Maximal tree size of few-qubit states

Le, Huy Nguyen; Cai, Yu; Wu, Xingyao; Rabelo, Rafael; Scarani, Valerio

doi:10.1103/physreva.89.062333

Cited by 4 publications

(7 citation statements)

References 20 publications

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“…Furthermore, the tools we proposed for entanglement characterization can also be useful as states complexity measures, since they share analogies with the tree size method presented in Refs. [36,37]. Indeed, the notion of tree size can be understood as the length of the shortest bracket representation of a state, which in turn is the tensor rank.…”

Section: Discussionmentioning

confidence: 99%

Fine-structure classification of multiqubit entanglement by algebraic geometry

Gharahi

Mancini

Ottaviani

2020

Phys. Rev. Research

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We present a fine-structure entanglement classification under stochastic local operation and classical communication (SLOCC) for multiqubit pure states. To this end, we employ specific algebraic-geometry tools that are SLOCC invariants, secant varieties, to show that for n-qubit systems there are 2 n n+1 entanglement families. By using another invariant,-multilinear ranks, each family can be further split into a finite number of subfamilies. Not only does this method facilitate the classification of multipartite entanglement but it also turns out to be operationally meaningful as it quantifies entanglement as a resource.

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Section: Discussionmentioning

confidence: 99%

Fine-structure classification of multiqubit entanglement by algebraic geometry

Gharahi

Mancini

Ottaviani

2020

Phys. Rev. Research

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“…With an obvious decomposition into GHZ and product states, TS = 6 for 3/7 < p ≤ 1, even though ρ ∈ W \ B when 3/7 < p ≤ p w and ρ ∈ GHZ \ W when p > p w , where p w ≈ 0.695 5427 [23].…”

Section: B Tree Size Of Few-qubit Statesmentioning

confidence: 99%

“…An exhaustive search [23] shows that TS(|W ) = 8, TS(|GHZ ) = 6, TS(|B ) = 5, and TS(|P ) = 3. So the TS of a three-qubit state can adopt only one of these four different values depending on what SLOCC class the state belongs to.…”

Section: B Tree Size Of Few-qubit Statesmentioning

confidence: 99%

“…3. Interestingly, this complexity class is unstable in the sense that an arbitrarily small deviation could bring the W state to a state in the GHZ class [23,24]. For studying how TS changes in the presence of fluctuation, we define a smoothed version of the tree size over a small neighbourhood of the desired state.…”

Section: B Tree Size Of Few-qubit Statesmentioning

confidence: 99%

“…In Ref. [23] it is shown that the maximal TS of four-qubit state is 16 and a set of criteria for identifying whether a given state has this maximal TS is given. The class of most complex four-qubit states were found to belong to a SLOCC class not described in previous inductive classifications [25].…”

Section: B Tree Size Of Few-qubit Statesmentioning

confidence: 99%

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State complexity and quantum computation

Cai

Scarani

2015

Annalen der Physik

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The complexity of a quantum state may be closely related to the usefulness of the state for quantum computation. We discuss this link using the tree size of a multiqubit state, a complexity measure that has two noticeable (and, so far, unique) features: it is in principle computable, and non-trivial lower bounds can be obtained, hence identifying truly complex states. In this paper, we first review the definition of tree size, together with known results on the most complex three and four qubits states. Moving to the multiqubit case, we revisit a mathematical theorem for proving a lower bound on tree size that scales superpolynomially in the number of qubits. Next, states with superpolynomial tree size, the Immanant states, the Deutsch-Jozsa states, the Shor's states and the subgroup states, are described. We show that the universal resource state for measurement based quantum computation, the 2D-cluster state, has superpolynomial tree size. Moreover, we show how the complexity of subgroup states and the 2D cluster state can be verified efficiently. The question of how tree size is related to the speed up achieved in quantum computation is also addressed. We show that superpolynomial tree size of the resource state is essential for measurement based quantum computation. The necessary role of large tree size in the circuit model of quantum computation is still a conjecture; and we prove a weaker version of the conjecture.

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Classifying entanglement by algebraic geometry

Gharahi

2024

Int. J. Quantum Inform.

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Quantum Entanglement is one of the key manifestations of quantum mechanics that separate the quantum realm from the classical one. Characterization of entanglement as a physical resource for quantum technology became of uppermost importance. While the entanglement of bipartite systems is already well understood, the ultimate goal to cope with the properties of entanglement of multipartite systems is still far from being realized. This paper covers characterization of multipartite entanglement using algebraic-geometric tools. First, we establish an algorithm to classify multipartite entanglement by [Formula: see text]-secant varieties of the Segre variety and [Formula: see text]-multilinear ranks that are invariant under Stochastic Local Operations with Classical Communication (SLOCC). We present a fine-structure classification of multiqubit and tripartite entanglement based on this algorithm. Another fundamental problem in quantum information theory is entanglement transformation that is quite challenging regarding to multipartite systems. It is captivating that the proposed entanglement classification by algebraic geometry can be considered as a reference to study SLOCC and asymptotic SLOCC interconversions among different resources based on tensor rank and border rank, respectively. In this regard, we also introduce a new class of tensors that we call persistent tensors and construct a lower bound for their tensor rank. We further cover SLOCC convertibility of multipartite systems considering several families of persistent tensors.

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Maximal tree size of few-qubit states

Cited by 4 publications

References 20 publications

Fine-structure classification of multiqubit entanglement by algebraic geometry

Fine-structure classification of multiqubit entanglement by algebraic geometry

State complexity and quantum computation

Classifying entanglement by algebraic geometry

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