We consider two memory nodes of a quantum network connected by flying qubits. We are particularly interested in the case where a flying qubit produced by one node has to be transformed before it can interface efficiently with the next node. Such transformations can be utilized as a key part of the distribution of quantum states and hence entanglement between the nodes of a hybrid quantum network linking together different quantum technologies. We show how and why the probability of interfacing successfully is determined by the overlap of the spectral shape of the actual flying qubit and the ideal shape. This allows us to analytically and numerically analyze how the probability of success is impacted by realistic errors, and show the utility of our scheme (in consonance with known error correction methods) in connecting hybrid nodes of a quantum network. We focus here on a concrete implementation in which the memory nodes consist of three-level atoms in cavities and the flying qubits are photons.
The Partial Information Decomposition (PID) takes one step beyond Shannon's theory in decomposing the information two variables $A,B$ possess about a third variable $T$ into distinct parts: unique, shared (or redundant) and synergistic information. Here we show how these concepts can be defined in a quantum setting. We apply a quantum PID to scrambling in quantum many-body systems, for which a quantum-theoretic description has been proven productive. Unique information in particular provides a finer description of scrambling than does the so-called tri-information.
Being able to reliably transfer the quantum state from one system to another is crucial to developing quantum networks. A standard way to accomplish this transfer of information is by making use of an intermediate information carrier (e.g., a photon) that is emitted by the first system and absorbed by the second. For such a scenario one can develop an effective description by eliminating the intermediate degrees of freedom, which yields an effective direct coupling between the two systems. If, however, the spectral properties of the two systems are different, the photon's time-frequency shape needs to be appropriately modified before it reaches the second system. We study here the effective description that results when we thus manipulate the intermediate photon. We examine a unitary transformation, $U$, that time reverses, frequency translates, and stretches the photon wave packet. We find that the concomitant modifications to the effective description can best be understood in terms of a change to the state's time argument, $\rho(t) = \rho_1(\tilde{t}) \otimes \rho_2(t)$, where $\tilde{t}$ is a fictitious time for the first system that is stretched and runs backward. We apply this theory to three-level $\Lambda$-systems inside optical cavities, and we numerically illustrate how performing the unitary transformation $U$ results in improved quantum state transfer.
We apply input-output theory with quantum pulses [AH Kiilerich, K Mølmer, Phys. Rev. Lett. \bf 123, 123604 (2019)] to a model of a new type of two-photon detector consisting of one molecule that can detect two photons arriving sequentially in time. The underlying process is distinct from the usual two-photon absorption process where two photons arriving simultaneously and with frequencies adding up to the resonance frequency are absorbed by a single molecule in one quantum jump. Our detector model includes a Hamiltonian description of the amplification process necessary to convert the microscopic change in the single molecule to a macroscopic signal.
We consider the quantum theory of paraxial non-relativistic electron beams in non-uniform magnetic fields, such as the Glaser field. We find the wave function of an electron from such a beam and show that it is a joint eigenstate of two ($z$-dependent) commuting gauge-independent operators. This generalized Laguerre-Gaussian vortex beam has a phase that is shown to consist of two parts, each being proportional to the eigenvalue of one of the two conserved operators and each having different symmetries. We also describe the dynamics of the angular momentum and cross-sectional area of any mode and how a varying magnetic field can split a mode into a superposition of modes. By a suitable change in frame of reference all of our analysis also applies to an electron in a quantum Hall system with a time-dependent magnetic field.
We construct a class of Hamiltonians that describe the photodetection process from beginning to end. Our Hamiltonians describe the creation of a photon, how the photon travels to an absorber (such as a molecule), how the molecule absorbs the photon, and how the molecule after irreversibly changing its configuration triggers an amplification process---at a wavelength that may be very different from the photon's wavelength---thus producing a macroscopic signal. We use a simple prototype Hamiltonian to describe the single-photon detection process analytically in the Heisenberg picture, which neatly separates desirable from undesirable effects. Extensions to more complicated Hamiltonians are pointed out.
A recurring problem in quantum mechanics is to estimate either the state of a quantum system or the measurement operator applied to it. If we wish to estimate both, then the difficulty is that the state and the measurement always appear together: to estimate the state, we must use a measurement; to estimate the measurement operator, we must use a state. The data of such quantum estimation experiments come in the form of measurement frequencies. Ideally, the measured average frequencies can be attributed to an average state and an average measurement operator. If this is not the case, we have correlated state-preparation-and-measurement (SPAM) errors. We extend some tests developed to detect such correlated errors to apply to a cryptographic scenario in which two parties trust their individual states but not the measurement performed on the joint state.
The time-frequency degree of freedom of the electromagnetic field is the final frontier for single-photon measurements. The temporal and spectral distribution a measurement retrodicts (that is, the state it projects onto) is determined by the detector's intrinsic resonance structure. In this paper, we construct ideal and more realistic positive operator-valued measures (POVMs) that project onto arbitrary single-photon wavepackets with high efficiency and low noise. We discuss applications to super-resolved measurements and quantum communication. In doing so we will give a fully quantum description of the entire photo detection process, give prescriptions for (in principle) performing single-shot Heisenberg-limited time-frequency measurements of single photons, and discuss fundamental limits and trade-offs inherent to single-photon detection.
Suppose a classical electron is confined to move in the $xy$ plane under the influence of a constant magnetic field in the positive $z$ direction. It then traverses a circular orbit with a fixed positive angular momentum $L_z$ with respect to the center of its orbit. It is an underappreciated fact that the quantum wave functions of electrons in the ground state (the so-called lowest Landau level) have an azimuthal dependence $\propto \exp(-im\phi) $ with $m\geq 0$, seemingly in contradiction with the classical electron having positive angular momentum. We show here that the gauge-independent meaning of that quantum number $m$ is not angular momentum, but that it quantizes the distance of the center of the electron's orbit from the origin, and that the physical angular momentum of the electron is positive and independent of $m$ in the lowest Landau levels. We note that some textbooks and some of the original literature on the fractional quantum Hall effect do find wave functions that have the seemingly correct azimuthal form $\propto\exp(+im\phi)$ but only on account of changing a sign (e.g., by confusing different conventions) somewhere on the way to that result.
Single photon detection generally consists of several stages: the photon has to interact with one or more charged particles, its excitation energy will be converted into other forms of energy, and amplification to a macroscopic signal must occur, thus leading to a "click." We focus here on the part of the detection process before amplification (which we have studied in a separate publication). We discuss how networks consisting of coupled discrete quantum states and structured continua (e.g. band gaps) provide generic models for that first part of the detection process. The input to the network is a continuum (the continuum of single-photon states), the output is again a continuum describing the next irreversible step. The process of a single photon entering the network, its energy propagating through that network and finally exiting into another output continuum of modes can be described by a single dimensionless complex transmission amplitude, $T(\omega)$. We discuss how to obtain from $T(\omega)$ the photo detection efficiency, how to find sets of parameters that maximize this efficiency, as well as expressions for other input-independent quantities such as the frequency-dependent group delay and spectral bandwidth. We then study a variety of networks and discuss how to engineer different transmission functions $T(\omega)$ amenable to photo detection.
The actual gate performed on, say, a qubit in a quantum computer may depend, not just on the actual laser pulses and voltages we programmed to implement the gate, but on its \em context as well. For example, it may depend on what gate has just been applied to the same qubit, or on how much a long series of previous laser pulses has been heating up the qubit's environment. This paper analyzes several tests to detect such context-dependent errors (which include various types of non-Markovian errors). A key feature of these tests is that they are robust against both state preparation and measurement (SPAM) errors and gate-dependent errors. Since context-dependent errors are expected to be small in practice, it becomes important to carefully analyze the effects of statistical fluctuations and so we investigate the power and precision of our tests as functions of the number of repetitions and the length of the sequences of gates. From our tests an important quantity emerges: the logarithm of the determinant (log-det) of a probability (relative frequency) matrix $\mathcal{P}.$ For this reason, we derive the probability distribution of the log-det estimates which we then use to examine the performance of our tests for various single- and two-qubit sets of measurements and initial states. Finally, we emphasize the connection between the log-det and the degree of reversibility (the unitarity) of a context-independent operation.
The phase factor $(-1)^{2s}$ that features in the exchange symmetry for identical spin-$s$ fermions or bosons is not simply and automatically equal to the phase factor one can observe in an interference experiment that involves physically exchanging two such particles. The observable phase contains, in general, single-particle geometric and dynamical phases as well, induced by both spin and spatial exchange transformations. By extending the analysis to (non-abelian) anyons it is argued that, similarly, there are single-anyon geometric and dynamical contributions in addition to purely topological unitary transformations that accompany physical exchanges of anyons. Work remains to be done in order to demonstrate---if it is still true---that those additional contributions to the gates in anyonic topological quantum computers do not destroy the inherent robustness of the ideal gates. This negative result is described most clearly in terms of the Berry matrix.
How do indistinguishable identical bosons manage to obey Bose-Einstein statistics---and hence be correlated---even when they do not interact with each other? Part of the answer is that the bosons have to interact indirectly with each other by interacting with the same environment. A joint measurement interaction provides a good example. Thermalization occurs whenever there are two competing processes, one diagonal in the energy basis (namely, reversible Hamiltonian evolution), the other irreversible and diagonal in a complementary basis (for example, a measurement in a spatially localized basis). Correlations arise only from initial states in which the bosons start in different (orthogonal) states.
We show that detection of single photons is not subject to the fundamental limitations that accompany quantum linear amplification of bosonic mode amplitudes, even though a photodetector does amplify a few-photon input signal to a macroscopic output signal. Alternative limits are derived for \emphnonlinear photon-number amplification schemes with optimistic implications for single-photon detection. Four commutator-preserving transformations are presented: one idealized (which is optimal) and three more realistic (less than optimal). Our description makes clear that nonlinear amplification takes place, in general, at a different frequency $\omega'$ than the frequency $\omega$ of the input photons. This can be exploited to suppress thermal noise even further up to a fundamental limit imposed by amplification into a single bosonic mode. A practical example that fits our description very well is electron-shelving.
We experimentally demonstrate that loop state-preparation-and-measurement (SPAM) tomography is capable of detecting correlated errors in a two-qubit system. We prepare photon pairs in a state that approximates a Werner state, which may or may not be entangled. By performing measurements with multiple different detector settings we are able to detect correlated errors between two single-qubit measurements performed in different locations. No assumptions are made concerning either the state preparations or the measurements, other than that the dimensions of the states and the positive-operator-valued measures describing the detectors are known. The only other needed information is experimentally measured expectation values, which are analyzed for self-consistency. This demonstrates that loop SPAM tomography is a useful technique for detecting errors that would degrade the performance of multiple-qubit quantum information processors.
It is possible for two parties, Alice and Bob, to establish a secure communication link by sharing an ensemble of entangled particles, and then using these particles to generate a secret key. One way to establish that the particles are indeed entangled is to verify that they violate a Bell inequality. However, it might be the case that Bob is not trustworthy and wishes Alice to believe that their communications are secure, when in fact they are not. He can do this by managing to have prior knowledge of Alice's measurement device settings and then modifying his own settings based upon this information. In this case it is possible for shared particle states that must satisfy a Bell inequality to appear to violate this inequality, which would also make the system appear secure. When Bob modifies his measurement settings, however, he produces false correlations. Here we demonstrate experimentally that Alice can detect these false correlations, and uncover Bob's trickery, by using loop-state-preparation-and-measurement (SPAM) tomography. More generally, we demonstrate that loop SPAM tomography can detect false correlations (correlated errors) in a two-qubit system without needing to know anything about the prepared states or the measurements, other than the dimensions of the operators that describe them.
Single-photon wave packets can carry quantum information between nodes of a quantum network. An important general operation in photon-based quantum information systems is blind reversal of a photon's temporal wave-packet envelope, that is, the ability to reverse an envelope without knowing the temporal state of the photon. We present an all-optical means for doing so, using nonlinear-optical frequency conversion driven by a short pump pulse. This scheme allows for quantum operations such as a temporal-mode parity sorter. We also verify that the scheme works for arbitrary states (not only single-photon ones) of an unknown wave packet.
In many experiments on microscopic quantum systems, it is implicitly assumed that when a macroscopic procedure or "instruction" is repeated many times -- perhaps in different contexts -- each application results in the same microscopic quantum operation. But in practice, the microscopic effect of a single macroscopic instruction can easily depend on its context. If undetected, this can lead to unexpected behavior and unreliable results. Here, we design and analyze several tests to detect context-dependence. They are based on invariants of matrix products, and while they can be as data intensive as quantum process tomography, they do not require tomographic reconstruction, and are insensitive to imperfect knowledge about the experiments. We also construct a measure of how unitary (reversible) an operation is, and show how to estimate the volume of physical states accessible by a quantum operation.
A photodetector may be characterized by various figures of merit such as response time, bandwidth, dark count rate, efficiency, wavelength resolution, and photon-number resolution. On the other hand, quantum theory says that any measurement device is fully described by its POVM, which stands for Positive-Operator-Valued Measure, and which generalizes the textbook notion of the eigenstates of the appropriate hermitian operator (the "observable") as measurement outcomes. Here we show how to define a multitude of photodetector figures of merit in terms of a given POVM. We distinguish classical and quantum figures of merit and issue a conjecture regarding trade-off relations between them. We discuss the relationship between POVM elements and photodetector clicks, and how models of photodetectors may be tested by measuring either POVM elements or figures of merit. Finally, the POVM is advertised as a platform-independent way of comparing different types of photodetectors, since any such POVM refers to the Hilbert space of the incoming light, and not to any Hilbert space internal to the detector.
Suppose we measure the time-dependent spectrum of a single photon. That is, we first send the photon through a set of frequency filters (which we assume to have different filter frequencies but the same finite bandwidth $\Gamma$), and then record at what time (with some finite precision $\Delta t$, and with some finite efficiency $\eta$) and after passing what filter the photon is detected. What is the POVM (Positive-Operator Valued Measure, the most general description of a quantum measurement) corresponding to such a measurement? We show how to construct the POVM in various cases, with special interest in the case $\Gamma\Delta t\ll 1$ (time-frequency uncertainty still holds, even in that limit). One application of the formalism is to heralding single photons. We also find a Hong-Ou-Mandel type of interference effect with two photons entering a frequency filter.
In the context of quantum tomography, quantities called a partial determinants\citejackson2015detecting were recently introduced. PDs (partial determinants) are explicit functions of the collected data which are sensitive to the presence of state-preparation-and-measurement (SPAM) correlations. In this paper, we demonstrate further applications of the PD and its generalizations. In particular we construct methods for detecting various types of SPAM correlation in multiqudit systems | e.g. measurement-measurement correlations. The relationship between the PDs of each method and the correlations they are sensitive to is topological. We give a complete classification scheme for all such methods but focus on the explicit details of only the most scalable methods, for which the number of settings scale as $\mathcal{O}(d^4)$. This paper is the second of a two part series where the first paper[2] is about theoretical perspectives of the PD and its interpretation as a holonomy.
In the context of quantum tomography, we recently introduced a quantity called a partial determinant \citejackson2015detecting. PDs (partial determinants) are explicit functions of the collected data which are sensitive to the presence of state-preparation-and-measurment (SPAM) correlated errors. As such, PDs bypass the need to estimate state-preparation or measurement parameters individually. In the present work, we suggest a theoretical perspective for the PD. We show that the PD is a holonomy and that the notions of state, measurement, and tomography can be generalized to non-holonomic constraints. To illustrate and clarify these abstract concepts, direct analogies are made to parallel transport, thermodynamics, and gauge field theory. This paper is the first of a two part series where the second paper [2] is about scalable applications of the PD to multiqudit systems.
We have performed an experiment demonstrating that loop state-preparation-and-measurement (SPAM) tomography [C. Jackson and S. J. van Enk, Phys. Rev. A 92, 042312 (2015)] is capable of detecting correlated errors between the preparation and the measurement of a quantum system. Specifically, we have prepared pure and mixed states of single qubits encoded in the polarization of heralded individual photons. By performing measurements using multiple state preparations and multiple measurement device settings we are able to detect if there are any correlated errors between them, and are also able to determine which state preparations are correlated with which measurements. This is accomplished by going around a 'loop' in parameter space, which allows us to check for self-consistency. No assumptions are made concerning either the state preparations or the measurements, other than that the dimensions of the states and the positive-operator-valued measures (POVM) describing the detector are known. In cases where no correlations are found we are able to perform quantum state tomography of the polarization qubits by using knowledge of the detector POVMs, or quantum detector tomography by using knowledge of the state preparations.
In the context of quantum tomography, we recently introduced a quantity called a partial determinant \citejackson2015detecting. PDs (partial determinants) are explicit functions of the collected data which are sensitive to the presence of state-preparation-and-measurement (SPAM) correlations. Importantly, this is done without any need to estimate state-preparation or measurement parameters. In the present work, we wish to better explain our theoretical perspective behind the PD. Further, we would like to demonstrate that there is an overwhelming variety of applications and generalizations of the PD. In particular we will construct methods for detecting SPAM correlations in multiqudit systems. The relationship between the PDs of each method and the correlations they are sensitive to is topological. We give a classification of all such methods but focus on explicitly detailing only the most scalable methods, $\mathcal{O}(d^4)$.
Two-photon interference effects, such as the Hong-Ou-Mandel (HOM) effect, can be used to characterize to what extent two photons are identical. Furthermore, these interference effects underly linear optics quantum computation. We show here how nonlinear optical effects, such as those mediated by atoms or quantum dots in a cavity, degrade the interference. This implies that, on the one hand, nonlinearities are to be avoided if one wishes to utilize the interference, but on the other hand, one may be able to measure or detect nonlinearities by observing the disappearance of the interference.
Whereas in standard quantum state tomography one estimates an unknown state by performing various measurements with known devices, and whereas in detector tomography one estimates the POVM elements of a measurement device by subjecting to it various known states, we consider here the case of SPAM (state preparation and measurement) tomography where neither the states nor the measurement device are assumed known. For $d$-dimensional systems measured by $d$-outcome detectors, we find there are at most $d^2(d^2-1)$ "gauge" parameters that can never be determined by any such experiment, irrespective of the number of unknown states and unknown devices. For the case $d=2$ we find new gauge-invariant quantities that can be accessed directly experimentally and that can be used to detect and describe SPAM errors. In particular, we identify conditions whose violations detect the presence of correlations between SPAM errors. From the perspective of SPAM tomography, standard quantum state tomography and detector tomography are protocols that fix the gauge parameters through the assumption that some set of fiducial measurements is known or that some set of fiducial states is known, respectively.
I explore the possibility that a quantum system S may be described completely by the combination of its standard quantum state $|\psi\rangle$ and a (hidden) quantum state $|\phi\rangle$ (that lives in the same Hilbert space), such that the outcome of any standard projective measurement on the system S is determined once the two quantum states are specified. I construct an algorithm that retrieves the standard quantum-mechanical probabilities, which depend only on $|\psi\rangle$, by assuming that the (hidden) quantum state $|\phi\rangle$ is drawn at random from some fixed probability distribution Pr(.) and by averaging over Pr(.). Contextuality and Bell nonlocality turn out to emerge automatically from this algorithm as soon as the dimension of the Hilbert space of S is larger than 2. If $|\phi\rangle$ is not completely random, subtle testable deviations from standard quantum mechanics may arise in sequential measurements on single systems.
Two-photon interference effects, such as the Hong-Ou-Mandel (HOM) effect, can be used to characterize to what extent two photons are identical. Identical photons are necessary for both linear optics quantum computing and single-photon quantum cryptography. We study here how storage and delay of photons in coupled cavity arrays, which inevitably will lead to changes in the photons' spectral and temporal profiles, affects their HOM interference. In addition we consider various types of entanglement that occur naturally in such a context.
We consider how a single photon can probe the quantum nature of a moving mirror in the context of quantum optomechanics. In particular, we demonstrate how the single-photon spectrum reveals resonances that depend on how many phonons are created as well as on the strength of the mirror-photon interaction. A dressed-state picture is used to explain positions and relative strengths of those resonances. The time-dependent spectrum shows how the resonances are built up over time by the photon interacting with the moving mirror.
We consider the concept of "the permutationally invariant (PI) part of a density matrix," which has proven very useful for both efficient quantum state estimation and entanglement characterization of $N$-qubit systems. We show here that the concept is, in fact, basis-dependent, but that this basis dependence makes it an even more powerful concept than has been appreciated so far. By considering the PI part $\rho^{{\rm PI}}$ of a general (mixed) $N$-qubit state $\rho$, we obtain: (i) strong bounds on quantitative nonseparability measures, (ii) a whole hierarchy of multi-partite separability criteria (one of which entails a sufficient criterion for genuine $N$-partite entanglement) that can be experimentally determined by just $2N+1$ measurement settings, (iii) a definition of an efficiently measurable degree of separability, which can be used for quantifying a novel aspect of decoherence of $N$ qubits, and (iv) an explicit example that shows there are, for increasing $N$, genuinely $N$-partite entangled states lying closer and closer to the maximally mixed state. Moreover, we show that if the PI part of a state is $k$-nonseparable, then so is the actual state. We further argue to add as requirement on any multi-partite entanglement measure $E$ that it satisfy $E(\rho)\geq E(\rho^{{\rm PI}})$, even though the operation that maps $\rho\rightarrow\rho^{{\rm PI}}$ is not local.
A pulsed scheme for generating robust optical entanglement via the coupling of two optical modes to a mechanical oscillator is proposed. This scheme is inspired by the Sørensen-Mølmer approach for entangling trapped ions in a thermal environment and is based on the use of optical driving pulses that are slightly detuned from the respective sideband resonance. We show that for certain pulse durations, the optomechanical interaction can return the mechanical oscillator to its initial state. The corresponding entanglement generation is robust against thermal mechanical noise in the weak as well as the strong coupling regimes. Significant optical entanglement can be generated in the weak coupling regime, even in the presence of a large thermal phonon occupation.
Threshold theorems for fault-tolerant quantum computing assume that errors are of certain types. But how would one detect whether errors of the "wrong" type occur in one's experiment, especially if one does not even know what type of error to look for? The problem is that for many qubits a full state description is impossible to analyze, and a full process description is even more impossible to analyze. As a result, one simply cannot detect all types of errors. Here we show through a quantum state estimation example (on up to 25 qubits) how to attack this problem using model selection. We use, in particular, the Akaike Information Criterion. The example indicates that the number of measurements that one has to perform before noticing errors of the wrong type scales polynomially both with the number of qubits and with the error size.
We show how the input-output formalism for cascaded quantum systems combined with the quantum trajectory approach yields a compact and physically intuitive description of single photons propagating through a coupled cavity array. As a new application we obtain the time-dependent spectrum of such a single photon, which directly reflects the fact that only certain frequency components of single-photon wavepackets are trapped inside the cavities and hence are delayed in time. We include in our description the actual generation of the single photon, by assuming we have a single emitter in one of the resonators.
In continuous-variable quantum information processing detectors are necessarily coarse grained and of finite range. We discuss how especially the latter feature is a bug and may easily lead to overoptimistic estimates of entanglement and of security, when missed data outside the detector range are ignored. We show that entropic separability or security criteria are much superior to variance-based criteria for mitigating the negative effects of this bug.
The principle behind quantum tomography is that a large set of observations -- many samples from a "quorum" of distinct observables -- can all be explained satisfactorily as measurements on a single underlying quantum state or process. Unfortunately, this principle may not hold. When it fails, any standard tomographic estimate should be viewed skeptically. Here we propose a simple way to test for this kind of failure using Akaike's Information Criterion (AIC). We point out that the application of this criterion in a quantum context, while still powerful, is not as straightforward as it is in classical physics. This is especially the case when future observables differ from those constituting the quorum.
While it is known that Tr(\rho^n) can be measured directly (i.e., without first reconstructing the density matrix) by performing joint measurements on n copies of the same state rho, it is shown here that random measurements on single copies suffice, too. Averaging over the random measurements directly yields estimates of Tr(\rho^n), even when it is not known what measurements were actually performed (so that one cannot reconstruct \rho).
We point out that the Sagnac effect can be measured by means of the Hong-Ou-Mandel effect. The latter is not sensitive to phase shifts, and thus the Hong-Ou-Mandel Sagnac effect hinges on the fact that the Sagnac effect is, fundamentally, a time delay, not a phase shift.
Recently several more efficient versions of quantum state tomography have been proposed, with the purpose of making tomography feasible even for many-qubit states. The number of state parameters to be estimated is reduced by tentatively introducing certain simplifying assumptions on the form of the quantum state, and subsequently using the data to rigorously verify these assumptions. The simplifying assumptions considered so far were (i) the state can be well approximated to be of low rank, or (ii) the state can be well approximated as a matrix product state. We add one more method in that same spirit: we allow in principle any model for the state, using any (small) number of parameters (which can, e.g., be chosen to have a clear physical meaning), and the data are used to verify the model. The proof that this method is valid cannot be as strict as in above-mentioned cases, but is based on well-established statistical methods that go under the name of "information criteria." We exploit here, in particular, the Akaike Information Criterion (AIC). We illustrate the method by simulating experiments on (noisy) Dicke states.
We propose and analyze a method to detect and characterize the drift of a nonstationary quantum source. It generalizes a standard measurement for detecting phase diffusion of laser fields to quantum systems of arbitrary Hilbert space dimension, qubits in particular. We distinguish diffusive and systematic drifts, and examine how quickly one can determine that a source is drifting. We show that for single-photon wavepackets our measurement is implemented by the Hong-Ou-Mandel effect.
The Hong-Ou-Mandel interference dip is caused by an entangled state, a delocalized bi-photon state. We propose a method of detecting this entanglement by utilizing inverse Hong-Ou-Mandel interference, while taking into account vacuum and multi-photon contaminations, phase noise, and other imperfections. The method uses just linear optics and photodetectors, and for single-mode photodetectors we find a lower bound on the amount of entanglement.
We propose one and a half criteria for determining how many measurements are needed to quantify entanglement reliably. We base these criteria on Bayesian analysis of measurement results, and apply our methods to four-qubit entanglement, but generalizations to more qubits are straightforward.
Quantum networks are composed of quantum nodes that interact coherently by way of quantum channels and open a broad frontier of scientific opportunities. For example, a quantum network can serve as a `web' for connecting quantum processors for computation and communication, as well as a `simulator' for enabling investigations of quantum critical phenomena arising from interactions among the nodes mediated by the channels. The physical realization of quantum networks generically requires dynamical systems capable of generating and storing entangled states among multiple quantum memories, and of efficiently transferring stored entanglement into quantum channels for distribution across the network. While such capabilities have been demonstrated for diverse bipartite systems (i.e., N=2 quantum systems), entangled states with N > 2 have heretofore not been achieved for quantum interconnects that coherently `clock' multipartite entanglement stored in quantum memories to quantum channels. Here, we demonstrate high-fidelity measurement-induced entanglement stored in four atomic memories; user-controlled, coherent transfer of atomic entanglement to four photonic quantum channels; and the characterization of the full quadripartite entanglement by way of quantum uncertainty relations. Our work thereby provides an important tool for the distribution of multipartite entanglement across quantum networks.
Suppose an experimentalist wishes to verify that his apparatus produces entangled quantum states. A finite amount of data cannot conclusively demonstrate entanglement, so drawing conclusions from real-world data requires statistical reasoning. We propose a reliable method to quantify the weight of evidence for (or against) entanglement, based on a likelihood ratio test. Our method is universal in that it can be applied to any sort of measurements. We demonstrate the method by applying it to two simulated experiments on two qubits. The first measures a single entanglement witness, while the second performs a tomographically complete measurement.
We consider the interference of two photons with different colors in the context of a Hong-Ou-Mandel experiment, in which single photons enter each of the input ports of a beam splitter, and exit in the same, albeit undetermined, output port. Such interference is possible if one uses an active (energy-non-conserving) beam splitter. We find scenarios in which one "red" and one "blue" photon enter the beam splitter, and either two red or two blue photons exit, but never one of each color. We show how the precise form of the active beam-splitter transformation determines in what way the spectral degrees of freedom of the input photons should be related to each other for perfect destructive interference of the different-color components in the output. We discuss two examples of active beam splitters: one is a gedanken experiment involving a moving mirror and the other is a more realistic example involving four-wave mixing in an optical fiber.
We discuss how to characterize entanglement sources with finite sets of measurements. The measurements do not have to be tomographically complete, and may consist of POVMs rather than von Neumann measurements. Our method yields a probability that the source generates an entangled state as well as estimates of any desired calculable entanglement measures, including their error bars. We apply two criteria, namely Akaike's information criterion and the Bayesian information criterion, to compare and assess different models (with different numbers of parameters) describing entanglement-generating devices. We discuss differences between standard entanglement-verificaton methods and our present method of characterizing an entanglement source.
Bell inequalities were meant to test quantum mechanics vs local hidden variable models, but can also be used to verify entanglement. For entanglement verification purposes one assumes the validity of quantum mechanics as well as quantum descriptions of one's measurements. With the help of these assumptions it is possible to derive a strengthened Bell inequality whose violation implies entanglement. We generalize known examples of such inequalities by relating the expectation value of the Bell operator to a particular quantitative measure of entanglement, namely the negativity. Moreover, we obtain statistics illustrating the fact that violating a given (strengthened or not) Bell inequality is a much more rare feat for a quantum state of two qubits than it is to be entangled.
We show that when an electron or photon propagates in a cylindrically symmetric waveguide, its spin angular momentum (SAM) and its orbital angular momentum (OAM) interact. Remarkably, we find that the dynamics resulting from this spin-orbit interaction are quantitatively described by a single expression applying to both electrons and photons. This leads to the prediction of several novel rotational effects: the spatial or time evolution of either particle's spin/polarization vector is controlled by the sign of its OAM quantum number, or conversely, its spatial wavefunction is controlled by its SAM. We show that the common origin of these effects in electrons and photons is a universal geometric phase. We demonstrate how these phenomena can be used to reversibly transfer entanglement between the SAM and OAM degrees of freedom of two-particle states.
We construct a method for verifying mode entanglement of N-mode W states. The ideal W state contains exactly one excitation symmetrically shared between N modes, but our method takes the existence of higher numbers of excitations into account, as well as the vacuum state and other deviations from the ideal state. Moreover, our method distinguishes between full N-party entanglement and states with M-party entanglement with M<N, including mixtures of the latter. We specialize to the case N=4 for illustrative purposes. In the optical case, where excitations are photons, our method can be implemented using linear optics.
So-called direct measurements of entanglement are collective measurements on multiple copies of a (bipartite or multipartite) quantum system that directly provide one a value for some entanglement measure, such as the concurrence for bipartite states. Multiple copies are needed since the entanglement of a mixed state is not a linear function of the density matrix. Unfortunately, so far all experimental implementations of direct measurements made unverified assumptions about the form of the states, and, therefore, do not qualify as entanglement verification tests. I discuss how a direct measurement can be turned into a quantitative entanglement verification test by exploiting a recent theorem by Renner (R. Renner, Nature Physics 3, 645 (2007)).
There is a constraining relation between the reliability of a quantum measurement and the extent to which the measurement process is, in principle, reversible. The greater the information that is gained, the less reversible the measurement dynamics become. To illustrate this relation, we develop a simple physical model for quantum measurement, as well as a hypothetical scheme by which the experimenters can determine the reliability and reversibility. We derive an "uncertainty" (constraining) relation between reliability and reversibility, which holds even when there is no interaction with any external environment other than the fundamental information recording device.
I consider two identical quantum particles in two boxes. We can split each box, and thereby the wavefunction of each particle, into two parts. When two half boxes are interchanged and combined with the other halves, where do the two particles end up? I solve this problem for two identical bosons and for two identical fermions. The solution can be used to define a measurement that yields some information about the relative phase between the two parts of a split wavefunction.
We show how a straightforward Bayesian updating procedure allows one to detect and quantify entanglement from any finite set of measurement results. The measurements do not have to be tomographically complete, and may consist of POVMs rather than von Neumann measurements. One obtains a probability that one's state is entangled and an estimate of any desired entanglement measure, including their error bars. As an example we consider (tomographically incomplete) spin correlation measurements on both 2-qubit and 3-qubit states. As byproducts we obtain estimates of the volume of entangled states vs. states that violate a given Bell inequality for both pure and mixed states, and an inequality that relates the expectation value of the Bell operator to the negativity.
Whereas single- and two-photon wave packets are usually treated as pure states, in practice they will be mixed. We study how entanglement created with mixed photon wave packets is degraded. We find in particular that the entanglement of a delocalized single-photon state of the electro-magnetic field is determined simply by its purity. We also discuss entanglement for two-photon mixed states, as well as the influence of a vacuum component.
We propose a quantum theory of rotating light beams and study some of its properties. Such beams are polychromatic and have either a slowly rotating polarization or a slowly rotating transverse mode pattern. We show there are, for both cases, three different natural types of modes that qualify as rotating, one of which is a new type not previously considered. We discuss differences between these three types of rotating modes on the one hand and non-rotating modes as viewed from a rotating frame of reference on the other. We present various examples illustrating the possible use of rotating photons, mostly for quantum information processing purposes. We introduce in this context a rotating version of the two-photon singlet state.
The toy model used by Spekkens [R. Spekkens, Phys. Rev. A 75, 032110 (2007)] to argue in favor of an epistemic view of quantum mechanics is extended by generalizing his definition of pure states (i.e. states of maximal knowledge) and by associating measurements with all pure states. The new toy model does not allow signaling but, in contrast to the Spekkens model, does violate Bell-CHSH inequalities. Negative probabilities are found to arise naturally within the model, and can be used to explain the Bell-CHSH inequality violations.
Dec 09 2006
quant-ph arXiv:quant-ph/0612057v3
We propose a scheme of generating and verifying mesoscopic-level entanglement between two atomic ensembles using non-resonant stimulated Raman scattering. Entanglement can be generated by direct detection or balanced homodyne detection of the Stokes fields from the two cells, after they interfere on a beam splitter. The entanglement of the collective atomic fields can be transferred to the anti-Stokes fields in a readout process. By measuring the operator moments of the anti-Stokes fields, we can verify the presence of entanglement. We model the effects of practical factors such as Stokes field detector quantum efficiency and additive thermal noise in the entanglement generating process, and anti-Stokes field losses in the entanglement verification process, and find achievable regimes in which entanglement can be verified at the levels of tens to hundreds of atomic excitations in the ensembles.
We give an overview of different types of entanglement that can be generated in experiments, as well as of various protocols that can be used to verify or quantify entanglement. We propose several criteria that, we argue, should be applied to experimental entanglement verification procedures. Explicit examples demonstrate that not following these criteria will tend to result in overestimating the amount of entanglement generated in an experiment or in infering entanglement when there is none. We distinguish protocols meant to refute or eliminate hidden-variable models from those meant to verify entanglement.
Jun 02 2006
quant-ph arXiv:quant-ph/0606017v1
I raise some doubts concerning a protocol recently applied in an experiment (Walborn et al, Nature) to measure entanglement. The protocol is much simpler than other known entanglement-verification methods, but, I argue, needs assumptions (namely that the state generated is known and pure) that are too strong to be allowed and that are not justified in most experiments. An extension of the protocol suggested in quant-ph/0605250 is much harder to implement and still relies on assumptions not justified in entanglement-verification protocols, as demonstrated by an explicit example.
Feb 09 2006
quant-ph arXiv:quant-ph/0602079v2
We consider quantum communication in the case that the communicating parties not only do not share a reference frame but use imperfect quantum communication channels, in that each channel applies some fixed but unknown unitary rotation to each qubit. We discuss similarities and differences between reference frames within that quantum communication model and gauge fields in gauge theory. We generalize the concept of refbits and analyze various quantum communication protocols within the communication model.
Oct 11 2005
quant-ph arXiv:quant-ph/0510055v1
A critical requirement for diverse applications in Quantum Information Science is the capability to disseminate quantum resources over complex quantum networks. For example, the coherent distribution of entangled quantum states together with quantum memory to store these states can enable scalable architectures for quantum computation, communication, and metrology. As a significant step toward such possibilities, here we report observations of entanglement between two atomic ensembles located in distinct apparatuses on different tables. Quantum interference in the detection of a photon emitted by one of the samples projects the otherwise independent ensembles into an entangled state with one joint excitation stored remotely in 10^5 atoms at each site. After a programmable delay, we confirm entanglement by mapping the state of the atoms to optical fields and by measuring mutual coherences and photon statistics for these fields. We thereby determine a quantitative lower bound for the entanglement of the joint state of the ensembles. Our observations provide a new capability for the distribution and storage of entangled quantum states, including for scalable quantum communication networks .
Jul 20 2005
quant-ph arXiv:quant-ph/0507189v4
I give a simple argument that demonstrates that the state |0>|1>+|1>|0>, with |0> denoting a state with 0 particles and |1> a 1-particle state, is entangled in spite of recent claims to the contrary. I also discuss new viewpoints on the old controversy about whether the above state can be said to display single-particle or single-photon nonlocality.
Mar 29 2005
quant-ph arXiv:quant-ph/0503207v1
For entangled states of light both the amount of entanglement and the sensitivity to noise generally increase with the number of photons in the state. The entanglement-sensitivity tradeoff is investigated for a particular set of states, multi-dimensional entangled coherent states. Those states possess an arbitrarily large amount of entanglement $E$ provided the number of photons is at least of order $2^{2E}$. We calculate how fast that entanglement decays due to photon absorption losses and how much entanglement is left. We find that for very small losses the amount of entanglement lost is equal to $2/\log(2)\approx 2.89$ ebits per absorbed photon, irrespective of the amount of pure-state entanglement $E$ one started with. In contrast, for larger losses it tends to be the remaining amount of entanglement that is independent of $E$. This may provide a useful strategy for creating states with a fixed amount of entanglement.
Mar 16 2005
quant-ph arXiv:quant-ph/0503140v2
We discuss certain relations between cloning and the NOT operation that can be derived from conservation laws alone. Those relations link the limitations on cloning and the NOT operation possibly imposed by \em other laws of Nature. Our result is quite general and holds both in classical and quantum-mechanical worlds, for both optimal and suboptimal operations, and for bosons as well as fermions.
Jan 03 2005
quant-ph arXiv:quant-ph/0412221v2
We study how photon absorption losses degrade the bipartite entanglement of entangled states of light. We consider two questions: (i) what state contains the smallest average number of photons given a fixed amount of entanglement? and (ii) what state is the most robust against photon absorption? We explain why the two-mode squeezed state is the answer to the first question but not quite to the second question.
Oct 12 2004
quant-ph arXiv:quant-ph/0410083v3
We define a new quantity called refbit, which allows one to quantify the resource of sharing a reference frame in quantum communication protocols. By considering both asymptotic and nonasymptotic protocols we find relations between refbits and other communication resources. We also consider the same resources in encoded, reference-frame independent, form. This allows one to rephrase and unify previous work on phase references, reference frames, and superselection rules.
Mar 17 2004
quant-ph arXiv:quant-ph/0403119v1
These notes are more or less a faithful representation of my talk at the Workshop on ``Quantum Coding and Quantum Computing'' held at the University of Virginia. As such it is an introduction for non-physicists to the topics of the quantum theory of light and entangled states of light. In particular, I discuss the photon concept and what is really entangled in an entangled state of light (it is not the photons). Moreover, I discuss an example that highlights the peculiar behavior of entanglement in an infinite-dimensional Hilbert space.
Jul 30 2003
quant-ph arXiv:quant-ph/0307216v1
We describe the resonant interaction of an atom with a strongly focused light beam by expanding the field in multipole waves. For a classical field, or when the field is described by a coherent state, we find that both intensity pattern and photon statistics of the scattered light are fully determined by a small set of parameters. One crucial parameter is the overlap of the field with the appropriate dipole wave corresponding to the relevant dipole transition in the atom. We calculate this overlap for a particular set of strongly focused longitudinally polarized light beams, whose spot size is only $0.1\lambda^2$, as discussed in S. Quabis, et al., Appl. Phys. B \bf 72, 109 (2001).
Apr 29 2003
quant-ph arXiv:quant-ph/0304179v2
We present reduction theorems for the problem of optimal unambiguous state discrimination (USD) of two general density matrices. We show that this problem can be reduced to that of two density matrices that have the same rank $n$ and are described in a Hilbert space of dimensions $2n$. We also show how to use the reduction theorems to discriminate unambiguously between N mixed states (N \ge 2).
Mar 18 2003
quant-ph arXiv:quant-ph/0303096v1
We discuss how continous-variable quantum states such as coherent states and two-mode squeezed states can be encoded in phase-reference independent ways.
Feb 12 2003
quant-ph arXiv:quant-ph/0302086v2
An example is given of an interaction that produces an infinite amount of entanglement in an infinitely short time, but only a finite amount in longer times. The interaction arises from a standard Kerr nonlinearity and a 50/50 beamsplitter, and the initial state is a coherent state. For certain finite interaction times multi-dimensional generalizations of entangled coherent states are generated, for which we construct a teleportation protocol. Similarities between probabilistic teleportation and unambiguous state discrimination are pointed out.
Braneshop, and
the US National Science Foundation.
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