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Verifying Entanglement of Constrained Bipartite Quantum Systems

Jones, Steven

Published Jan 1, 2009• 0 citations
Publisher

Abstract

Microscopic physical systems requiring a quantum description are increasingly being harnessed to develop useful technologies. The appeal of such an approach lies in the fact that quantum systems possess inherently different properties from those observed on a macroscopic scale. Uniquely quantum phenomena, such as entanglement, offer new tools for performing experimental tasks. For instance, the use of quantum systems for implementing cryptographic protocols and performing computation has caused significant excitement in recent years as it seems likely that the quantum approach offers an advantage over purely classical implementations. The development of quantum technologies leads to the concept of quantum resources. One key quantum resource that lies at the heart of the divergence between quantum and classical descriptions is that of entanglement. However, for entanglement to act as a resource, it must be possible for the parties (we will restrict ourselves to only two) sharing the entanglement to verify its presence. This may seem like a simple task, however, verifying the properties of quantum systems can be nontrivial. To make matters more complicated, there are often constraints under which the parties sharing the quantum systems find themselves which further restrict the ability of the quantum systems to act as a resource. One family of constraints arises when the two parties are dealing with quantum systems which have properties constrained by a conservation law, or some other constraint which limits knowledge of the properties of the local subsystems. These constraints can be expressed using the well studied formalism of superselection rules. When a superselection rule applies to quantum systems it can be ambiguous as to whether entanglement survives between the systems. This thesis considers a situation that arises naturally in a nuclear magnetic resonance experiment where two parties share many copies of identical quantum systems. In this case a superselection rule associated with the finite symmetric group applies due to the indistinguishability of the systems. The concept of a reference frame for the symmetric group is used to alleviate the effect of the superselection rule and demonstrate when entanglement survives despite the constraint of indistinguishability. In addition to this, a situation which goes beyond the superselection formalism is considered, one where the operations performable by the two parties are further constrained. It is shown that despite this stronger constraint, it is still possible in principle to verify the existence of entanglement experimentally as demonstrated by the violation of a Bell inequality. Methods for verifying the presence of entanglement can also prove useful in characterising the properties of entangled quantum systems. Such characterisation may offer insight into the possible advantages offered by quantum systems over their classical counterparts. It becomes possible to more accurately characterise entangled systems by considering a conceptual constraint under which the two parties may find themselves. This is the situation that arises when one of the parties, say Bob, distrusts the other, Alice. In this situation, Alice can only convince Bob of the presence of their entanglement if the entangled system they share possesses a certain property. We term this property steerability, after Schrödinger’s term “steering” [Proc. Camb. Phil. Soc. 31, 553 (1935)] for the Einstein-Podolsky-Rosen effect. This thesis offers a rigorous characterisation of steering in a quantum information context and probes a number of questions relating to this property. First, a number of examples are presented which demonstrate state steerability which is a uniquely quantum property possessed by certain entangled states. Second, the task of experimentally detecting steerability, and thus entanglement, is addressed through the introduction of steerability criteria. Finally two more detailed examples are explored which illustrate various aspects of the measurement dependence of steerability in realistic experiments.

DOI
10.25904/1912/2654

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Microscopic physical systems requiring a quantum description are increasingly being harnessed to develop useful technologies. The appeal of such an approach lies in the fact that quantum systems possess inherently different properties from those observed on a macroscopic scale. Uniquely quantum phenomena, such as entanglement, offer new tools for performing experimental tasks. For instance, the use of quantum systems for implementing cryptographic protocols and performing computation has caused significant excitement in recent years as it seems likely that the quantum approach offers an advantage over purely classical implementations. The development of quantum technologies leads to the concept of quantum resources. One key quantum resource that lies at the heart of the divergence between quantum and classical descriptions is that of entanglement. However, for entanglement to act as a resource, it must be possible for the parties (we will restrict ourselves to only two) sharing the entanglement to verify its presence. This may seem like a simple task, however, verifying the properties of quantum systems can be nontrivial. To make matters more complicated, there are often constraints under which the parties sharing the quantum systems find themselves which further restrict the ability of the quantum systems to act as a resource. One family of constraints arises when the two parties are dealing with quantum systems which have properties constrained by a conservation law, or some other constraint which limits knowledge of the properties of the local subsystems. These constraints can be expressed using the well studied formalism of superselection rules. When a superselection rule applies to quantum systems it can be ambiguous as to whether entanglement survives between the systems. This thesis considers a situation that arises naturally in a nuclear magnetic resonance experiment where two parties share many copies of identical quantum systems. In this case a superselection rule associated with the finite symmetric group applies due to the indistinguishability of the systems. The concept of a reference frame for the symmetric group is used to alleviate the effect of the superselection rule and demonstrate when entanglement survives despite the constraint of indistinguishability. In addition to this, a situation which goes beyond the superselection formalism is considered, one where the operations performable by the two parties are further constrained. It is shown that despite this stronger constraint, it is still possible in principle to verify the existence of entanglement experimentally as demonstrated by the violation of a Bell inequality. Methods for verifying the presence of entanglement can also prove useful in characterising the properties of entangled quantum systems. Such characterisation may offer insight into the possible advantages offered by quantum systems over their classical counterparts. It becomes possible to more accurately characterise entangled systems by considering a conceptual constraint under which the two parties may find themselves. This is the situation that arises when one of the parties, say Bob, distrusts the other, Alice. In this situation, Alice can only convince Bob of the presence of their entanglement if the entangled system they share possesses a certain property. We term this property steerability, after Schrödinger’s term “steering” [Proc. Camb. Phil. Soc. 31, 553 (1935)] for the Einstein-Podolsky-Rosen effect. This thesis offers a rigorous characterisation of steering in a quantum information context and probes a number of questions relating to this property. First, a number of examples are presented which demonstrate state steerability which is a uniquely quantum property possessed by certain entangled states. Second, the task of experimentally detecting steerability, and thus entanglement, is addressed through the introduction of steerability criteria. Finally two more detailed examples are explored which illustrate various aspects of the measurement dependence of steerability in realistic experiments.
In recent years it has become apparent that constraints on possible quantum operations, such as those constraints imposed by superselection rules (SSRs), have a profound effect on quantum information theoretic concepts like bipartite entanglement. This paper concentrates on a particular example: the constraint that applies when the parties (Alice and Bob) cannot distinguish among certain quantum objects they have. This arises naturally in the context of ensemble quantum information processing such as in liquid NMR. We discuss how a SSR for the symmetric group can be applied, and show how the extractable entanglement can be calculated analytically in certain cases, with a maximum bipartite entanglement in an ensemble of NNN Bell-state pairs scaling as log(N)\mathrm{log}(N)log(N) as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We discuss the apparent disparity with the asymptotic $(N\ensuremath{\rightarrow}\ensuremath{\infty})$ recovery of unconstrained entanglement for other sorts of superselection rules, and show that the disparity disappears when the correct notion of applying the symmetric group SSR to multiple copies is used. Next we discuss reference frames in the context of this SSR, showing the relation to the work of von Korff and Kempe [Phys. Rev. Lett. 93, 260502 (2004)]. The action of a reference frame can be regarded as the analog of activation in mixed-state entanglement. We also discuss the analog of distillation: there exist states such that one copy can act as an imperfect reference frame for another copy. Finally we present an example of a stronger operational constraint, that operations must be noncollective as well as symmetric. Even under this stronger constraint we, nevertheless, show that Bell nonlocality (and hence entanglement) can be demonstrated for an ensemble of NNN Bell-state pairs no matter how large NNN is. This last work is a generalization of that of Mermin [Phys. Rev. D 22, 356 (1980)].
Entanglement is a resource central to quantum information (QI). In particular, entanglement shared between two distant parties allows them to do certain tasks that would otherwise be impossible. In this context, we study the effect on the available entanglement of physical restrictions on the local operations that can be performed by the two parties. We enforce these physical restrictions by generalized superselection rules (SSRs), which we define to be associated with a given group of physical transformations. Specifically the generalized SSR is that the local operations must be covariant with respect to that group. Then we operationally define the entanglement constrained by a SSR, and show that it may be far below that expected on the basis of a naive (or ``fluffy bunny'') calculation. We consider two examples. The first is a particle number SSR. Using this we show that for a two-mode BEC (with Alice owning mode AAA and Bob mode BBB), the useful entanglement shared by Alice and Bob is identically zero. The second, a SSR associated with the symmetric group, is applicable to ensemble QI processing such as in liquid-NMR. We prove that even for an ensemble comprising many pairs of qubits, with each pair described by a pure Bell state, the entanglement per pair constrained by this SSR goes to zero for a large ensemble.
Bipartite entanglement may be reduced if there are restrictions on allowed local operations. We introduce the concept of a generalized superselection rule to describe such restrictions, and quantify the entanglement constrained by it. We show that ensemble quantum information processing, where elements in the ensemble are not individually addressable, is subject to the superselection rule associated with the symmetric group (the group of permutations of elements). We prove that even for an ensemble comprising many pairs of qubits, each pair described by a pure Bell state, the entanglement per element constrained by this superselection rule goes to zero for a large number of elements.
Superselection rules severely constrain the operations which can be implemented on a distributed quantum system. While the restriction to local operations and classical communication gives rise to entanglement as a nonlocal resource, particle number conservation additionally confines the possible operations and should give rise to a new resource. In Phys. Rev. Lett. 92, 087904 (2004) we showed that this resource can be quantified by a single additional number, the superselection-induced variance (SIV)(\mathrm{SIV})(SIV), without changing the concept of entanglement. In this paper, we give the results on pure states in greater detail; additionally, we provide a discussion of mixed-state nonlocality with superselection rules where we consider both formation and distillation. Finally, we demonstrate that SIV\mathrm{SIV}SIV is indeed a resource, i.e., that it captures how well a state can be used to overcome the restrictions imposed by the superselection rule.
Despite the success of quantum mechanics in predicting the outcomes of experiments in many branches of physics, the foundations of the theory have remained subject of research and dispute. At the basis of this struggle with the theory lie the phenomena of nonlocality and entanglement. Since it was first predicted by Bell in 1964, nonlocality was not only verified in numerous experiments, but also identified as a useful resource for quantum information processing. Thus, the study of nonlocality is important both from a fundamental point of view and with respect to new applications in quantum information theory, such as secure cryptography and randomness generation. The identification of entanglement as a resource for quantum information led to a strong theoretical effort devoted to its characterisation and detection. Many of the resulting mathematical tools find application in several domains of physics. Although the only known way to create nonlocal correlations is to measure entangled quantum systems, it has been shown that entanglement and nonlocality constitute two inequivalent properties. Therefore, in the light of the success of entanglement theory, it is of interest to also devise a resource theory of nonlocality. The task of this thesis is to develop such a theoretical framework for the characterisation of nonlocality as a resource. To gain a better understanding of nonlocal correlations it will be necessary to investigate correlation scenarios that go beyond the situation originally considered by Bell. In doing so, this thesis provides new description of nonlocality that also have implications for the characterisation of quantum correlations and the detection of new forms of nonlocality. The first question we address is how nonlocality can consistently be defined in a scenario of arbitrarily many parties that may collaborate among each other. To this end we recognise which are the allowed physical operations in this situation and then define nonlocality as the resource that cannot be created by these operations. Our approach shows that the conventional definition of multipartite nonlocality, adopted by the community so far, is inconsistent with this operational definition; we further propose and analyse new models that do not suffer from these inconsistencies. Furthermore, we show that our findings have implications for the characterisation of quantum correlations. A recent approach to describe the set of quantum correlation consists in using principles inspired from information theory. By using a special instance of the models we defined earlier we show a fundamental limitation of this approach: no bipartite information principle is sufficient to single out the set of quantum correlations from the set of nonsignalling correlations. We then developed a description of nonlocality in an even more generalised scenario of several parties. Motivated by a result of Popescu we study scenarios where the parties are allowed to perform not only a single but sequences of measurements on their systems. Characterising nonlocality also in this scenario in operational terms and defining local models compatible with this definition, we show that a new form of nonlocality can be detected Lastly, we examine the problem of detecting the presence of nonlocality in a multipartite scenario when one is given only partial access to the global system. We find that one can verify that the total system must display nonlocality, even though the accessible subsystems only exhibit local correlations.
The phenomenon of quantum entanglement is a fundamental feature of quantum mechanics which, as a counterintuitive and inherently ”quantum” phenomenon (with no classical analogue) has been the subject of much study, especially in quantum information theory. One fruitful approach to the description of entanglement has been in its operational description - that is, in the consideration of what can be achieved using entangled states under certain restrictions, typically the regime of local operations and classical communications.\n\nWe present results here related to the operational characterisation of entanglement in the resource model, in both bipartite and multipartite cases. First, we consider the conversion between pure bipartite entangled states in terms of an often-ignored resource - the classical communication cost. Using prior results for more specific conversions, we derive lower bounds on this cost (and the related quantity of the conversion inefficiency) for general bipartite pure states.\n\nWe also consider pure-state conversions of multipartite entanglement, in particular the class of protocols in which multipartite states are converted to states shared between fewer parties. We have found a previously-unconsidered variety of such conversions, in which the target state of the conversion is a state shared between a random subset of the parties. We find that when such post-selection of parties in the protocol is permitted allows for a wider variety of achievable target states; certain states which can not be reliably obtained between predetermined parties (even some where the probability of doing so is arbitrarily small) can be obtained between random parties. We consider a variety of states in which this phenomenon occurs, as well as bounds on such protocols can achieve.\n\nFinally we consider a practical use of entanglement as a resource, in an experimental implementation of a multipartite QKD protocol. This is based on the tripartite GHZ entangled state, but can be implemented using only bipartite entanglement. We adapt existing QKD results for both the bipartite and multipartite case to derive a secure key rate for this implementation, taking into account the ways in which it differs from the idealised theoretical case.