We develop a quantum learning scheme for binary discrimination of coherent states of light. This is a problem of technological relevance for the reading of information stored in a digital memory. In our setting, a coherent light source is used to illuminate a memory cell and retrieve its encoded bit by determining the quantum state of the reflected signal. We consider a situation where the amplitude of the states produced by the source is not fully known, but instead this information is encoded in a large training set comprising many copies of the same coherent state. We show that an optimal global measurement, performed jointly over the signal and the training set, provides higher successful identification rates than any learning strategy based on first estimating the unknown amplitude by means of Gaussian measurements on the training set, followed by an adaptive discrimination procedure on the signal. By considering a simplified variant of the problem, we argue that this is the case even for non-Gaussian estimation measurements. Our results show that, even in absence of entanglement, collective quantum measurements yield an enhancement in the readout of classical information, which is particularly relevant in the operating regime of low-energy signals.
We develop a quantum learning scheme for binary discrimination of coherent states of light. This is a problem of technological relevance for the reading of information stored in a digital memory. In our setting, a coherent light source is used to illuminate a memory cell and retrieve its encoded bit by determining the quantum state of the reflected signal. We consider a situation where the amplitude of the states produced by the source is not fully known, but instead this information is encoded in a large training set comprising many copies of the same coherent state. We show that an optimal global measurement, performed jointly over the signal and the training set, provides higher successful identification rates than any learning strategy based on first estimating the unknown amplitude by means of Gaussian measurements on the training set, followed by an adaptive discrimination procedure on the signal. By considering a simplified variant of the problem, we argue that this is the case even for non-Gaussian estimation measurements. Our results show that, even in absence of entanglement, collective quantum measurements yield an enhancement in the readout of classical information, which is particularly relevant in the operating regime of low-energy signals.
The problem of discriminating the state of a quantum system among a number of hypothetical states is usually addressed under the assumption that one has perfect knowledge of the possible states of the system. In this thesis, I analyze the role of the prior information available in facing such problems, and consider scenarios where the information regarding the possible states is incomplete. In front of a complete ignorance of the possible states' identity, I discuss a quantum "programmable" discrimination machine for qubit states that accepts this information as input programs using a quantum encoding, rather than as a classical description. The optimal performance of these machines is studied for general qubit states when several copies are provided, in the schemes of unambiguous, minimum-error, and error-margin discrimination. Then, this type of automation in discrimination tasks is taken further. By realizing a programmable machine as a device that is trained through quantum information to perform a specific task, I propose a quantum "learning" machine for classifying qubit states that does not require a quantum memory to store the qubit programs and, nevertheless, performs as good as quantum mechanics permits. Such learning machine thus allows for several optimal uses with no need for retraining. A similar learning scheme is also discussed for coherent states of light. I present it in the context of the readout of a classical memory by means of classically correlated coherent signals, when these are produced by an imperfect source. I show that, in this case, the retrieval of information stored in the memory can be carried out more accurately when fully general quantum measurements are used. Finally, as a transversal topic, I propose an efficient algorithmic way of decomposing any quantum measurement into convex combinations of simpler (extremal) measurements.
Recent results have established dramatic advantages in learning properties of quantum states when a quantum computer is available to process or jointly measure multiple copies of the unknown quantum state. Learning tasks can be accomplished with exponentially fewer copies of the state when compared to optimized classical learning strategies that are restricted to measuring one copy of the state at a time. While these results were established in abstract settings and for artificial learning tasks, they motivate the application of quantum computers to imaging and sensing of weak electromagnetic fields since these settings are ultimately concerned with the learning of unknown quantum states. In this work we apply these new results in quantum learning to the problem of learning Gaussian states of the electromagnetic field, which are germane since they describe most fields used in imaging and sensing. In order to connect with quantum learning theory, we consider the transduction of an n-mode Gaussian state into a register of qubits on a quantum computer followed by optimized measurements on these qubits to extract the parameters defining the original Gaussian state. We rigorously bound the number of copies of the Gaussian state required to achieve worst-case additive error in parameter estimates. The scaling of this bound with n is exponentially better than naïve strategies for characterizing Gaussian states and matches recently derived bounds for characterization of Gaussian states using continuous-variable (CV) classical shadows. In addition, our bound has a polynomially better dependence on the energy of the multimode Gaussian state compared to the CV shadows protocol.
It has been shown recently S. Pirandola [Phys. Rev. Lett. 106, 090504 (2011)] that entangled light with Einstein-Podolsky-Rosen correlations retrieves information from digital memory better than any classical light. In identifying this, a model of digital memory with each cell consisting of a reflecting medium with two reflectivities (each memory cell encoding the binary numbers 0 or 1) is employed. The readout of binary memory essentially corresponds to discrimination of two bosonic attenuator channels characterized by different reflectivities. The model requires an entire mathematical paraphernalia of a continuous variable Gaussian setting for its analysis when arbitrary values of reflectivities $0\ensuremath{\le}{r}_{0},{r}_{1}\ensuremath{\le}1$ are considered. Here we restrict ourselves to a basic quantum readout mechanism with two different families of non-Gaussian entangled states of light, in which the binary channels to be discriminated are (i) ideal memory characterized by reflectivity r1=1 (identity channel) and (ii) a thermal noise channel---where the signal light illuminating the memory location gets completely lost (r0=0) and only a white thermal noise hitting the upper side of the memory reaches the decoder. We compare the quantum reading efficiency of two families of non-Gaussian entangled light [($m,{m}^{\ensuremath{'}}$) family of path-entangled photon states and entangled state obtained by mixing a single photon with coherent light in a 50:50 beam splitter] with any classical source of light in this model. We identify that the classes of non-Gaussian entangled transmitters studied here offer significantly better reading performance than any classical transmitters of light in the regime of low signal intensity. We also demonstrate that the ($m,{m}^{\ensuremath{'}}$) family of entangled light exhibits better reading performance than NOON states.
Pattern recognition is a central topic in learning theory, with numerous applications such as voice and text recognition, image analysis and computer diagnosis. The statistical setup in classification is the following: we are given an i.i.d. training set ( X 1 , Y 1 ), ... , ( X n , Y n ), where X i represents a feature and Y i ∊{0, 1} is a label attached to that feature. The underlying joint distribution of ( X , Y ) is unknown, but we can learn about it from the training set, and we aim at devising low error classifiers f: X → Y used to predict the label of new incoming features. In this paper, we solve a quantum analogue of this problem, namely the classification of two arbitrary unknown mixed qubit states. Given a number of 'training' copies from each of the states, we would like to 'learn' about them by performing a measurement on the training set. The outcome is then used to design measurements for the classification of future systems with unknown labels. We found the asymptotically optimal classification strategy and show that typically it performs strictly better than a plug-in strategy, which consists of estimating the states separately and then discriminating between them using the Helstrom measurement. The figure of merit is given by the excess risk equal to the difference between the probability of error and the probability of error of the optimal measurement for known states. We show that the excess risk scales as n −1 and compute the exact constant of the rate.
The readout of a classical memory can be modelled as a problem of quantum channel discrimination, where a decoder retrieves information by distinguishing the different quantum channels encoded in each cell of the memory (Pirandola 2011 Phys. Rev. Lett. 106 090504). In the case of optical memories, such as CDs and DVDs, this discrimination involves lossy bosonic channels and can be remarkably boosted by the use of nonclassical light (quantum reading). Here we generalize these concepts by extending the model of memory from single-cell to multi-cell encoding. In general, information is stored in a block of cells by using a channel-codeword, i.e. a sequence of channels chosen according to a classical code. Correspondingly, the readout of data is realized by a process of 'parallel' channel discrimination, where the entire block of cells is probed simultaneously and decoded via an optimal collective measurement. In the limit of a large block we define the quantum reading capacity of the memory, quantifying the maximum number of readable bits per cell. This notion of capacity is nontrivial when we suitably constrain the physical resources of the decoder. For optical memories (encoding bosonic channels), such a constraint is energetic and corresponds to fixing the mean total number of photons per cell. In this case, we are able to prove a separation between the quantum reading capacity and the maximum information rate achievable by classical transmitters, i.e. arbitrary classical mixtures of coherent states. In fact, we can easily construct nonclassical transmitters that are able to outperform any classical transmitter, thus showing that the advantages of quantum reading persist in the optimal multi-cell scenario.
A quantum learning machine for binary classification of qubit states that does not require quantum memory is introduced and shown to perform with the minimum error rate allowed by quantum mechanics for any size of the training set. This result is shown to be robust under (an arbitrary amount of) noise and under (statistical) variations in the composition of the training set, provided it is large enough. This machine can be used an arbitrary number of times without retraining. Its required classical memory grows only logarithmically with the number of training qubits, while its excess risk decreases as the inverse of this number, and twice as fast as the excess risk of an estimate-and-discriminate machine, which estimates the states of the training qubits and classifies the data qubit with a discrimination protocol tailored to the obtained estimates.