The work in this thesis can be divided into three parts. The first two parts deal with optical quantum communication protocols and the third part deals with quantum cryptography. The first part of the thesis is a step towards reformulating quantum protocols in terms of coherent states. Quantum communication protocols are typically formulated in terms of abstract qudit states and operations. This leaves the question of an experimental realization open. Direct translation of these protocols, say into single photons with some d-dimensional degree of freedom, are typically challenging to realize. Multi-mode coherent states, on the other hand, can be easily generated experimentally. Reformulation of protocols in terms of these states has been a successful strategy for implementation of quantum protocols. Quantum key distribution and the quantum fingerprinting protocol have both followed this route. In Chapter 3, we characterize the Gram matrices of multi-mode coherent states in an attempt to understand the class of communication protocols, which can be implemented using these states. We also derive the closure of the Gram matrices, which can be implemented in this way, so that we also characterize those matrices, which can be approximated arbitrarily well using multi-mode coherent states. \n \nIn the second part of the thesis, Chapter 4, we describe our collaboration with an experimental group to implement the quantum fingerprinting protocol and examine the tradeoffs between the resources required to implement such protocols. It is seen that it is difficult to implement the quantum fingerprinting protocol experimentally for large input sizes. This leads us to study the tradeoff between the two resources expended during optically implemented simultaneous message passing communication protocols: the duration of the protocol and the energy required to run it. We derive general bounds on the growth of these quantities which are valid for all optical protocols. We also develop tighter bounds for the growth of these resources for protocols implementing quantum fingerprinting with coherent states. \n \nFinally in Chapter 5, we venture into quantum cryptography. We introduce a new setting for two-party cryptography with temporarily trusted third parties. In this setting, in addition to Alice and Bob, there are third parties, which Alice and Bob both trust to be honest during the course of the protocol. However, once the protocol concludes, there is no guarantee over the behaviour of these third parties. It is possible that they collaborate and act adversarially against the honest parties. We implement a variant of bit commitment in this setting, which we call erasable bit commitment. In this primitive, Alice has the choice of either opening or erasing her commitment after the commit phase. The ability to ask for an erasure allows Alice to ask the trusted nodes to erase her commitment in case the trust period is about to expire. However, this ability also makes erasable bit commitment weaker than the standard version of bit commitment. In addition to satisfying the security requirements of bit commitment, our protocol also does not reveal any information about the commitment to the third parties. Lastly, our protocol for this primitive requires a constant number of trusted third parties and can tolerate a small number of dishonest trusted nodes as well as implementation errors.
Partial distinguishability affects multiphoton interference, which is central to large-scale photonic experiments such as Boson Sampling and to applications of multiphoton states across quantum technologies. Characterizing the internal degrees of freedom responsible for this behavior is challenging: in many scenarios they cannot be accessed directly and even their dimension or physical nature may be unknown \emph{a priori}. Here we introduce Gram-matrix tomography for independently prepared photons with pure or nearly pure internal states. For pure states, the Gram matrix contains all information required to predict the statistics of experiments that are insensitive to the internal degrees of freedom. We reconstruct this matrix, up to physically irrelevant phase choices, from a quadratic number of measurable Bargmann invariants (quantities encoding overlap magnitudes and collective phases). We access these quantities using two complementary approaches: targeted Fourier interferometers and two- and three-body photon-number correlations measured in a single random interferometer. We derive explicit error bounds that control the total-variation distance between predicted and actual outcome distributions for any subsequent experiment that is insensitive to the internal states. These guarantees account for finite sampling and deviations from purity. Crucially, the total sample complexity of the protocol is independent of the internal Hilbert-space dimension and polynomial under mild assumptions on overlaps of internal states. Our protocol uses standard interferometric measurements and turns them into a robust and operationally grounded characterization of partially distinguishable photons.
In the studies of quantum information and computation, the quantum correlations
present in a non-classical system are seen as natural resources. In this
thesis, we present a detailed study on quantification and characterization of
multipartite quantum entanglement from a geometric perspective. Furthermore,
we present protocols and algorithms where quantum correlations such
as entanglement, quantum Fisher information, and quantum superposition
have been used to provide advantages as well securities in quantum communication
tasks, quantum speed-up, and protection against the decoherence in
an open quantum system.
First, we use a geometric measure of entanglement quantification based
on Euclidean distance of the Hermitian matrices to obtain the minimum distance
between the set of bipartite n- qudit density matrices with a positive
partial transpose and the maximally mixed state. This minimum distance coincides
with the radius of the largest separable ball. We present the interior
geometry of the set of all positive semidefinite matrices and identify a particular
class of Werner states for which the Peres-Horodecki criterion is both
necessary and sufficient for separability even in the dimensions greater than
six.
We introduce several families of quantum fingerprinting protocols to evaluate the equality function on two
n-bit strings in the simultaneous message passing model. The original quantum fingerprinting protocol uses a tensor product of a small number of
O(logn)-qubit high-dimensional signals [H. Buhrman et al., Phys. Rev. Lett. 87, 167902 (2001)], whereas a recently proposed optical protocol uses a tensor product of
O(n) single-qubit signals, while maintaining the
O(logn) information leakage of the original protocol [J. M. Arazola and N. L\"utkenhaus, Phys. Rev. A 89, 062305 (2014)]. We find a family of protocols which interpolate between the original and optical protocols while maintaining the
O(logn) information leakage, thus demonstrating a tradeoff between the number of signals sent and the dimension of each signal. There has been interest in experimental realization of the recently proposed optical protocol using coherent states [F. Xu et al., Nat. Commun. 6, 8735 (2015); J.-Y. Guan et al., Phys. Rev. Lett. 116, 240502 (2016)], but as the required number of laser pulses grows linearly with the input size
n, eventual challenges for the long-time stability of experimental setups arise. We find a coherent state protocol which reduces the number of signals by a factor
21 while also reducing the information leakage. Our reduction makes use of a simple modulation scheme in optical phase space, and we find that more complex modulation schemes are not advantageous. Using a similar technique, we improve a recently proposed coherent state protocol for evaluating the Euclidean distance between two real unit vectors [N. Kumar et al., Phys. Rev. A 95, 032337 (2017)] by reducing the number of signals by a factor
21 and also reducing the information leakage.
Nowadays, the realization of quantum computations and communications based on continuous variables has attracted a significant attention due to a substantial expansion of the system dimensionality. The main progress in this area is attributed to the implementation of multimode systems based on squeezed states of light. One of the simplest ways to generate such states relies upon their producing in a single-pass optical parametric amplifier (OPA) using ultrafast pumping. However, for homodyne detection of such multimode states, the profile of the local oscillator (LO) must perfectly match the profile of the measured mode. Usually, this is not the case; therefore a proper treatment of multimode squeezing is required. In this work, we study both theoretically and experimentally the multimode squeezed light generated in type-0 and type-II OPA. We characterize such sources and investigate the degree of squeezing in dependence on the LO spectral profile, employing a pulse shaping technique. The theoretical analysis is performed using the Schmidt-mode theory. This work might have a significant impact on the realization of multimode quantum protocols
We provide an example of a communication model and a distributed task, for which there exists a realistic quantum protocol that is asymptotically more efficient than any classical protocol, both in the communication and the information resources. For this, we extend a recently proposed coherent state mapping for quantum communication protocols, study the use of coherent state fingerprints over multiple channels, and show their role in the design of an efficient quantum protocol for estimating the Euclidean distance of two real vectors within a constant factor.