantum Mechanics (Revised Edition). Addison Wesley, 1993.
- When multiplying Eq. (1) by |A3⟩ketsubscript𝐴3|A_{3}\rangle| italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⟩ (which is the seed state |Ψs⟩ketsubscriptΨ𝑠|\Psi_{s}\rangle| roman_Ψ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ⟩ here) where g=Δ′⊗D′⊗𝟙𝑔tensor-productsuperscriptΔ′superscript𝐷′1g=\sqrt{\Delta^{\prime}}\otimes\sqrt{D^{\prime}}\otimes\mathbbm{1}italic_g = square-root start_ARG roman_Δ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG ⊗ square-root start_ARG italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG ⊗ blackboard_1 and h=Δ⊗D⊗𝟙ℎtensor-productΔ𝐷1h=\sqrt{\Delta}\otimes\sqrt{D}\otimes\mathbbm{1}italic_h = square-root start_ARG roman_Δ end_ARG ⊗ square-root start_ARG italic_D end_ARG ⊗ blackboard_1, the term g†∑qNq†Nqg|A3⟩=0superscript𝑔†subscript𝑞superscriptsubscript𝑁𝑞†subscript𝑁𝑞𝑔ketsubscript𝐴30g^{\dagger}\sum_{q}N_{q}^{\dagger}N_{q}g|A_{3}\rangle=0italic_g start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT italic_g | italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⟩ = 0 because all Nq∈𝒩gΨssubscript𝑁𝑞subscript𝒩𝑔subscriptΨ𝑠N_{q}\in\mathcal{N}_{g\Psi_{s}}italic_N start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ∈ caligraphic_N start_POSTSUBSCRIPT italic_g roman_Ψ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUBSCRIPT satisfy Nqg|A3⟩=0subscript𝑁𝑞𝑔ketsubscript𝐴30N_{q}g|A_{3}\rangle=0italic_N start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT italic_g | italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⟩ = 0 by definition.
- Alternatively, one can see this by showing that |A3⟩ketsubscript𝐴3|A_{3}\rangle| italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⟩ is the only state among all the MES candidates in Observation 11 that has a completely mixed single qutrit reduced density matrix for all 3 bipartite splittings. Applying Nielsen’s theorem [10] to all 3 bipartitions proves that |A3⟩ketsubscript𝐴3|A_{3}\rangle| italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⟩ is indeed not LOCC-reachable.