Generalized state spaces and nonlocality in fault-tolerant quantum-computing schemes

Ratanje, N.; Virmani, S.

doi:10.1103/physreva.83.032309

Cited by 12 publications

(24 citation statements)

References 39 publications

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“…The fact that EPR pairs have a local hidden variable model for Pauli measurements can be reinterpreted as a statement that the EPR pair can be considered to be separable (i.e. non-entangled) with respect to a more general set of single-system operators that consists of cubes of Bloch vectors enclosing the usual Bloch sphere [5]. While the operators that correspond to these 'cube' Bloch vectors are not always physical, as we shall shortly describe, they can be considered as valid state descriptions if measurements are restricted to the Pauli operators.…”

Section: Overviewmentioning

confidence: 99%

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Smallest state spaces for which bipartite entangled quantum states are separable

Anwar

Jevtic

Rudolph³

et al. 2015

New J. Phys.

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According to usual definitions, entangled states cannot be given a separable decomposition in terms of products of local density operators. If we relax the requirement that the local operators be positive, then an entangled quantum state may admit a separable decomposition in terms of more general sets of single-system operators. This form of separability can be used to construct classical models and simulation methods when only a restricted set of measurements is available. With these motivations in mind, we ask what are the smallest sets of local operators such that a pure bipartite entangled quantum state becomes separable? We find that in the case of maximally entangled states there are many inequivalent solutions, including for example the sets of phase point operators that arise in the study of discrete Wigner functions. We therefore provide a new way of interpreting these operators, and more generally, provide an alternative method for constructing local hidden variable models for entangled quantum states under subsets of quantum measurements.Due to its connection with the Born rule, this definition can be used to define operators that are 'positive' for subsets of quantum measurements.

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Section: Overviewmentioning

confidence: 99%

“… see for instance [5,[14][15][16]. With such applications in mind, our goal in this work will be to try to identify the smallest choices for A  and B  such that the maximally entangled state d f ñ | is  -separable.…”

mentioning

confidence: 99%

Smallest state spaces for which bipartite entangled quantum states are separable

Anwar

Jevtic

Rudolph³

et al. 2015

New J. Phys.

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“…Campbell and Browne [19,20] identified an analog to bound entanglement, with certain families of nonstabilizer states being undistillable for finite-sized computers. Ratanje and Virmani [21] considered resource theories that interpolate between separable states and stabilizer states and found new regimes that are efficiently classically simulable.This article explores the fundamental principles that govern magic states, and we uncover several new phenomena previarXiv:1010.0104v3 [quant-ph] …”

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confidence: 99%

Catalysis and activation of magic states in fault-tolerant architectures

Campbell

2011

Phys. Rev. A

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In many architectures for fault tolerant quantum computing universality is achieved by a combination of Clifford group unitary operators and preparation of suitable nonstabilizer states, the so-called magic states. Universality is possible even for some fairly noisy nonstabilizer states, as distillation can convert many noisy copies into fewer purer magic states. Here we propose novel protocols that exploit multiple species of magic states in surprising ways. These protocols provide examples of previously unobserved phenomena that are analogous to catalysis and activation well known in entanglement theory.PACS numbers: 03.67.Pp Quantum computers are capable of executing algorithms whilst tolerating modest rates of faults or errors. Stabilizer codes encode information in subspaces of larger Hilbert spaces and allow a proportion of errors to be actively detected and corrected [1]. Whereas some anyonic systems with topologically protected ground states provide a passive method of safely storing quantum information [2]. Research into anyonic systems has been stimulated by the recent discovery of alloys that are topological insulators [3,4], opening up a variety of readily available systems that may be suitable for anyonic quantum computing.However, fault tolerant quantum computing is not just about archiving quantum information, but also processing the information whilst stored in its protected form. However, by employing stabilizer codes and topological systems we restrict how the quantum information may be fault-tolerantly manipulated. Stabilizer codes only allow coherent implementation of a limited group of fault tolerant gates, the so-called transversal gates. Unfortunately, recent research has shown that no stabilizer code can both protect against generic errors and offer a universal set of transversal gates [5]. Similarly, topologically protected groups of gates, implemented by braiding anyons, are not universal for many species of anyons [6][7][8]. Theoretically, some exotic anyons do offer universal topologically protected gates, but these are more physically speculative [9]. Consequently, an alternative route to universal and fault tolerant quantum computing must be sought out.This obstacle is overcome by gate injection techniques. A suitable resource state is identified, and through fault tolerant gates and measurements, this resource is consumed in exchange for a new fault-tolerant unitary operator that promotes the group of gates to full universality. For both stabilizer codes and anyonic systems, the manifestly fault tolerant gates are often contained within the Clifford group, the group of unitary operators that conjugate the Pauli operators. What resource states might promote the Clifford group to universality? Since the Clifford group maps stabilizer states -eigenstates of Pauli operators -to other stabilizer states, and such evolutions are efficiently classically simulable [10], we know that stabilizer states fail to provide universality. However, numer- * Electronic address: earltcampbell@gmail.com...

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“…Indeed this can be done for the Bell state of equation (2), as its density matrix can be written in a 'cube-separable' form [10,18] | | , and consequently it supplies a local hidden variable model for Pauli measurements. In passing we note that any two qubit quantum state that has a local hidden variable model for Pauli measurements must have a cube-separable decomposition [28], and these constructions can be generalised to maximally entangled states of any dimension of particles, under mutually unbiased basis measurements [28,10,18].…”

Section: Motivating Example: Discrete Wigner Representation Of Bell Smentioning

confidence: 99%

“…Note that as demonstrated by the Werner states [6] in the case of projective quantum measurements, a lack of generalised separability for a given class of measurements does not necessarily imply that a state is non-local w.r.t. those measurements (although for a small enough number of measurements and measurement outcomes, generalised separability for appropriately chosen state spaces can be equivalent to the existence of a LHV model [28]).…”

Section: Appendix: Notions Of Positivitymentioning

confidence: 99%

Generalised versions of separable decompositions applicable to bipartite entangled quantum states

Anwar

Jevtic

Rudolph³

et al. 2019

New J. Phys.

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The investigation of separable states in quantum theory has been driven by the notion that they are highly classical, in that they do not demonstrate nonlocality, and are in some contexts unable to support non-classical computation. The converse question, the extent to which entangled states do or do not support non-classical information processing, is less well understood. Motivated by this question we extend the notion of quantum separability into the entangled quantum states, by constructing separable decompositions that describe them with the 'smallest' possible sets of nonphysical local operators. We consider a few ways to define the word 'smallest' and present techniques for obtaining them. The methods involve calculating certain forms of cross norm. The results generalise significantly the results obtained in our previous work on this topic (2015 New J. Phys. 17 093047), and can be be used to construct classical simulation methods and local hidden variable models for subsets of local measurements on entangled quantum states. r are local density operators, then it has a local hidden variable model [6]. The converse is not true, in that there are entangled, non-separable, quantum states for which there are local hidden variable models for all local measurements [1,6,7]. Similarly, in the context of computational complexity, it is known that if a device consists of non-entangling quantum gates acting on product input states [8], then the device can be efficiently simulated classically. Hence in the context of gatemodel quantum computation, entanglement is also necessary for non-classical computation, although in more general models of quantum computation this might not be true (e.g. even sampling thermal states of classical many-body systems cannot always be performed efficiently [9]).

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Generalized state spaces and nonlocality in fault-tolerant quantum-computing schemes

Cited by 12 publications

References 39 publications

Smallest state spaces for which bipartite entangled quantum states are separable

Smallest state spaces for which bipartite entangled quantum states are separable

Catalysis and activation of magic states in fault-tolerant architectures

Generalised versions of separable decompositions applicable to bipartite entangled quantum states

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