Nonorthogonal States Research Articles

Quantum-inspired clustering with light

This article introduces a novel approach to perform the simulation of a single qubit quantum-inspired algorithm using laser beams. Leveraging the polarization states of photonic qubits, and inspired by variational quantum eigensolvers, we develop a variational quantum-inspired algorithm implementing a clustering procedure following the approach proposed by some of us in SciRep 13, 13284 (2023). A key aspect of our research involves the utilization of non-orthogonal states within the photonic domain, harnessing the potential of polarization schemes to reproduce unitary circuits. By mapping these non-orthogonal states into polarization states, we achieve an efficient and versatile quantum information processing unit which serves as a clustering device for a diverse set of datasets.

Brief theory of multiqubit measurement

Peculiarities of multiqubit measurement are for the most part similar to peculiarities of measurement for qudit — quantum object with finite-dimensional Hilbert space. Three different interpretations of measurement concept are analyzed. One of those is purely quantum and is in collection, for a given state of the object to be measured, of incompatible observable measurement results in amount enough for reconstruction of the state. Two others make evident the difference between the reduced density matrix and the density matrices of physical objects involved in the measurement. It is shown that the von Neumann projectors, in combination with the concept of qudit phase space, produce an idea of a phase portrait of qudit state as a set of mathematical expectations for projectors on the possible pure states. The phase portrait is not a probability distribution since the projectors on nonorthogonal states are incompatible observables. Along with that, the phase portrait includes probability distributions for all the resolutions of identity of the qudit observable algebra. Additional peculiarities of measurement of the qudit degenerated observables, caused by the possibility of independent measurement of the observables for the particles, make possible to distinguish the local reduction of the qudit particle states from the entanglement of the local measurement results. The phase portrait of a composite system comprised by a qudit pair generates local and conditional phase portraits of particles. The entanglement is represented by the dependence of the shape of conditional phase portrait on the properties of the observable used in the measurement for the other particle. Analysis of the properties of a conditional phase portrait of a multiqubit qubits shows that absence of the entanglement is possible only in the case of substantial restrictions imposed on the method of multiqubit decomposition into qubits. Such a special method for determination of particles exists for each multiqubit state.

Necessary condition for information transfer under simulated parity-time-symmetric evolution

Parity-time (PT) symmetric quantum theory can broaden the scope of quantum dynamics beyond unitary evolution which may lead to numerous counter-intuitive phenomena, including single-shot discrimination of non-orthogonal states, faster evolution of state than the standard quantum speed limit, and violation of no-signaling principle. On the other hand, PT -symmetric evolution can be realized as reduced dynamics of a subsystem in real experiments within the scope of standard QT. In this experimental setup, if one side of a composite system is evolved according to a PT -symmetric way, a non-trivial information transfer can happen, i.e. the operation performed at one side can be gathered by the other side. By considering an arbitrary shared state between two parties situated in two distant locations and arbitrary measurements, we show that the PT -symmetric evolution of the reduced subsystem at one side is not sufficient for this information transfer to occur. Specifically, we prove that the information transfer can only happen when the density matrix and the corresponding measurements contain complex numbers. Moreover, we connect the entanglement content of the shared state with the efficacy of information transfer. We find evidence that the task becomes more efficient with the increase of dimension.

Optimal individual and collective measurements for nonorthogonal quantum key distribution signals

We consider how the theory of optimal quantum measurements determines the maximum information available to the receiving party of a quantum key distribution (QKD) system employing linearly independent but nonorthogonal quantum states. Such a setting is characteristic of several practical QKD protocols. Due to nonorthogonality, the receiver is not able to discriminate unambiguously between the signals. To understand the fundamental limits that this imposes, the quantity of interest is the maximum mutual information between the transmitter (Alice) and the receiver, whether legitimate (Bob) or an eavesdropper (Eve). To find the optimal measurement—taken individually or collectively—we use a framework based on operator algebra and general results derived from singular-value decomposition, achieving optimal solutions for von Neumann measurements and positive operator-valued measures (POVMs). The formal proof and quantitative analysis elaborated for two signals allow us to conclude that optimal von Neumann measurements are uniquely defined and provide a higher information gain compared to POVMs. Interestingly, collective measurements not only do not provide additional information gain with respect to individual ones but also suffer from a gain reduction in the case of POVMs. Published by the American Physical Society 2024

Minimum-Consumption Discrimination of Quantum States via Globally Optimal Adaptive Measurements.

Reducing the average resource consumption is the central quest in discriminating non-orthogonal quantum states for a fixed admissible error rate ϵ. The globally optimal fixed local projective measurement for this task is found to be different from that for previous minimum-error discrimination tasks [S. Slussarenko etal., Phys. Rev. Lett. 118, 030502 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.030502]. To achieve the ultimate minimum average consumption, here we develop a general globally optimal adaptive strategy (GOA) by subtly using the updated posterior probability, which works under any error rate requirements and any one-way measurement restrictions, and can be solved by a convergent iterative relation. First, under the local measurement restrictions, our GOA is solved to serve as the local bound, which saves 16.6 copies (24%) compared with the previously best globally optimal fixed local projective measurement. When the more powerful two-copy collective measurements are allowed, our GOA is experimentally demonstrated to beat the local bound by 3.9 copies (6.0%). By exploiting both adaptivity and collective measurements, our Letter marks an important step toward minimum-consumption quantum state discrimination.

Pure Decoherence of the Jaynes–Cummings Model: Initial Entanglement with the Environment, Spin Oscillations and Detection of Non-Orthogonal States

A model based on pure decoherence for the Jaynes–Cummings spin–boson system, coupled through its integral of motion to an infinite bosonic bath, is proposed and examined. The properties of the spin oscillation process suggest an initial entanglement between the environment and the spin–boson degrees of freedom. The study demonstrates that the potential applicability of the Jaynes–Cummings model in detecting non-orthogonal bosonic states is preserved in the presence of pure decoherence.

Quantum state discrimination with assisted entanglement

State discrimination is one of the key steps in quantum information tasks. This paper introduces the assisted quantum entanglement to achieve this step. For this purpose, the paper introduces some preliminary knowledge to help enter the research topic, such as the matrix rank, Kronecker product, quantum states and the calculation method of Schmidt decomposition and rank. The latter involves the Singular value decomposition in matrix factorization theory. The method of distinguishing orthogonal states in single and many-body systems is also introduced, by using the so-called positive operator-valued measurements. Then the paper proves the fact that single or multipartite system non-orthogonal states are indistinguishable regardless of whether there are auxiliary resources, which may be either separable or entangled states. The paper further illustrates the feasibility of maximum entangled states acting on two-body systems and gives corresponding examples. At the same time, the feasibility of auxiliary resources for smaller entanglement is also investigated.

Discriminating mixed qubit states with collective measurements

It is a central fact in quantum mechanics that non-orthogonal states cannot be distinguished perfectly. In general, the optimal measurement for distinguishing such states is a collective measurement. However, to the best our knowledge, collective measurements have not been used to enhance quantum state discrimination to date. One of the main reasons for this is the fact that, in the usual state discrimination setting with equal prior probabilities, at least three copies of a quantum state are required to be measured collectively to outperform separable measurements. This is very challenging experimentally. In this work, by considering unequal prior probabilities, we propose and experimentally demonstrate a protocol for distinguishing two copies of single qubit states using collective measurements which achieves a lower probability of error than can be achieved by any non-entangling measurement. Additionally, we implemented collective measurements on three and four copies of the unknown state and found they performed poorly.

Variational quantum and quantum-inspired clustering

Here we present a quantum algorithm for clustering data based on a variational quantum circuit. The algorithm allows to classify data into many clusters, and can easily be implemented in few-qubit Noisy Intermediate-Scale Quantum devices. The idea of the algorithm relies on reducing the clustering problem to an optimization, and then solving it via a Variational Quantum Eigensolver combined with non-orthogonal qubit states. In practice, the method uses maximally-orthogonal states of the target Hilbert space instead of the usual computational basis, allowing for a large number of clusters to be considered even with few qubits. We benchmark the algorithm with numerical simulations using real datasets, showing excellent performance even with one single qubit. Moreover, a tensor network simulation of the algorithm implements, by construction, a quantum-inspired clustering algorithm that can run on current classical hardware.

Coherence, superposition, and Löwdin symmetric orthogonalization

The notions of coherence and superposition are conceptually the same; however, an important distinction exists between their resource-theoretic formulations. Namely, while basis states are orthogonal in the resource theory of coherence, they are not necessarily orthogonal in the resource theory of superposition. Owing to the nonorthogonality, the manipulation and characterization of superposition states require significant efforts. Here, we demonstrate that the Löwdin symmetric orthogonalization (LSO) method offers a useful means for characterizing pure superposition states. The principal property of LSO is that the structure and symmetry of the original nonorthogonal basis states are preserved to the greatest extent possible, which prompts us to study the role of LSO in identifying the hierarchical relations of resource states. Notably, we reveal that the maximally coherent states turn into the states with maximal superposition with the help of LSO: in other words, they are equivalent under the action of symmetric orthogonalization. Our results facilitate further connections between coherence and superposition, where LSO is the main tool.

Improving the Performance of Quantum Cryptography by Using the Encryption of the Error Correction Data.

Security of quantum key distribution (QKD) protocols rely solely on quantum physics laws, namely, on the impossibility to distinguish between non-orthogonal quantum states with absolute certainty. Due to this, a potential eavesdropper cannot extract full information from the states stored in their quantum memory after an attack despite knowing all the information disclosed during classical post-processing stages of QKD. Here, we introduce the idea of encrypting classical communication related to error-correction in order to decrease the amount of information available to the eavesdropper and hence improve the performance of quantum key distribution protocols. We analyze the applicability of the method in the context of additional assumptions concerning the eavesdropper's quantum memory coherence time and discuss the similarity of our proposition and the quantum data locking (QDL) technique.

Variational quantum non-orthogonal optimization

Current universal quantum computers have a limited number of noisy qubits. Because of this, it is difficult to use them to solve large-scale complex optimization problems. In this paper we tackle this issue by proposing a quantum optimization scheme where discrete classical variables are encoded in non-orthogonal states of the quantum system. We develop the case of non-orthogonal qubit states, with individual qubits on the quantum computer handling more than one bit classical variable. Combining this idea with Variational Quantum Eigensolvers (VQE) and quantum state tomography, we show that it is possible to significantly reduce the number of qubits required by quantum hardware to solve complex optimization problems. We benchmark our algorithm by successfully optimizing a polynomial of degree 8 and 15 variables using only 15 qubits. Our proposal opens the path towards solving real-life useful optimization problems in today’s limited quantum hardware.

Protecting Quantum Modes in Optical Fibers

Polarization-preserving fibers maintain the two polarization states of an orthogonal basis. Quantum communication, however, requires sending at least two nonorthogonal states and these cannot both be preserved. We present an alternative scheme that allows for using polarization encoding in a fiber not only in the discrete, but also in the continuous-variable regime. For the example of a helically twisted photonic crystal fiber, we experimentally demonstrate that using appropriate nonorthogonal modes, the polarization-preserving fiber does not fully scramble these modes over the full Poincar\'e sphere, but that the output polarization will stay on a great circle; that is, within a one-dimensional protected subspace, which can be parametrized by a single variable. This allows for more efficient measurements of quantum excitations in nonorthogonal modes.

Quantum Lock: A Provable Quantum Communication Advantage

Physical unclonable functions(PUFs) provide a unique fingerprint to a physical entity by exploiting the inherent physical randomness. Gao et al. discussed the vulnerability of most current-day PUFs to sophisticated machine learning-based attacks. We address this problem by integrating classical PUFs and existing quantum communication technology. Specifically, this paper proposes a generic design of provably secure PUFs, called hybrid locked PUFs(HLPUFs), providing a practical solution for securing classical PUFs. An HLPUF uses a classical PUF(CPUF), and encodes the output into non-orthogonal quantum states to hide the outcomes of the underlying CPUF from any adversary. Here we introduce a quantum lock to protect the HLPUFs from any general adversaries. The indistinguishability property of the non-orthogonal quantum states, together with the quantum lockdown technique prevents the adversary from accessing the outcome of the CPUFs. Moreover, we show that by exploiting non-classical properties of quantum states, the HLPUF allows the server to reuse the challenge-response pairs for further client authentication. This result provides an efficient solution for running PUF-based client authentication for an extended period while maintaining a small-sized challenge-response pairs database on the server side. Later, we support our theoretical contributions by instantiating the HLPUFs design using accessible real-world CPUFs. We use the optimal classical machine-learning attacks to forge both the CPUFs and HLPUFs, and we certify the security gap in our numerical simulation for construction which is ready for implementation.

Training a quantum measurement device to discriminate unknown non-orthogonal quantum states

Here, we study the problem of decoding information transmitted through unknown quantum states. We assume that Alice encodes an alphabet into a set of orthogonal quantum states, which are then transmitted to Bob. However, the quantum channel that mediates the transmission maps the orthogonal states into non-orthogonal states, possibly mixed. If an accurate model of the channel is unavailable, then the states received by Bob are unknown. In order to decode the transmitted information we propose to train a measurement device to achieve the smallest possible error in the discrimination process. This is achieved by supplementing the quantum channel with a classical one, which allows the transmission of information required for the training, and resorting to a noise-tolerant optimization algorithm. We demonstrate the training method in the case of minimum-error discrimination strategy and show that it achieves error probabilities very close to the optimal one. In particular, in the case of two unknown pure states, our proposal approaches the Helstrom bound. A similar result holds for a larger number of states in higher dimensions. We also show that a reduction of the search space, which is used in the training process, leads to a considerable reduction in the required resources. Finally, we apply our proposal to the case of the phase flip channel reaching an accurate value of the optimal error probability.

Quantum key distribution over FSO channel using error reconciliation protocol

Quantum key distribution (QKD) has evolved as a robust way of secret key distribution based on renowned modern physics concepts. To transmit sensitive information in the government, private, and personal sectors, high-security levels are now necessary. All conventional cryptographic algorithms applied in the communication models, which rely on mathematical models and conceptual assumptions, are unsecure. Thus, QKD systems are the best choice for protecting this information as a countermeasure since they provide unconditional security. In this paper, a design is proposed for the FSO channel using the CASCADE protocol under different atmospheric condition such as haze, rain and snow. The suggested framework’s effectiveness and security in the context of QKD with two non-orthogonal photon states are assessed. Simulation results of this model show the percentage of original sequence recovery; the number of errors removed, optical spectrum at transmitter and receiver for a base frame length of 104 bits. Moreover, Error correction is calculated per iteration to improve signal quality at the receiver side and compared to previous work. The calculated value in this article shows a better result. Hence, both the sender and the receiver could achieve a security key using this method.

Characterization of the nonclassical relation between measurement outcomes represented by nonorthogonal quantum states

Quantum mechanics describes seemingly paradoxical relations between the outcomes of measurements that cannot be performed jointly. In Hilbert space, the outcomes of such incompatible measurements are represented by non-orthogonal states. In this paper, we investigate how the relation between outcomes represented by non-orthogonal quantum states differs from the relations suggested by a joint assignment of measurement outcomes that do not depend on the actual measurement context. The analysis is based on a well-known scenario where three statements about the impossibilities of certain outcomes would seem to make a specific fourth outcome impossible as well, yet quantum theory allows the observation of that outcome with a non-vanishing probability. We show that the Hilbert space formalism modifies the relation between the four measurement outcomes by defining a lower bound of the fourth probability that increases as the total probability of the first three outcomes drops to zero. Quantum theory thus makes the violation of non-contextual consistency between the measurement outcomes not only possible, but actually requires it as a necessary consequence of the Hilbert space inner products that describe the contextual relation between the outcomes of different measurements.

Toys can’t play: physical agents in Spekkens’ theory

Information is physical (Landauer 1961 IBM J. Res. Dev. 5 183–91), and for a physical theory to be universal, it should model observers as physical systems, with concrete memories where they store the information acquired through experiments and reasoning. Here we address these issues in Spekkens’ toy theory (Spekkens 2005 Phys. Rev. A 71 052108), a non-contextual epistemically restricted model that partially mimics the behaviour of quantum mechanics. We propose a way to model physical implementations of agents, memories, measurements, conditional actions and information processing. We find that the actions of toy agents are severely limited: although there are non-orthogonal states in the theory, there is no way for physical agents to consciously prepare them. Their memories are also constrained: agents cannot forget in which of two arbitrary states a system is. Finally, we formalize the process of making inferences about other agents’ experiments and model multi-agent experiments like Wigner’s friend. Unlike quantum theory (Nurgalieva and del Rio Lidia 2019 Electron. Proc. Theor. Comput. Sci. 287 267–97; Fraser et al 2020 Fitch’s knowability axioms are incompatible with quantum theory arXiv:2009.00321; Frauchiger and Renner 2018 Nat. Commun. 9 3711; Nurgalieva and Renner 2021 Contemp. Phys. 61 1–24; Brukner 2018 Entropy 20 350) or box world (Vilasini et al 2019 New J. Phys. 21 113028), in toy theory there are no inconsistencies when physical agents reason about each other’s knowledge.

Plug-and-play discrete modulation continuous variable quantum key distribution based on non-Gaussian state-discrimination detection

Plug-and-play discrete modulation continuous variable quantum key distribution can generate local oscillator light locally without using two independent lasers, and both signal light and local oscillator are generated from the same laser, which can effectively ensure the practical security of the system and have a completely identical frequency characteristic. In addition, this scheme has good compatibility with efficient error correction codes, and can achieve high reconciliation efficiency even at low signal-to-noise ratio. However, there exists large excess noise in the plug-and-play configuration based on the untrusted source model, which seriously limits the maximum transmission distance of the discrete modulation scheme. To solve this problem, we propose a plug-and-play discrete modulation continuous variable quantum key distribution based on non-Gaussian state-discrimination detection. That is to say, a non-Gaussian state-discrimination detector is deployed at the receiver. With adaptive measurement method and Bayesian inference, four non-orthogonal coherent states which are based on four-state discrete modulation can be unconditionally distinguished on condition that the error probability is lower than the standard quantum limit. We analyze the security of the proposed protocol by considering both asymptotic limit and finite-size effect. Simulation results show that the secret key rate and maximum transmission distance are significantly enhanced by using no-Gaussian state-discrimination detection even under the influence of the untrusted source noise compared with the original plug-and-play discrete modulation continuous variable quantum key distribution. These results indicate that the proposed scheme can effectively reduce the negative influence of the untrust source noise on the performance of the plug-and-play discrete modulation continuous variable quantum key distribution protocol. The proposed protocol can not only ensure the practical security of the system, but also achieve more efficient and longer transmission distance quantum key distribution.

Four-channel metasurface for multiplexing images under two nonorthogonal polarization states

Four-channel metasurface for multiplexing images under two nonorthogonal polarization states