Session Y53: Quantum Foundations and Information I

Forward-backward simulations of Einstein-Podolsky-Rosen correlations and Bell nonlocality reveal hidden causal loops

A quantum measurement of x is modelled as amplification.  We solve for the measurement dynamics in terms of stochastic phase space variables based on the Q function. The solutions lead to amplitudes x and p that propagate backward and forward in the time direction, respectively, with future and past boundary conditions. The trajectories are simulated, the backward-propagating variable requiring a future noise input at the vacuum level. We prove the equivalence between the joint density of amplitudes and the Q function. Causal relations are inferred from the simulations. For superpositions of eigenstates of x, we find that the backward and forward-propagating trajectories are connected by a conditional boundary condition at the initial time, which creates the origin of a causal loop for variables that can be shown to be “hidden” (i.e. not observable). For mixtures, this connection is lost. The simulations reveal a causal structure consistent with macroscopic realism, despite the inherent retrocausality. We simulate Einstein-Podolsky-Rosen entanglement and Bell nonlocality for continuous-variable measurements, revealing similar hidden causal loops. The simulations suggest that a weak form of local realism defined for systems after the measurement settings are fixed is valid, and that Bell nonlocality emerges as a breakdown of a subset of Bell’s local-realism assumptions.

 

 

    

Classifying Causal Structures: Ascertaining when Classical Correlations are Constrained by Inequalities

The classical causal relations between a set of variables, some observed and somelatent, caninduce both equality constraints (typically conditionalindependencies) as well as inequality constraints (Instrumental and Bell inequalities being prototypical examples) on their compatible distribution over the observed variables. Enumerating a causal structure's implied inequality constraints is generally far more difficult than enumerating its equalities. Furthermore, only inequality constraints ever admit violation by quantum correlations. For both those reasons, it is important to classify causal scenarios into those which impose inequality constraints versus those which do not. Here we develop methods for detecting such scenarios by appealing to d-separation, e-separation, and incompatible supports. Many (perhaps all?) scenarios with exclusively equality constraints can be detected via a condition articulated by Henson, Lal and Pusey (HLP). Considering all scenarios with up to 4 observed variables, which number in the thousands, we are able to resolve all but three causal scenarios, providing evidence that the HLP condition is, in fact, exhaustive.   

Error and Disturbance as Irreversibility with Applications: Unified Definition, Wigner--Araki--Yanase Theorem and Out-of-Time-Order Correlator

Error and disturbance are fundamental concepts in quantum measurements; therefore the question of how to formulate these two concepts has come to be considered so far. However, we still do not have the ultimate answer, hopefully unifying the existing definitions, to the question of what the error and disturbance are. Here we show that error and disturbance can be defined as special cases of the irreversibility of quantum processes. In our formulation, by applying a channel conversion---quantum comb in quantum information theory---to the measuring process, we convert its error and disturbance into the irreversibility of time evolution of an ancillary qubit system. The re-definitions provide fruitful byproducts: First, we can unify the existing definitions of error and disturbance as special aspects of irreversibility. Second, we extend the quantitative Wigner--Araki--Yanase theorem---a universal restriction on measurement implementation under a conservation law---to error and disturbance of arbitrary definitions. Third, we provide a novel treatment of out-of-time-orderd-correlator (OTOC)---a measure of quantum chaos in a quantum many-body system---as irreversibility, and its experimental evaluation method.   

Clifford Group and Unitary Designs in the Presence of Symmetry

In this talk, we present the symmetric generalization of the well-known statement that the Clifford group is a unitary 3-design, with the symmetric extension of unitary designs. Concretely, we show that a symmetric Clifford group is a symmetric unitary 3-design, if and only if the symmetry constraint is essentially given by some Pauli subgroup. Moreover, we also present a complete and unique construction method of symmetric Clifford groups with simple quantum gates for Pauli symmetries. For a comprehensive understanding, we also take physically relevant U(1) and SU(2) symmetries as examples of non-Pauli symmetries, and show that the symmetric Clifford groups are only symmetric unitary 1-designs and not 2-designs under those symmetries. Finally, for the verification of our results, we present the numerical results about the frame potentials, which measure the difference with respect to randomness between the uniform ensemble over a symmetric unitary group and its subgroup. This work is expected to open up a new perspective into quantum information processing techniques such as randomized benchmarking, and to provide a deep understanding to many-body systems such as monitored random circuits.   

Random unitaries with SU(d) Symmetry: OTOCs and Covariant Codes

Quantum information processing in continuous symmetry is of wide importance and exhibits many novel physical and mathematical phenomena. Continuous symmetry is always associated with some conserved quantity from Noether's theorem on which much modern understanding of physics is built. In recent years, continuous symmetries have drawn considerable interest in quantum information, ranging from quantum hydrodynamics, quantum error correction, quantum communication, and quantum machine learning. Central to many applications is the ability that local interactions can still lead to sufficient randomness in the presence of continuous symmetry, which is characterized by the notion of Harr randomness in the late-time using tools such as the out-of-time-ordered correlator (OTOC) and entanglement. For instance, the late-time OTOC value with respect to a single site for Haar random circuits is isotropic and decrease exponentially in finite-size circuit, which can be understood classically. Surprisingly, charge conservation on random circuits could drastically change the circuit behavior and often renders power-law decay in OTOCs in the finite-size circuit, and an outstanding question is the behavior of random circuits under non-Abelian charges—for which SU(d) symmetry provides a canonical example. We provide the first step towards this direction by showing the under SU(d) conservation law, the finite-size OTOC with respect to a given site decays in on qubits. Based on this result, we further explore applications of random SU(d)-symmetric circuits in approximate covariant quantum error correction, where we show that random SU(d)-symmetric unitaries with constant encoding are nearly optimal covariant codes, saturating the fundamental limit imposed by the approximate Eastin-Knill theorem. Our work invites further research on quantum information with continuous symmetries, where the mathematical tools developed in this work are expected to be useful.   

Information Dynamics in Random Quantum Circuits

Determining where the information flows is of fundamental importance in understanding quantum many-body physics. In chaotic systems, information spreads into entangled form, showing non-trivial dynamics, including the light-cone structure and the scrambling phenomenon. The detailed process between these two cases is less well-understood and involves rich phenomena. In this talk, I will discuss our two works on information dynamics in random quantum circuits, one of the minimally structured models that capture generic chaotic behavior. In our first work, we adopt infinitely long circuits and discover phase transitions in the information retrieved from different final probe regions. In the other one, we study the full-time dynamics of information and discover a rich set of dynamical phase transitions in the information flow. Such dynamical phase transitions vary with the initial position where the information is located and the final probe region, showing ubiquitous behaviors. These results will help us better understand the information propagation and scrambling in this quantum many-body model.   

Extracting Randomness from "Magic" States

Magic is a fundamental description of the non-classicality of quantum states and plays a crucial role in fault-tolerant quantum computation. Many constructions in quantum information science rely on ensembles of random unitaries or states, prompting intense study on the topic of randomness within the context of the field. In this presentation, we show that there is a direct mathematical relationship between the magic of a multiqubit pure state and the randomness of the post-measurement ensemble obtained from measuring out subsystems of this state. Here, we quantify randomness using the framework of approximate state 2-designs and present compelling numerical evidence that our result extends to higher-order approximate state designs as well. To illustrate the relationship between magic and randomness, we numerically simulate subsystem measurements on the output of a random Clifford circuit initialized to a product state and show how the randomness of the resulting ensemble depends on the magic of the circuit's initial state. Our results further demonstrate how magic states act as a quantum resource, and suggest a practical method for leveraging magic to generate approximate state designs.   

Local Weak Measurements as a Probe of the Spectrum of Quantum Magnets

Weak measurements are a powerful but under-explored tool in physics. In essence, weak measurements indirectly probe a system by way of an ancillary degree of freedom in order to avoid total collapse of the measured state at the cost of gaining less information about it. Their unique ability to preserve a degree of coherence while extracting information from a system in a tunable tradeoff opens up possibilities for a vast array of interesting ways to interact with and study quantum systems. In this work, we explore how sequences of local weak measurements can allow us to induce and then simultaneously detect excitations in quantum magnets. In particular, we examine how repeated weak measurements of a spin on the end of a Heisenberg chain give rise to oscillations connected to features of the spectrum that depend on the entire chain. We will discuss the origin and symmetry considerations of these oscillations, and comment on how this and similar measurement schemes could be practically useful.   

Two-Source Topological Phases and A New Method for Testing Bell-CHSH Inequalities

Quantum topological phases arise under the influence of vector potential or vector potential-like physical quantities that enter as a complex phase factor into the wave functions of particles moving along closed trajectories around singularities created by using electric or magnetic field sources, without the effects of any classical force. The well-known examples of the topological phases are the Aharonov-Bohm (AB) and Aharonov-Casher (AC) physical processes. The effects of topological phases on entangled quantum mechanical systems are of great importance for revealing the non-local properties of quantum mechanics. In this regard, the joint probabilities of particle arrival at each of two spatially separated detectors for momentum-correlated states can be calculated by considering two-source AB and AC systems in EPR-Bohm type hybrid gedanken experimental setups. From this point of view, in this work, the effect of topological phases on quantum entangled states will be investigated and it will be shown that the system to be designed using beam splitters and phase retarders can be used as an analog for joint spin measurements within the framework of entirely quantum mechanics. As a result, this setup will be proposed as a new experimental method for testing the Bell-CHSH inequalities.   

Entanglement Entropy of Interacting Fermions from Correlation functions

Entanglement measures such as entanglement entropy (EE) exhibit characteristic scaling behaviour with subsystem size in a variety of novel quantum states. However, analytical methods to calculate EE have been limited to non-interacting theories, or theories with conformal symmetry in one spatial dimension. Numerical methods applicable to more generic interacting systems can access small sizes limited by the exponentially growing computational complexity. Adapting recent Wigner-characteristic based techniques, we show that Renyi EE of interacting fermions in arbitrary dimensions can be represented as a Schwinger-Keldysh free energy on replicated manifolds with a current between the replicas. The current is local in real space and present only in the subsystem of interest. This lets us construct a diagrammatic representation of EE in terms of connected correlators in the standard unreplicated field theory. We further decompose EE into "particle" contributions which depend on the one-particle correlator, two-particle connected correlator and so on. For repulsively interacting fermions in two and three dimensions, we find the one particle contribution to entanglement picks up a leading volume scaling which is entirely determined by the incoherent piece of the one-particle momentum distribution function. The coefficient of the now subleading log-enhanced area piece is seen to decrease with increasing interaction strength.   

Separability criterion using one observable for special states: Entanglement detection via quantum quench

Detecting entanglement in many-body quantum systems is crucial but challenging, typically requiring multiple measurements. Here, we establish the class of states where measuring connected correlations in just basis is sufficient and necessary to detect bipartite separability, provided the appropriate basis and observables are chosen. This methodology leverages prior information about the state, which, although insufficient to reveal the complete state or its entanglement, enables our one basis approach to be effective. We discuss the possibility of one observable entanglement detection in a variety of systems, including those without conserved charges, such as the Transverse Ising model, reaching the appropriate basis via quantum quench. This provides a much simpler pathway of detection than previous works. It also shows improved sensitivity from Pearson Correlation detection techniques.   

Quantum Entanglement Phase Transitions and Computational Complexity: Insights from Ising Models

In this paper, we construct 2-dimensional bipartite cluster states and perform single-qubit measurements on the bulk qubits. We explore the entanglement scaling of the unmeasured 1-dimensional boundary state and show that under certain conditions, the boundary state can undergo a volume-law to an area-law entanglement transition driven by variations in the measurement angle. We bridge this boundary state entanglement transition and the measurement-induced phase transition in the non-unitary 1+1-dimensional circuit via the transfer matrix method. We also explore the application of this entanglement transition on the computational complexity problems. Specifically, we establish a relation between the boundary state entanglement transition and the sampling complexity of the bipartite 2d cluster state, which is directly related to the computational complexity of the corresponding Ising partition function with complex parameters. By examining the boundary state entanglement scaling, we numerically identify the parameter regime for which the 2d quantum state can be efficiently sampled, which indicates that the Ising partition function can be evaluated efficiently in such a region.   

Optimal lower bound of the average indeterminate length lossless block encoding

Consider a general quantum source that emits at discrete 

time steps quantum pure states which are chosen from a finite alphabet according to some probability distribution 

which may depend on the whole history.  Also, fix two 

positive integers $m$ and $l$.

We encode any

tensor product of $ml$ many states emitted by the quantum source by 

breaking it into $m$ many blocks where each block has 

length $l$, and considering 

sequences of $m$ many isometries so that each isometry

maps one of these blocks into the Fock space, followed by  

the concatenation of their images. We only consider certain 

sequences of such isometries that we call 

``special block codes"

in order to ensure that the concatenation is also an 

isometry, and hence the string of alphabet states is uniquely decodable. We compute the minimum average length

of these encodings which depends on the quantum source

and the integers $m$, $l$, among all possible special block codes.

Our result extends the result of

[Bellomo, Bosyk, Holik and Zozor,  Scientific Reports 7.1 (2017): 14765] where the minimum was computed for one block,

($m=1$).   

Boson-Assisted Quantum Error-Correcting Codes

Quantum error-correcting codes (QECC), which detect and correct errors that occur during quantum computations, are a crucial avenue for scaling up quantum computing. Originally, these codes encoded information using spin-½ physical qubits, but they were later extended to apply to bosonic systems as well [1-2]. In this study, we present a hybrid QECC that leverages both spin and bosonic degrees of freedom in systems combining both spins and bosons. We investigate the application of these codes in trapped-ion quantum processors and discuss their fault-tolerant implementation. 

[1] R. Calderbank and P. Shor (1996). "Good quantum error-correcting codes exist". Physical Review A. 54 (2): 1098-1105; P. W. Shor, "Fault-tolerant quantum computation," Proceedings of 37th Conference on Foundations of Computer Science, Burlington, VT, USA, 1996, pp. 56-65.

[2] D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator”, Physical Review A 64, (2001).