Quantum computation - the use of quantum systems as bits, or qubits, to perform computation - has been proven to be exponentially faster than what is thought possible with classical computation for several tasks. Quantum computers have the unique ability to store an amount of information that is exponential in the number of constituent particles used to encode that information. This property, which is arguably where quantum computers derive their power, makes it very difficult to characterize quantum systems. In order to fully characterize an unknown quantum state, a series of measurements must be performed on it and the state deduced from the results of those measurements. This process, called quantum tomography, is known to require exponential resources in the number of qubits. Both the number of measurements - the measurement complexity - and the number of times each measurement must be repeated - the sample complexity - present serious challenges when performing tomography. In addition, the measurement apparatus itself, which is usually a quantum system, must be characterized leading to even more experimental complexity. Here, I present three experiments which aim to reduce the experimental complexity of quantum tomography. The first aims to reduce the number of measurements needed to locate the decoherence-free subspaces of a noisy quantum channel. The process of locating these subspaces, which would normally require full process tomography, is performed with quadratically fewer measurement settings. The second experiment aims to characterize both an uncalibrated measurement apparatus and an unknown quantum state at the same time. With a specific model of measurement, it is found that an uncalibrated parameter of the measurement can be directly extracted using a single measurement. In the third experiment, I perform an experiment in which the optimal sample complexity for quantum state tomography is obtained using adaptivity: the ability to change measurement settings part way through an experiment. All three experiments were performed using the polarization degree of freedom of photons generated by spontaneous parametric downconversion. The source of photon pairs and the measurement techniques used are described in detail, along with the theory of quantum distinguishability and its relevance to tomography.
Quantum computation - the use of quantum systems as bits, or qubits, to perform computation - has been proven to be exponentially faster than what is thought possible with classical computation for several tasks. Quantum computers have the unique ability to store an amount of information that is exponential in the number of constituent particles used to encode that information. This property, which is arguably where quantum computers derive their power, makes it very difficult to characterize quantum systems. In order to fully characterize an unknown quantum state, a series of measurements must be performed on it and the state deduced from the results of those measurements. This process, called quantum tomography, is known to require exponential resources in the number of qubits. Both the number of measurements - the measurement complexity - and the number of times each measurement must be repeated - the sample complexity - present serious challenges when performing tomography. In addition, the measurement apparatus itself, which is usually a quantum system, must be characterized leading to even more experimental complexity. Here, I present three experiments which aim to reduce the experimental complexity of quantum tomography. The first aims to reduce the number of measurements needed to locate the decoherence-free subspaces of a noisy quantum channel. The process of locating these subspaces, which would normally require full process tomography, is performed with quadratically fewer measurement settings. The second experiment aims to characterize both an uncalibrated measurement apparatus and an unknown quantum state at the same time. With a specific model of measurement, it is found that an uncalibrated parameter of the measurement can be directly extracted using a single measurement. In the third experiment, I perform an experiment in which the optimal sample complexity for quantum state tomography is obtained using adaptivity: the ability to change measurement settings part way through an experiment. All three experiments were performed using the polarization degree of freedom of photons generated by spontaneous parametric downconversion. The source of photon pairs and the measurement techniques used are described in detail, along with the theory of quantum distinguishability and its relevance to tomography.
Quantum state tomography is a technique in quantum information science used to reconstruct the density matrix of an unknown quantum state, providing complete information about the quantum state. It is of significant importance in fields such as quantum computation, quantum communication, and quantum simulation. However, as the size of the quantum system increases, the number of measurement settings and sampling requirements for quantum state tomography grow exponentially with the number of qubits. This not only makes experimental design and implementation more complex, but also exacerbates the consumption of experimental resources. These limitations severely hinder the application of state tomography in large-scale quantum systems. To reduce measurement settings and improve sampling efficiency, this study proposes a state tomography method based on auxiliary systems. This method can be implemented through either entanglement between the quantum system to be measured and a quantum auxiliary system or through correlation between the quantum system and a probabilistic classical auxiliary system. Measurements on the entire joint system enable more efficient extraction of information about the quantum state to be measured. This method relies on standard quantum gate operations and requires only two measurement settings, with a total sampling complexity of O(d2), significantly simplifying experimental operations and measurement processes. Additionally, this study provides two schemes for measuring purity based on the proposed circuit, one of which achieves measurement precision at the Heisenberg limit. This study validates the effectiveness of the proposed method through a detailed theoretical analysis, a series of numerical simulations, and experiments.
This thesis presents the results of a series of four photonic experiments on the topic of quantum measurement. The first two experiments relate to quantum metrology, and the use of quantum states to increase the precision of measurements beyond what is possible with classical systems; first to detect and characterize decoherence, and then in the context of quantum imaging. The third experiment studies a fundamental question in quantum mechanics: "How much must a quantum system be disturbed by a measurement?". We use weak measurement to confirm a recent theoretical result, showing that if a particle's state is already sufficiently uncertain we can perform a measurement with very little disturbance -- contrary to common explanations of Heisenberg's uncertainty principle. The fourth experiment falls in the category of quantum computation. In quantum mechanics having multiple copies of an identical system allows us to extract more information than we can extract from a single copy (since quantum mechanics allows each system to be measured only once before collapsing). We present and experimentally implement a quantum algorithm to compress all of the "extractable information" present in an ensemble of identical copies of quantum bits into exponentially fewer quantum bits. The research presented here samples from a variety of topics in quantum information, showing in several contexts how fascinating quantum effects can be exploited to gain a "quantum enhancement".To enable these experiments two sources of entangled photons were built, and "hybrid" quantum systems (encoding information in multiple degrees of freedom of a photon) were used to implement quantum circuits. This thesis will present the details of one of these sources (a novel and practical source of entangled N00N states), which was used in a four-photon quantum metrology experiment. The other, more standard, source of polarization-entangled photon pairs will only briefly be reviewed to leave room for the discussion of hybrid quantum logic.
Characterization of quantum processes is a preliminary step necessary in the development of quantum technology. The conventional method uses standard quantum process tomography, which requires d2 input states and d4 quantum measurements for a d-dimensional Hilbert space. These experimental requirements are compounded by the complexity of processing the collected data, which can take several orders of magnitude longer than the experiment itself. In this paper we propose an alternative self-guided algorithm for quantum process tomography, tuned for the task of finding an unknown unitary process. Our algorithm is a fully automated and adaptive process characterization technique. The advantages of our algorithm are the following: it has an inherent robustness to both statistical and technical noise; it requires less space and time since there is no postprocessing of the data; it requires only a single input state and measurement; and it provides on-the-fly diagnostic information while the experiment is running. Numerical results show our algorithm achieves the same 1/n scaling as standard quantum process tomography when n uses of the unknown process are used. We also present experimental results wherein the algorithm and its advantages are realized for the task of finding an element of SU(2).
How many samples of a quantum state are required to learn a complete description of it? As we will see in this thesis, the fine-grained answer depends on the measurements available to the learner, but in general it is at least Ω(d^2/ϵ^2) where d is the dimension of the state and ϵ the trace distance accuracy. Optimal algorithms for this task -- known as quantum state tomography -- make use of powerful, yet highly impractical entangled measurements, where some joint measurement is performed on all copies of the state. What can be accomplished without such measurements, where one must perform measurements on individual copies of the states? \n \nIn Chapter 2 we show a relationship between the recently proposed quantum online learning framework and quantum state tomography. Specifically, we show that tomography can be accomplished using online learning algorithms in a black-box manner and O(d^4/ϵ^4) two-outcome measurements on separate copies of the state. The interpretation of this approach is that the experimentalist uses informative measurements to teach the learner by helping it make "mistakes" on measurements as early as possible. \n \nWe move on to proving lower bounds on tomography in Chapter 3. First, we review a known lower bound for entangled measurements as well as a Ω(d^3/ϵ^2) lower bound in the setting where non-entangled measurements are made non-adaptively, both due to Ref.[18]. We then derive a novel bound of Ω(d^4/ϵ^2) samples when the learner is further restricted to observing a constant number of outcomes (e.g., two-outcome measurements). This implies that the folklore "Pauli tomography" algorithm is optimal in this setting. \n \nUnderstanding the power of adaptive measurements, where measurement choices can depend on previous outcomes, is currently an open problem. In Chapter 4 we present two scenarios in which adapting on previous outcomes makes no difference to the number of samples required. In the first, the learner is limited to adapting on at most o(d^2/ϵ^2) of the previous outcomes. In the second, measurements are drawn from some set of at most exp(O(d)) measurements. In particular, this second lower bound implies that adaptivity makes no difference in the regime of efficiently implementable measurements, in the context of quantum computing. \n \nFinally, we apply the above technique to the problems of classical shadows and shadow tomography to obtain similar lower bounds. Here, one is interested only in determining the expectations of some fixed set of observables. We once again find that, for the worst-case input of observables, adaptivity makes no difference to the sample complexity when considering efficient, non-entangled measurements. As a corollary, we find a straightforward algorithm for shadow tomography is optimal in this setting.
This thesis fathoms the possibilities and limitations of measurements in quantum optics. After a short introduction to the general aspects of measurements in quantum mechanics, it presents three examples of a measurement process in concrete applications to quantum optics. For one, while operators describing quantities of interest might have a simple theoretical form, it is often hard to actually implement a measurement of their expectation values experimentally. This situation is encountered in the first part which is devoted to the measurement of quantum statistical properties of a light beam. In contrast to classical optics, these statistics are not fully determined by first- and second-order correlations. Instead, higher-order correlations, usually described by non-Hermitian operators, are necessary to characterize the quantum state completely. A particular example is the N-photon Fock state, for which only a measurement of the Nth-order correlations allows for a full characterization. Here, we present a new method how to measure an important class of Nth-order correlations for arbitrary N in a single spatial mode with two polarizations with limited experimental resources. The second part is again centered around the realization of specific measurement operators for photon correlation measurements. However, in contrast to the previous chapter, these measurements are not an end in their own, but are designed to create specific quantum states as a consequence of the measurement process: A suitable measurement of photons emitted by a number of atoms allows to transfer the atoms into long-lived entangled states. We find that with this method it is possible to entangle two remote atomic qubits to an arbitrary and well-defined degree. Furthermore, we show how to generate two families of quantum states in an arbitrary number of remote atomic qubits: one family consists of all -- symmetric and non-symmetric -- total angular momentum eigenstates in N remote qubits, the other is a family of cluster states. It is also systematically investigated whether the technique of quantum state engineering by projective measurements allows for the creation of any arbitrary state. We find explicit solutions for the case of two and three qubits, while it is shown that no solution exist for N > 6. We conclude the section by proving that with some changes in the setup, this measurement-based quantum state engineering technique allows to create the same states among the scattered photons themselves rather than among the emitting atoms. The last part is devoted to the amount of information one is able to gain about a given system when performing a measurement. The corresponding limits of information are investigated at the example of the duality that arises in a two-way interferometer between the visibility of the interference pattern and the knowledge about the path of the interfering object. We show that there exist correlations between the phase difference the interfering object acquires on its way through the interferometer and the amount of which-way information retrievable from certain observables of the which-way detector. In this way, the which-way information becomes a phase-dependent quantity. In particular, we find that for certain values of the phase shift, the amount of extractable which-way information can be larger than allowed by phase-independent measurement. This property is put to use to find the optimal observable for the which-way detector readout in dependence on the phase of the interfering object. Following this strategy, we are able to gain more which-way information than previously thought possible.
The initialization of a quantum system into a certain state is a crucial aspect of quantum information science. While a variety of measurement strategies have been developed to characterize how well the system is initialized, for a given one, there is in general a trade-off between its efficiency and the accessible information of the quantum state. Conventional quantum state tomography can characterize unknown states while requiring exponentially expensive time-consuming postprocessing. Alternatively, recent theoretical breakthroughs show that quantum state verification provides a technique to quantify the prepared state with significantly fewer samples, especially for multipartite entangled states. In this Letter, we modify the original proposal to be robust to practical imperfections, and experimentally implement a scalable quantum state verification on two-qubit and four-qubit entangled states with nonadaptive local measurements. For all the tested states, the estimated infidelity is inversely proportional to the number of samples, which illustrates the power to characterize a quantum state with a small number of samples. Compared to the globally optimal strategy which requires nonlocal measurements, the efficiency in our experiment is only worse by a small constant factor (<2.5). We compare the performance difference between quantum state verification and quantum state tomography in an experiment to characterize a four-photon Greenberger-Horne-Zeilinger state, and the results indicate the advantage of quantum state verification in both the achieved efficiency and precision.