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Journal of Physics A Mathematical and General

Searching for highly entangled multi-qubit states

Iain D K Brown, Susan Stepney, Anthony Sudbery, Samuel L. Braunstein

Published Jan 20, 2005• 198 citations
Publisher

Abstract

We present a simple numerical optimization procedure to search for highly entangled states of 2, 3, 4 and 5 qubits. We develop a computationally tractable entanglement measure based on the negative partial transpose criterion, which can be applied to quantum systems of an arbitrary number of qubits. The search algorithm attempts to optimize this entanglement cost function to find the maximal entanglement in a quantum system. We present highly entangled 4-qubit and 5-qubit states discovered by this search. We show that the 4-qubit state is not quite as entangled, according to two separate measures, as the conjectured maximally entangled Higuchi–Sudbery state. Using this measure, these states are more highly entangled than the 4-qubit and 5-qubit GHZ states. We also present a conjecture about the NPT measure, inspired by some of our numerical results, that the single-qubit reduced states of maximally entangled states are all totally mixed.

DOI
10.1088/0305-4470/38/5/013

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We present a simple numerical optimization procedure to search for highly entangled states of 2, 3, 4 and 5 qubits. We develop a computationally tractable entanglement measure based on the negative partial transpose criterion, which can be applied to quantum systems of an arbitrary number of qubits. The search algorithm attempts to optimize this entanglement cost function to find the maximal entanglement in a quantum system. We present highly entangled 4-qubit and 5-qubit states discovered by this search. We show that the 4-qubit state is not quite as entangled, according to two separate measures, as the conjectured maximally entangled Higuchi–Sudbery state. Using this measure, these states are more highly entangled than the 4-qubit and 5-qubit GHZ states. We also present a conjecture about the NPT measure, inspired by some of our numerical results, that the single-qubit reduced states of maximally entangled states are all totally mixed.
Recent works by Brown et al and Borras et al have explored numerical optimisation procedures to search for highly entangled multi-qubit states according to some computationally tractable entanglement measure. We present an alternative scheme based upon the idea of searching for states having not only high entanglement but also simple algebraic structure. We report results for 4, 5, 6, 7 and 8 qubits discovered by this approach, showing that many of such states do exist. In particular, we find a maximally entangled 6-qubit state with an algebraic structure simpler than the best results known so far. For the case of 7, we discover states with high, but not maximum, entanglement and simple structure, as well as other desirable properties. Some preliminary results are shown for the case of 8 qubits.
Numerical optimization methods such as hillclimbing and simulated annealing have been applied to search for highly entangled multi-qubit states. Here the genetic algorithm is applied to this optimization problem -- to search not only for highly entangled states, but also for the corresponding quantum circuits creating these states. Simple quantum circuits for maximally (highly) entangled states are discovered for 3, 4, 5, and 6-qubit systems; and extension of the method to systems with more qubits is discussed. Among other results we have found explicit quantum circuits for maximally entangled 5 and 6-qubit circuits, with only 8 and 13 quantum gates respectively. One significant advantage of our method over previous ones is that it allows very simple construction of quantum circuits based on the quantum states found.
We revisit the criterion of multi-particle entanglement based on the overlaps of a given quantum state ρρρ with maximally entangled states. For a system of mmm particles, each with NNN distinct states, we prove that ρρρ is mmm-particle negative partial transpose (NPT) entangled, if there exists a maximally entangled state ∣MES>|{\rm MES}>∣MES>, such that >1/N>{1}/{N}>1/N. While this sufficiency condition is weaker than the Peres-Horodecki criterion in all cases, it applies to multi-particle systems, and becomes especially useful when the number of particles (mmm) is large. We also consider the converse of this criterion and illustrate its invalidity with counter examples.
We first present a generalized criterion for maximally entangled states of 2–8 and in theory to arbitrary-number qubits. By this criterion, some known highly entangled multi-qubit states are examined and a new genuine eight-qubit maximally entangled state is obtained. For the four-, seven- and eight-qubit system in which no perfect maximally multi-qubit entangled state (MMES) was thought to exist before, we find that the proven four- and eight-qubit MMESs and the suspected seven-qubit MMES, are not completely mixed in subsystems with two, four and three qubits, respectively, but are completely mixed in subsystems with fewer qubits. The new criterion and MMES can play important roles in quantum information technology, such as teleportation and dense coding.
The main topic of this compilation thesis is the investigation of multipartite entanglement of finite dimensional systems. We developed a numerical algorithm that detects if a multipartite state is entangled or separable in a finite number of steps of a semi-definite optimization task. This method is an extension of previously known semi-definite methods, which are inconclusive when the state is separable. In our case, if the state is separable, an explicit decomposition into a mixture of separable states can be extracted. This was achieved by mapping the entanglement problem onto the mathematically well studied truncated moment problem. Additionally, a new way of writing the partially transposed state for symmetric multi-qubit states was developed which simplifies many results previously known in entanglement theory. This new way of writing the partial transpose criterion unifies different interpretations and alternative formulations of the partial transpose criterion and it is also a part in the aforementioned semi-definite algorithm. The geometric properties of entangled symmetric states of two qubits were studied in detail: We could answer the question of how far a given pure state is from the convex hull of symmetric separable states, as measured by the Hilbert-Schmidt distance, by giving an explicit formula for the closest separable symmetric state. For mixed states we could provide a numerical upper and analytical lower bound for this distance. For a larger number of qubits we investigated the ball of absolutely classical states, i.e. symmetric multi-qubit states that stay separable under any unitary transformation. We found an analytical lower bound for the radius of this ball around the maximally mixed symmetric state and gave a numerical upper bound on this radius, by searching for an entangled state as close as possible to the maximally mixed symmetric state. The tensor representation of a symmetric multi-qubit state, or spin-j state, allowed us to study entanglement properties based on the spectrum of the tensor via tensor eigenvalues. The definiteness of this tensor relates to the entanglement of the state it represents and, hence, the smallest tensor eigenvalue can be used to detect entanglement. However, the tensor eigenvalues are more difficult to determine than the familiar matrix eigenvalues which made the investigation computationally more challenging. The relationship between the value of the smallest tensor eigenvalue and the amount of entanglement in the state was also investigated. It turned out that they are strongly correlated for small system sizes, i.e. for up to six qubits. However, to investigate this correlation we needed an independent way to gauge the amount of entanglement of a state and in order to do so we improved existing numerical methods to determine the distance of a state to the set of separable symmetric states, using a combination of linear and quadratic programming. The tensor representation of symmetric multi-qubit states was also used to formally define a new tensor class of regularly decomposable tensors that corresponds to the set of separable symmetric multi-qubit states.
In chapter 2 various parameterizations for the orbits under local unitary transformations of three-qubit pure states are analyzed. It is shown that the entanglement monotones of any multipartite pure state uniquely determine the orbit of that state. It follows that there must be an entanglement monotone for three-qubit pure states which depends on the Kempe invariant defined in [1]. A form for such an entanglement monotone is proposed. A theorem is proved that significantly reduces the number of entanglement monotones that must be looked at to find the maximal probability of transforming one multipartite state to another. In chapter 3 Grover's unstructured quantum search algorithm is generalized to use an arbitrary starting superposition and an arbitrary unitary matrix. A formula for the probability of the generalized Grover's algorithm succeeding after n iterations is derived. This formula is used to determine the optimal strategy for using the unstructured quantum search algorithm. The speedup obtained illustrates that a hybrid use of quantum computing and classical computing techniques can yield a performance that is better than either alone. The analysis is extended to the case of a society of k quantum searches acting in parallel. In chapter 4 the positive map Г : p → (Trρ) - ρ is introduced as a separability criterion. Any separable state is mapped by the tensor product of Г and the identity in to a non-negative operator, which provides a necessary condition for separability. If Г acts on a two-dimensional subsystem, then it is equivalent to partial transposition and therefore also sufficient for 2 x 2 and 2 x 3 systems. Finally, a connection between this map for two qubits and complex conjugation in the magic basis [2] is displayed.