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Prediction is the making of statements, usually probabilistic, about future events based on current information. Retrodiction is the making of statements about past events based on current information. We present the foundations of quantum retrodiction and highlight its intimate connection with the Bayesian interpretation of probability. The close link with Bayesian methods enables us to explore controversies and misunderstandings about retrodiction that have appeared in the literature. To be clear, quantum retrodiction is universally applicable and draws its validity directly from conventional predictive quantum theory coupled with Bayes' theorem., Comment: New version, accepted by journal: Symmetry

The results of experiments in quantum mechanics can be predicted correctly either by assigning a forward-evolving state to the system based on the preparation outcome or by assigning a state that evolves backwards in time based on the measurement outcome. The latter picture admits some retrocausality without allowing messages to be sent at a faster speed than that of light. This retrocausality allows some standard quantum paradoxes to be examined from a different viewpoint. It also allows closed causal cycles to be examined in the context of laboratory experiments. For a particular experiment, we find agreement with the principle that inconsistent causal loops have zero probability of occurring, that is, only self-consistent loops can occur.

A retrocausal interpretation of quantum mechanics is examined and is applied to the problem of measuring an optical qubit before the qubit is actually created. Although the predictions of the retrocausal interpretation are the same as for the conventional causal picture, it provides a new perspective which should give a useful way of understanding some quantum mechanical processes.

We show explicitly how the causal arrow of time that follows from quantum mechanics has already been inserted at a deeper level by the choice of normalisation conditions. This prohibits information being sent backwards in time but does not determine a time direction for state propagation.

We use the impossibility of sending information faster than light to show that it is impossible to distinguish between various types of laser cavity field state, such as a Fock state of uncertain energy and any superposition of Fock states of unknown free evolution time, by means of the physically measurable properties of the laser beam. Superpositions can include coherent states, squeezed states and cat states, indeed any pure intracavity state will produce the characteristic properties of the beam such as intrinsic phase coherence.

Methods proposed so far for measuring the canonical phase probability distribution of light are very difficult to implement. In particular the method of projection synthesis, which synthesizes a phase state projector, involves the use of a beam splitter and a specially engineered reference state. Here we extend this method by use of an additional beam splitter to synthesize phase sine and cosine operators. This allows the measurement of all the moments needed to construct the phase distribution. The reference state required for operator synthesis is, remarkably, just an easily prepared coherent state.

Buschʼs theorem deriving the standard quantum probability rule can be regarded as a more general form of Gleasonʼs theorem. Here we show that a further generalization is possible by reducing the number of quantum postulates used by Busch. We do not assume that the positive measurement outcome operators are effects or that they form a probability operator measure. We derive a more general probability rule from which the standard rule can be obtained from the normal laws of probability when there is no measurement outcome information available, without the need for further quantum postulates. Our general probability rule has prediction–retrodiction symmetry and we show how it may be applied in quantum communications and in retrodictive quantum theory.

The results of experiments designed to measure the operational phase cosine and sine variances of weak states of light disagree with the variances predicted by canonical phase formalisms. As these variances are fundamental manifestations of the quantum nature of phase, it is important to be able to measure the canonical variances also. A recent suggestion to do so, based on the use of a two-component probe, involves the difficult preparation of exotic states of light which have not yet been produced. In this paper we show how the variances can be measured with simple coherent state inputs. The retrodictive formalism of quantum mechanics provides useful insight into the physics involved.

Mathematically, the simplest state of light containing phase information is a superposition of the vacuum and the one-photon state and, as we show in this paper, such a state is reasonably simple to measure. We investigate how the information contained in a more complicated pure state of light, in particular the ratio of successive number-state coefficients, can be transferred selectively to fields in this two-state superposition for subsequent measurement. By this means the number-state representation of the more complicated state can be ascertained, provided there are no gaps in the number state distribution. We also discuss how to correct for the effect of non-unit efficiencies of the photodetectors involved in the transferral process.

The usual formalism of quantum mechanics is predictive with state vectors being assigned on the basis of past preparation procedures. The retrodictive formalism, in which some state vectors are assigned on the basis of future measurement results is less usual, although it has been known for some time. Although at first sight this formalism may appear to be anticausal and subject to paradoxes, if used carefully it still predicts the same measurable correlations as the more usual approach despite different assignments of state vectors to states in the interval between preparation and measurement of the system. Here we use a procedure which involves retrodiction to examine some quantum optical processes, including a recent state truncation process. The collapse of the state vector occurs at a different time from that for the usual approach underlining the difficulty in regarding the collapse as a real physical process.

A state of light which is a superposition of the vacuum and the one-photon number state is the simplest state containing phase information. Recently we have shown how a field in such a state might be generated and here we explore its usefulness as a probe for measuring unknown states of light. We find that this probe can be used reasonably simply both to determine completely some pure states of light and to measure the diagonal and nearest off-diagonal elements of the density matrix in the number state basis and hence to obtain the mean sine and cosine of the phase of an unknown mixed state. We suggest further how a field in a superposition of the vacuum and the two-photon number state might be generated and how this can be used as a probe, both to measure the off-diagonal matrix elements second nearest to the diagonal of a mixed state density matrix and to measure the variance of the cosine and the sine of the phase. We also examine the experimentally more likely case where the probe fields are ...

The much-studied energy-time uncertainty relation has well-known difficulties that are exacerbated for a system with discrete energy levels. The difficulty in representing time in the abstract sense by an operator raises the related question of whether or not there is some other quantity that is complementary to the Hamiltonian of a quantum system. Such a quantity would have dimensions of time but would be a property of the system itself. We examine this question for a system with discrete energy eigenstates for which the ratios of the energy differences are rational. We find that such a quantity does exist and can be represented both by a probability-operator measure and by an Hermitian operator, but in a state space larger than the minimal space needed to include the states of the system. The uncertainty relation with the energy is slightly more complicated than the momentum-position uncertainty relation, but is readily interpretable. To describe such a quantity the name ``age'' is suggested.

We provide an expression for the state of a single field mode, conditioned only on the outcome of an imperfect photon-counting measurement. This suggests that we can view such a measurement as projection onto one of a non-orthogonal set of mixed states. We briefly discuss some of the implications of this result for state preparation.

We show how the recently-introduced method of projection synthesis can be applied to find the phase probability distribution for a single-mode field. The realization of such a measurement scheme requires the production of reciprocal-binomial states of light or suitable alternative states. We describe how such states might be produced in the laboratory.

In the theory of the interaction of radiation with atoms, the concept of a dressed atom is well known. In this paper we explore aspects of the antithesis of this, the concept of a dressed field, that is, a field dressed by the atoms of the medium. In particular we use the concept to study the quantum mechanical action of a retarding plate. This simple physical process has fundamental implications on the quantum theory of phase. Although it is usually accepted that a retarding plate will shift the phase of light fields, including weak fields in the quantum domain, whether or not this happens depends on what quantum phase actually is. Conversely, if we assume that a retrading plate is a quantum, as well as a classical, phase shifter, then the known quantum mechanical action of the plate imposes restrictions on possible descriptions of quantum phase. We find that accepting the plate as a phase shifter supports the notion of complementarity of photon number and phase in that a field in a photon number state, including the vacuum, must have a completely random phase. This result, although simple, is non-trivial because while some recent attempts to provide a quantum mechanical description of the phase of light are consistent with the complementarity of phase and photon number, others are not.

One of the long term interests of George Series was the construction of a theory of spontaneous emission which does not involve field quantisation. His approach was written in terms of atomic operators only and he drew a parallel with the Wheeler-Feynman absorber theory of radiation. By making a particular extra postulate, he was able to obtain the correct spontaneous emission rate and the Lamb shift reasonably simply and directly. An examination of his approach indicates that this postulate is physically reasonable and the need for it arises because quantisation in his theory occurs after the response of the absorber has been accounted for by means of the radiative reaction field. We review briefly an alternative absorber theory approach to spontaneous emission based on the direct action between the emitting atom and a quantised absorber, and outline some applications to more recent effects of interest in quantum optics.

A construction of covariant quantum phase observables, for Hamiltonians with a finite number of energy eigenvalues, has been recently given by D. Arsenovic et al. [Phys. Rev. A 85, 044103 (2012)]. For Hamiltonians generating periodic evolution, we show that this construction is just a simple rescaling of the known canonical 'time' or 'age' observable, with the period T rescaled to 2\pi. Further, for Hamiltonians generating quasiperiodic evolution, we note that the construction leads to a phase observable having several undesirable features, including (i) having a trivially uniform probability density for any state of the system, (ii) not reducing to the periodic case in an appropriate limit, and (iii) not having any clear generalisation to an infinite energy spectrum. In contrast, we note that a covariant time observable has been previously defined for such Hamiltonians, which avoids these features. We also show how this 'quasiperiodic' time observable can be represented as the well-defined limit of a sequence of periodic time observables., Comment: 4 pages, accepted version

Standard information theory deals with alphabets and their transmission over communication channels. Here we examine the novel features introduced by allowing the alphabet symbols to be quantum states. A simple device for communication of one bit of information is discussed and the transition between quantum and classical behaviour is highlighted. A further level of complexity is introduced when we allow the communication to take place with quantum-correlated states. We show, by the simple expedient of constructing a suitable local hidden variable theory, that many of the novel features of such communication are compatible with the concept of local realism. We introduce a convenient parameter for characterizing the contribution of the quantum entanglement to the communication.

We examine the effects of linear amplification and attenuation on the quantum-mechanical phase properties of light for fields with mean photon numbers of at least the order of 10. The phase probability density is found to satisfy a diffusion equation for both phase-insensitive and phase-sensitive amplifiers and attenuators. The solution is a convolution of the initial phase probability density with an infinite series of expanding Gaussians which clearly illustrates the diffusion of the phase. In particular, we find that for phase-insensitive amplification the phase of the field undergoes time-dependent uniform diffusion. In the limit of large amplification the diffusion ceases and the phase variance of the amplified light is given by the input phase variance plus an extra term which is equal to the phase variance of a coherent state of the same intensity as the initial field. We show that the reduced phase variance of phase-optimized states (relative to coherent states of the same intensity) is lost for power gains greater than the photon-cloning value of 2. In contrast, phase-sensitive amplifiers give rise to time-dependent nonuni form phase diffusion. The amount of phase diffusion depends on the relative phase angle between the light and the amplifier. If the peak of a relatively narrow phase probability density is near a minimum in the phase diffusion coefficient, then the phase noise added by the amplifier will be less than that found for a phase-insensitive amplifier. Further squeezing of the amplifier reduces the added phase noise proportionally. We find that it is possible, using phase-sensitive amplifiers, to amplify phaseoptimized states by power gains considerably larger than 2 and still retain a reduced phase variance.

Previous studies of the phase properties of optical linear amplifiers and attenuators have been restricted to situations where the effect on the phase of single-mode light is found to be diffusive. In this paper we give an analysis of phase-sensitive linear amplification and attenuation in the weak-field regime that clearly reveals a non-diffusive effect on the phase. Indeed we show that a weak field which has initially a uniform phase probability density can inherit non-random phase properties from both phase-sensitive attenuators and amplifiers. In contrast, we find that phase-insensitive amplifiers and attenuators have merely a diffusive effect on the phase of the light.

The applicability is examined of the measured-phase operator describing a simple homodyne measurement scheme to states of the quantum optical field which have the minimum possible phase fluctuations. Such states necessarily have huge intensity fluctuations, so it is expected that this operator should be inapplicable because intensity fluctuations are deliberately ignored in its derivation. However, the surprising result found is that this operator can still provide a reasonably good approximation to the phase properties of fields even in this extreme limit.

We review the formulation of the operator for rotation angles and the corresponding uncertainty relation. The orbital angular momentum of light allows us to test these ideas and also to explore angular entanglement.

In the Ψ-space limiting procedure limits of expectation values, rather than operator or states, are found as the state space dimensionality tends to infinity. This approach has been applied successfully to the calculation of the phase properties of various states of light, but its status as a valid quantum mechanical thory equivalent to the usual infinite Hilbert space (H-space) approach has not yet been fully accepted. Here we address this issue by investigating the formal relationship between the two approaches. We establish the consistency between the Ψ-space and H-space approaches for observables which are amenable to an H-space treatment. Such observables are represented in H by operators which are strong limits of Ψ-space operators and which obey the same algebra as the corresponding Ψ-space operators. The phase operator, however, exists in H only as a weak limit of a Ψ-space operator. For such limits the Ψ-space operator algebra is not preserved, which is the fundamental reason for the difficulties in constructing a consistent quantum description of phase in H. We show that for the phase observable the Ψ-space approach is consistent with the probability-operator measure (POM) method with the important distinction that, whereas the relation between non-orthogonal POMs and probability has to be accepted in the latter method as a postulate, the corresponding relation is derived in the Ψ-space approach. We conclude that the Ψ-space approach is not only equivalent to, but is also more fundamental than both the H-space and POM approaches.

We examine some of the attempts to describe the phase of a single field mode by a quantum operator acting in the conventional infinite Hilbert space. These operators lead to bizarre properties such as non-random phases for the number states and experience consistency difficulties when used to obtain a phase probability density. Moreover, in these approaches operator functions of phase are not simply functions of a phase operator. We show that these peculiarities do not arise when the Hermitian optical phase operator is employed. In our opinion, the problems associated with the descriptions of phase in conventional infinite Hilbert space arise from the nature of the limiting process.

We investigate the relationship between squeezing and reduced phase fluctuations for various states of the single-mode electromagnetic field, including the strongly-squeezed vacuum and phase states. We find that, although squeezing the fluctuations of the electric field that arise from the vacuum guarantees a more well-defined phase, reducing phase fluctuations does not guarantee a squeezed electric field. We also investigate the evolution of the electric field and its fluctuations for a phase state. Our results show that even though the electric field fluctuations never vanish for a phase state, the times when the electric field changes sign are precisely defined. We also discuss why it is not always possible to attribute physical properties to certain states, such as simple superpositions of phase states.

We examine the phase relationships between different packets of a beam of light emitted from a cavity through a partially transmitting mirror. We find that the beam has a pronounced phase coherence even when the intracavity field has a completely indeterminate phase such as, for example, when it is in a pure photon number state. In this case, the phase coherence can be directly attributed to entanglement caused by the partially transmitting mirror and, even though the external light has phase coherence, it has no identifiable phase. Consequently there is no phase diffusion but there is diffusion of the phase difference between a packet and the intracavity field. Remarkably, we find that the phase coherence of the beam is almost independent of the intracavity state, provided this has a reasonably narrow number-state distribution about a large mean value.

In quantum mechanics it is usual to represent physical reality as a vector in Hilbert space at a particular time, with evolution being governed by Schrodinger's equation which involves an externally imposed time parameter. This leads to difficulties if one wishes to regard the universe as a quantum mechanical system, because there should no time external to such a system. The approach in this paper is to represent the totality of reality as a vector in Hilbert space. The author shows how time evolution follows, where time now is defined in terms of the states of a quantum mechanical clock which is part of the system. Rather than the correlation between the clock states and the states of the rest of the system arising because both are governed by an imposed law involving an external time parameter, it is seen that this correlation is of the separation-independent Einstein-Podolsky-Rosen type. The total reality vector, which incorporates the whole history of the system, is shown to be a zero-energy eigenstate of the system Hamiltonian. He discusses systems of finite and infinite lifetime, and is able to answer the question: what was the state before the initial state? He concludes that the quantum mechanical system of this paper is a reasonable representation of the observed universe.

Ban [Opt. Commun. 143 (1977) 225] demonstrated an equivalence between a lossless beam splitter with one of the output modes in the vacuum state and a nondegenerate parametric amplifier with a vacuum input. We show that this equivalence can be extended to apply to all states.

We show that the physically absurd state obtained by Vorontsov and Rembovsky [Phys. Lett. A 254 (1999) 7] results from an unphysical choice of approximate phase measurement and not from the phase formalism itself as claimed. Thus their assertion that it is impossible to perform even an approximate measurement of phase is unfounded.

We extend the theory of the Hermitian optical phase operator to analyze the quantum phase properties of pairs of electromagnetic field modes. The operators representing the sum and difference of the two single-mode phases are simply the sum and difference of the two single-mode phase operators. The eigenvalue spectra of the sum and difference operators have widths of 4\ensuremath{\pi}, but phases differing by 2\ensuremath{\pi} are physically indistinguishable. This means that the phase sum and difference probability distributions must be cast into a 2\ensuremath{\pi} range. We obtain mod(2\ensuremath{\pi}) probability distributions for the phase sum and difference that unambiguously reveal the signatures of randomness, phase correlations, and phase locking. We use our approach to investigate the phase sum and difference properties for uncorrelated modes in random and partial phase states and the phase-locked properties of the two-mode squeezed vacuum states. We reveal the fundamental property of two-mode squeezed states that the phase sum is locked to the argument of the squeezing parameter. The variance of the phase sum depends dilogarithmically on 1+tanhr, where r is the magnitude of the squeezing parameter, vanishing in the large squeezing limit.

Observations of optical nutations from the NaD2 line excited by resonant, π-polarized light have indicated that the oscillations consist of essentially one frequency. Considering the complexity of the energy-level configuration associated with this line, the experimental results are surprising. However, it is shown that this pseudo-two-state behaviour is predicted by both a semiclassical theory and a completely quantum electrodynamic approach when the intensity of the light is above a moderately valued threshold, namely, when the power broadening is greater than the hyperfine splitting of the upper states.

We use a new limiting procedure, developed to study quantum-optical phase, to examine canonically conjugate operators in general. We find that Dirac's assumption that photon number and phase should be canonically conjugate variables, similar to momentum and position, is essentially correct. The difficulties with Dirac's approach are shown to arise through use of a form of the canonical commutator which, although the only possible form in the usual infinite Hilbert space approach, is not sufficiently general to be used as a model for a number-phase commutator. The approach in this paper unifies the theory of conjugate operators, which include photon number and phase, angular momentum and angle, and momentum and position as particular cases. The usual position-momentum commutator is regained from a more generally applicable expression by means of a domain restriction which cannot be used for the phase-number commutator.

We find the states of light which have minimum phase variance both for a given maximum energy state component and for a given mean energy. When these states contain sufficiently many photon number state components, the number state coefficients approximate sinusoidal and Airy functions respectively. The phase of these new states of light is much more sharply defined than is the phase of a coherent state with the same mean energy.

The formulation of the quantum description of the rotation angle of the plane rotator has been beset by many of the long-standing problems associated with harmonic-oscillator phases. We apply methods recently developed for oscillator phases to the problem of describing a rotation angle by a Hermitian operator. These methods involve use of a finite, but arbitrarily large, state space of dimension 2l+1 that is used to calculate physically measurable quantum properties, such as expectation values, as a function l. Physical results are then recovered in the limit as l tends to infinity. This approach removes the indeterminacies caused by working directly with an infinite-dimensional state space. Our results show that the classical rotation angle observable does have a corresponding Hermitian operator with well-determined and reasonable properties. The existence of this operator provides deeper insight into the quantum-mechanical nature of rotating systems.

The recently introduced Hermitian phase operator allows a phase-state representation of the single-mode light field. We find the requirement that a light field is in a physical state, that is, has finite energy moments, imposes strict and simple continuity conditions on the phase-amplitude distribution. We exploit these conditions to examine the physical intelligent and minimum-uncertainty states associated with the number-phase uncertainty relations, including those involving the Hermitian phase operator and its sine and cosine forms. The single number-state is found to be the only physical exact intelligent state and also the only physical exact minimum-uncertainty state for all the uncertainty relations considered. We construct states which are both physical states and approximately intelligent states. Under certain conditions coherent states, ideal squeezed states and the number-phase intelligent states associated with the Susskind-Glogower cosine and sine operators are found to be both physi...

We show how the number-state expansion of an optical state can be truncated so as to leave only its vacuum and one photon components. This can be achieved using a ``quantum scissors'' device, the operation of which relies on a nonlocal quantum effect.

We extend and generalize previous work on the interference of light from independent cavities that began with the suggestion of Pfleegor and Mandel [Phys. Rev. 159, 1084 (1967)] that their observed interference of laser beams should not be associated too closely with particular states of the beams but more with the detection process itself. In particular we examine how the detection of interference induces a nonrandom-phase difference between internal cavity states with initial random phases for a much broader range of such states than has previously been considered. We find that a subsequent interference measurement should give results consistent with the induced phase difference. The inclusion of more cavities in the interference measurements enables the construction in principle of a laboratory in the sense used by Aharonov and Susskind, made up of cavity fields that can serve as frames of phase reference. We also show reasonably simply how intrinsic phase coherence of a beam of light leaking from a single cavity arises for any internal cavity state, even a photon number state. Although the work presented here may have some implications for the current controversy over whether or not a typical laboratory laser produces a coherent state, it is notmore » the purpose of this paper to enter this controversy; rather it is to examine the interesting quantum physics that arises for cavities with more general internal states.« less

We derive a master equation describing the evolution of a quantum system subjected to a sequence of observations. These measurements occur randomly at a given rate and can be of a very general form. As an example, we analyse the effects of these measurements on the evolution of a two-level atom driven by an electromagnetic field. For the associated quantum trajectories we find Rabi oscillations, Zeno-effect type behaviour and random telegraph evolution spawned by mini quantum jumps as we change the rates and strengths of measurement., 14 pages and 8 figures, Optics Communications in press

We develop and discuss noveltypes of generalisedquantum measurement. The associated non-unitary time evolution enables the manipulation of quantum states in ways that are not achievable by unitary operations. Particular emphasis is given to quantum optical realisations.

All compositions of a mixed-state density operator are equivalent for the prediction of the probabilities of future outcomes of measurements. For retrodiction, however, this is not the case. The retrodictive formalism of quantum mechanics provides a criterion for deciding that some compositions are fictional. Fictional compositions do not contain preparation device operators, that is operators corresponding to states that could have been prepared. We apply this to Molmer's controversial conjecture that optical coherences in laser light are a fiction and find agreement with his conjecture. We generalise Molmer's derivation of the interference between two lasers to avoid the use of any fictional states. We also examine another possible method for discriminating between conerent states and photon number states in laser light and find that it does not work, with the equivalence for prediction saved by entanglement.

Although it has been known for some time that quantum mechanics can be formulated in a way that treats prediction and retrodiction on an equal footing, most attention in engineering quantum states has been devoted to predictive states, that is, states associated with the a preparation event. Retrodictive states, which are associated with a measurement event and propagate backwards in time, are also useful, however. In this paper we show how any retrodictive state of light that can be written to a good approximation as a finite superposition of photon number states can be generated by an optical multiport device. The composition of the state is adjusted by controlling predictive coherent input states. We show how the probability of successful state generation can be optimised by adjusting the multiport device and also examine a versatile configuration that is useful for generating a range of states., Comment: 14 pages, 1 figure

We show that the eight-port interferometer used by Noh, Foug\`{e}res, and Mandel [Phys. Rev. Lett. {\bf 71}, 2579 (1993)] to measure their operational phase distribution of light can also be used to measure the canonical phase distribution of weak optical fields, where canonical phase is defined as the complement of photon number. A binomial reference state is required for this purpose, which we show can be obtained to an excellent degree of approximation from a suitably squeezed state. The proposed method requires only photodetectors that can distinguish among zero, one and more than one photons and is not particularly sensitive to photodetector imperfections., Comment: 8 Pages, 4 figures

Retrodictive quantum states are states that propagate backwards in time from a measurement event. Although retrodictive quantum mechanics appears to be very different from the usual predictive formalism, in that propagation of states into the past appears to violate causality, this is not so. Indeed causality is not manifest in the time direction of propagation of quantum states at all. Instead, causality is ensured by the different normalization conditions applied to the preparation and measurement device operators. It is this difference that introduces the arrow of time into quantum mechanics. Retrodictive states are useful for applications such as measurement, predictive quantum state engineering and quantum communication. Here we show how any optical retrodictive state that can be expressed to a good approximation in a finite‐dimensional Hilbert space can be generated from predictive coherent states, a lossless multiport device and photodetectors. The composition of the retrodictive state can be contr...