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Reference Frames, Superselection Rules, and Quantum Information

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Reference Frames, Superselection Rules, and Quantum Information Reference frames, superselection rules, and quantum information Stephen D. Bartlett,1 Terry Rudolph,2,3 and Robert W. Spekkens4 1School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia 2Optics Section, Blackett Laboratory, Imperial College London, London SW7 2BW, United Kingdom 3Institute for Mathematical Sciences, Imperial College London, London SW7 2BW, United Kingdom 4Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom (Dated: 4 April 2007) Recently, there has been much interest in a new kind of “unspeakable” quantum information that stands to regular quantum information in the same way that a direction in space or a mo- ment in time stands to a classical bit string: the former can only be encoded using particular degrees of freedom while the latter are indifferent to the physical nature of the information car- riers. The problem of correlating distant reference frames, of which aligning Cartesian axes and synchronizing clocks are important instances, is an example of a task that requires the exchange of unspeakable information and for which it is interesting to determine the fundamental quantum limit of efficiency. There have also been many investigations into the information theory that is appropriate for parties that lack reference frames or that lack correlation between their reference frames, restrictions that result in global and local superselection rules. In the presence of these, quantum unspeakable information becomes a new kind of resource that can be manipulated, de- pleted, quantified, etcetera. Methods have also been developed to contend with these restrictions using relational encodings, particularly in the context of computation, cryptography, communica- tion, and the manipulation of entanglement. This article reviews the role of reference frames and superselection rules in the theory of quantum information processing. Contents B. General approach to aligning reference frames 30 C. Maximum likelihood estimation 31 I. Introduction – Why consider reference frames in 1. Maximum likelihood estimation of a phase quantum information? 2 reference 32 2. Maximum likelihood estimation of a Cartesian II. Formalizing reference frames and superselection frame 32 rules 4 D. General figures of merit 33 A. Reference frames in quantum theory 4 1. Fidelity of aligning a phase reference 34 B. Lacking a phase reference implies a photon-number 2. Fidelity of aligning a Cartesian frame 35 superselection rule 4 E. Reference frames associated with coset spaces 36 C. A general framework for reference frames and 1. Aligning a direction 36 superselection rules 6 F. Relation to phase/parameter estimation 38 G. Communication complexity of alignment 38 III. Quantum information without a shared reference H. Clock synchronization 40 frame 9 I. Other instances of alignment 41 A. Communication without a shared reference frame 10 J. Private communication of unspeakable information 41 1. Communication using photons without a shared K. Dense coding of unspeakable information 42 phase reference 10 L. Error correction of unspeakable information? 43 2. Communication without a shared Cartesian frame 11 VI. Quantum information with bounded reference arXiv:quant-ph/0610030v3 10 Apr 2007 3. Consequences for quantum information processing 13 B. QKD without a shared reference frame 14 frames 43 C. Entanglement without a shared reference frame 15 A. Measurements and state estimation with bounded 1. Entanglement without a shared phase reference 15 reference frames 43 2. Activation and entanglement distillation 16 1. A directional example 44 3. Quantifying bi-partite entanglement without a 2. Measuring relational degrees of freedom 44 shared reference frame 17 B. Quantum computation with bounded reference frames 45 4. Extensions and application to other systems 18 1. Precision of quantum gates 45 D. Private shared reference frames as cryptographic key 18 2. Degradation of a quantum reference frame 46 C. Quantum cryptography with bounded reference IV. Quantum treatment of reference frames 20 frames 47 A. Relational descriptions of phase 21 1. Data hiding with a superselection rule 48 1. Quantization of a phase reference 21 2. Ancilla-free bit commitment 48 2. Dequantization of a phase reference 22 D. Quantifying bounded shared reference frames 49 3. The optical coherence controversy 23 E. Purification of bounded shared reference frames? 51 4. Generalization to composite systems 24 F. Treating bounded reference frames as decoherence 51 B. Quantization of a general reference frame 24 C. Are certain superselection rules fundamental? 25 VII. Outlook 52 D. Superselection rules and quantum cryptography 26 Acknowledgments 53 V. Aligning reference frames 28 A. Example: sending a direction with two spins 28 References 53 2 I. INTRODUCTION – WHY CONSIDER REFERENCE It is critical to note that when one has a system encod- FRAMES IN QUANTUM INFORMATION? ing directional information, such as a spin-1/2 particle in a pure state, the direction is not defined with respect Classical information theory is typically concerned to any purported absolute Newtonian space, but rather with fungible information, that is, information for which with respect to another system, for instance, a set of the means of encoding is not important. Shannon’s cod- gyroscopes in the laboratory. Similarly, a system that ing theorems, for instance, are indifferent to whether the contains phase information, such as a two-level atom in two values “0” and “1” of a classical bit correspond to a coherent superposition of ground and excited states, is two values of magnetization on a tape, two voltages on not defined relative to any purported absolute time, but a transmission line, or two positions of a bead on an rather relative to a clock. We refer to the systems with abacus. Most information-processing tasks of interest to respect to which unspeakable/nonfungible information is computer scientists and information theorists are of this defined, clocks, gyroscopes, metre sticks and so forth, as sort, whether they be communication tasks such as data reference frames. The tasks we have highlighted thus far compression, cryptographic tasks such as key distribu- can all be described as the alignment of reference frames. tion, or computational tasks such as factoring. Nonethe- Nonfungible information is nonfungible precisely because less, there are many tasks that cannot be achieved with it can only be defined with respect to a particular type fungible information but that are also aptly described of reference frame. as “information processing” tasks. Examples include the Even a quantum information theorist who is uninter- synchronization of distant clocks, the alignment of dis- ested in tasks such as clock synchronization and Carte- tant Cartesian frames, and the determination of one’s sian frame alignment must necessarily consider physical global position. Imagine for instance that Alice and Bob systems which make use of reference frames. The reason are in separate spaceships with no shared Cartesian frame is that although fungible information can be encoded into (in particular, no access to the fixed stars). There is any degree of freedom, and thus defined with respect to clearly no way for Alice to describe a direction in space any reference frame, it is still the case that some degree to Bob abstractly, that is, using nothing more than a of freedom must be chosen, and consequently some refer- string of classical bits. Rather, she must send to Bob a ence frame is required. For instance, if a two-level atomic system that can point in some direction, a token of one qubit is being used for some task, one still requires a clock of the axes of her own Cartesian frame. This token can- in the background in order to implement arbitrary prepa- not be spherically symmetric; it must have a degree of rations and measurements on this qubit even if the task freedom that can encode directional information. On the is to perform abstract quantum information-processing other hand, if she wishes to synchronize her clock with rather than as a means of distributing phase informa- Bob’s by sending him a token system, she will need to tion. In this example, one can change the relative phase make use of a system that has a natural oscillation. The between the ground and excited states of a two-level atom information that is communicated in these sorts of tasks by a specified amount by turning on a static electric field is said to be nonfungible. These two sorts of information, for a specific time interval, but this requires a suitably fungible and nonfungible, have also been referred to as precise clock as well as alignment of the field with the speakable and unspeakable (Peres and Scudo, 2002b). atomic dipole moment. The relatively young field of quantum information It follows that to lack a reference frame for a particu- theory has been primarily concerned with developing lar degree of freedom has an impact on the success with a quantum theory of speakable information. Investi- which one can perform certain quantum information pro- gators have sought to determine the degree of success cessing tasks. On several occasions there has been consid- with which various abstract information-processing tasks erable controversy over the performance of certain tasks can be achieved assuming that the systems used to im- because this impact was ignored, or not treated properly. plement these tasks obey the laws of quantum theory. As we will see, the lack of
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