arXiv subjects

Alberto Montina

Publications and source records attributed to Alberto Montina.

At least 19 recordsLinked to original sources

arXiv2025

Local theories with parallel realities and the epistemic view of the quantum state

Hidden-variable theories effectively solve the measurement problem. However, a serious issue of this route towards a realistic completion of quantum theory is raised by Bell's proof that the resulting theories are nonlocal. A possible resolution is to reject the assumption that measurements have single actual outcomes. Indeed, relaxing this premise, Deutsch and Hayden showed that Bell's theorem can be evaded by delaying the buildup of the correlations until the parties compare their outcomes at a meeting point. However, the Deutsch-Hayden theory, which is deterministic and psi-ontic, leads to an infinite information flow towards the meeting point. Furthermore, alternative branches are weighted by amplitudes, leading to interpretative issues. In this paper, we introduce a general framework that combines the randomness of single-world theories with the coexistence of diverse instances, as found in many-worlds theory. This framework incorporates the existing theories as limiting cases. We explore how this hybrid approach addresses key challenges of single-world and Deutsch-Hayden theories. On one hand, the multiplicity of coexisting instances allows us to circumvent nonlocality and, possibly, contextuality. On the other hand, randomness makes it possible to derive quantum probabilities from unweighted counts of instances and ensemble averages. Furthermore, it can lead to a reduction of the information flow. We illustrate this framework with a local model for two spatially separate maximally entangled qubits. The model requires two unweighted instances and a finite information flow -- just one bit per measurement is communicated to the meeting point. Setting aside its foundational motivations, this framework has also relevance in quantum communication complexity and leads to novel technical questions, potentially providing new insights into some peculiarities of entanglement.

quant-ph↗

arXiv2023

Quantum-foundational implications of information erasure upon measurement

A projective measurement cannot decrease the von Neumann entropy if the outcome is ignored. However, under certain sound assumptions and using the quantum violation of Leggett-Garg inequalities, we have previously demonstrated that this property is not inherited by a classical simulation of such a measurement process. In the simulation, a measurement erases prior information by partially resetting the system, suggesting that the quantum-state update following a measurement cannot be entirely epistemic. The erasure of information has been proved by assuming that the maximally mixed quantum state corresponds to maximal ignorance of the classical state. A more intricate proof employed the weaker hypothesis that the entropy is finite at some stage of the simulation. In this paper, we focus on the quantum-foundational implications of this theorem. We first provide a simple proof by directly using the second hypothesis. Second, we identify information erasure as the mechanism breaking the time symmetry in ontological theories. This symmetry break has been previously proved by Pusey and Leifer. Third, we show that information erasure and, thus, symmetry break can be avoided by employing a branching a la many-worlds theory. The information flow and the time asymmetry are transferred to the measurement devices and the subsequent comparison of results, which inherently involve time-asymmetric processes. Thus, causality and the absence of information erasure suggest that measurements have multiple actual outcomes. Similarly, Deutsch and Hayden argued that Bell's theorem leads to the same conclusion if locality is given for granted. We conclude by showing that the problem of the clumsiness loophole in an experimental Leggett-Garg test of macrorealism is mitigated by the information-erasure theorem.

quant-ph↗

arXiv2022

An Algebraic-Geometry Approach to Prime Factorization

New algorithms for prime factorization that outperform the existing ones or take advantage of particular properties of the prime factors can have a practical impact on present implementations of cryptographic algorithms that rely on the complexity of factorization. Currently used keys are chosen on the basis of the present algorithmic knowledge and, thus, can potentially be subject to future breaches. For this reason, it is worth to investigate new approaches which have the potentiality of giving a computational advantage. The problem has also relevance in quantum computation, as an efficient quantum algorithm for prime factorization already exists. Thus, better classical asymptotic complexity can provide a better understanding of the advantages offered by quantum computers. In this paper, we reduce the factorization problem to the search of points of parametrizable varieties, in particular curves, over finite fields. The varieties are required to have an arbitrarily large number of intersection points with some hypersurface over the base field. For a subexponential or poly- nomial factoring complexity, the number of parameters have to scale sublinearly in the space dimension n and the complexity of computing a point given the parameters has to be subexponential or polynomial, respectively. We outline a procedure for building these varieties, which is illustrated with two constructions. In one case, we show that there are varieties whose points can be evaluated efficiently given a number of parameters not greater than n/2. In the other case, the bound is dropped to n/3. Incidentally, the first construction resembles a kind of retro-causal model. Retro-causality is considered one possible explanation of quantum weirdness.

cs.CR↗

arXiv2016

Can non-local correlations be discriminated in polynomial time?

In view of the importance of quantum non-locality in cryptography, quantum computation and communication complexity, it is crucial to decide whether a given correlation exhibits non-locality or not. In the light of a theorem by Pitowski, it is generally believed that this problem is computationally intractable. In this paper, we first prove that the Euclidean distance of given correlations from the local polytope can be computed in polynomial time with arbitrary fixed error, granted the access to a certain oracle. Namely, given a fixed error, we derive two upper bounds on the running time. The first bound is linear in the number of measurements. The second bound scales as the number of measurements to the sixth power. The former is dominant only for a very high number of measurements and is never saturated in the performed numerical tests. We then introduce a simple algorithm for simulating the oracle. In all the considered numerical tests, the simulation of the oracle contributes with a multiplicative factor to the overall running time and, thus, does not affect the sixth-power law of the oracle-assisted algorithm.

quant-ph↗

arXiv2016

Optimal measurements for nonlocal correlations

A problem in quantum information theory is to find the experimental setup that maximizes the nonlocality of correlations with respect to some suitable measure such as the violation of Bell inequalities. The latter has however some drawbacks. First and foremost it is unfeasible to determine the whole set of Bell inequalities already for a few measurements and thus unfeasible to find the experimental setup maximizing their violation. Second, the Bell violation suffers from an ambiguity stemming from the choice of the normalization of the Bell coefficients. An alternative measure of nonlocality with a direct information-theoretic interpretation is the minimal amount of classical communication required for simulating nonlocal correlations. In the case of many instances simulated in parallel, the minimal communication cost per instance is called nonlocal capacity, and its computation can be reduced to a convex-optimization problem. This quantity can be computed for a higher number of measurements and turns out to be useful for finding the optimal experimental setup. Focusing on the bipartite case, in this paper, we present a simple method for maximizing the nonlocal capacity over a given configuration space and, in particular, over a set of possible measurements, yielding the corresponding optimal setup. Furthermore, we show that there is a functional relationship between Bell violation and nonlocal capacity. The method is illustrated with numerical tests and compared with the maximization of the violation of CGLMP-type Bell inequalities on the basis of entangled two-qubit as well as two-qutrit states. Remarkably, the anomaly of nonlocality displayed by qutrits turns out to be even stronger if the nonlocal capacity is employed as a measure of nonlocality.

quant-ph↗

arXiv2016

Information-based measure of nonlocality

Quantum nonlocality concerns correlations among spatially separated systems that cannot be classically explained without post-measurement communication among the parties. Thus, a natural measure of nonlocal correlations is provided by the minimal amount of communication required for classically simulating them. In this paper, we present a method to compute the minimal communication cost, which we call nonlocal capacity, for any general nonsignaling correlations. This measure turns out to have an important role in communication complexity and can be used to discriminate between local and nonlocal correlations, as an alternative to the violation of Bell's inequalities.

quant-ph↗

arXiv2015

Communication complexity and the reality of the wave-function

In this review, we discuss a relation between quantum communication complexity and a long-standing debate in quantum foundation concerning the interpretation of the quantum state. Is the quantum state a physical element of reality as originally interpreted by Schrodinger? Or is it an abstract mathematical object containing statistical information about the outcome of measurements as interpreted by Born? Although these questions sound philosophical and pointless, they can be made precise in the framework of what we call classical theories of quantum processes, which are a reword of quantum phenomena in the language of classical probability theory. In 2012, Pusey, Barrett and Rudolph (PBR) proved, under an assumption of preparation independence, a theorem supporting the original interpretation of Schrodinger in the classical framework. Recently, we showed that these questions are related to a practical problem in quantum communication complexity, namely, quantifying the minimal amount of classical communication required in the classical simulation of a two-party quantum communication process. In particular, we argued that the statement of the PBR theorem can be proved if the classical communication cost of simulating the communication of n qubits grows more than exponentially in 'n'. Our argument is based on an assumption that we call probability equipartition property. This property is somehow weaker than the preparation independence property used in the PBR theorem, as the former can be justified by the latter and the asymptotic equipartition property of independent stochastic sources. The equipartition property is a general and natural hypothesis that can be assumed even if the preparation independence hypothesis is dropped. In this review, we further develop our argument into the form of a theorem.

quant-ph↗

arXiv2014

Necessary and sufficient optimality conditions for classical simulations of quantum communication processes

We consider the process consisting of preparation, transmission through a quantum channel, and subsequent measurement of quantum states. The communication complexity of the channel is the minimal amount of classical communication required for classically simulating it. Recently, we reduced the computation of this quantity to a convex minimization problem with linear constraints. Every solution of the constraints provides an upper bound on the communication complexity. In this paper, we derive the dual maximization problem of the original one. The feasible points of the dual constraints, which are inequalities, give lower bounds on the communication complexity, as illustrated with an example. The optimal values of the two problems turn out to be equal (zero duality gap). By this property, we provide necessary and sufficient conditions for optimality in terms of a set of equalities and inequalities. We use these conditions and two reasonable but unproven hypotheses to derive the lower bound $n 2^{n-1}$ for a noiseless quantum channel with capacity equal to $n$ qubits. This lower bound can have interesting consequences in the context of the recent debate on the reality of the quantum state.

quant-ph↗

arXiv2014

Lower bounds on the communication complexity of two-party (quantum) processes

The process of state preparation, its transmission and subsequent measurement can be classically simulated through the communication of some amount of classical information. Recently, we proved that the minimal communication cost is the minimum of a convex functional over a space of suitable probability distributions. It is now proved that this optimization problem is the dual of a geometric programming maximization problem, which displays some appealing properties. First, the number of variables grows linearly with the input size. Second, the objective function is linear in the input parameters and the variables. Finally, the constraints do not depend on the input parameters. These properties imply that, once a feasible point is found, the computation of a lower bound on the communication cost in any two-party process is linearly complex. The studied scenario goes beyond quantum processes and includes the communication complexity scenario introduced by Yao. We illustrate the method by analytically deriving some non-trivial lower bounds. Finally, we conjecture the lower bound $n 2^n$ for a noiseless quantum channel with capacity $n$ qubits. This bound can have an interesting consequence in the context of the recent quantum-foundational debate on the reality of the quantum state.

quant-ph↗

arXiv2013

Exponential communication gap between weak and strong classical simulations of quantum communication

The most trivial way to simulate classically the communication of a quantum state is to transmit the classical description of the quantum state itself. However, this requires an infinite amount of classical communication if the simulation is exact. A more intriguing and potentially less demanding strategy would encode the full information about the quantum state into the probability distribution of the communicated variables, so that this information is never sent in each single shot. This kind of simulation is called weak, as opposed to strong simulations, where the quantum state is communicated in individual shots. In this paper, we introduce a bounded-error weak protocol for simulating the communication of an arbitrary number of qubits and a subsequent two-outcome measurement consisting of an arbitrary pure state projector and its complement. This protocol requires an amount of classical communication independent of the number of qubits and proportional to Delta^{-1}, where Delta is the error and a free parameter of the protocol. Conversely, a bounded-error strong protocol requires an amount of classical communication growing exponentially with the number of qubits for a fixed error. Our result improves a previous protocol, based on the Johnson-Lindenstrauss lemma, with communication cost scaling as Delta^{-2} log Delta^{-1}.

quant-ph↗

arXiv2012

Epistemic view of quantum states and communication complexity of quantum channels

The communication complexity of a quantum channel is the minimal amount of classical communication required for classically simulating a process of state preparation, transmission through the channel and subsequent measurement. It establishes a limit on the power of quantum communication in terms of classical resources. We show that classical simulations employing a finite amount of communication can be derived from a special class of hidden variable theories where quantum states represent statistical knowledge about the classical state and not an element of reality. This special class has attracted strong interest very recently. The communication cost of each derived simulation is given by the mutual information between the quantum state and the classical state of the parent hidden variable theory. Finally, we find that the communication complexity for single qubits is smaller than 1.28 bits. The previous known upper bound was 1.85 bits.

quant-ph↗

arXiv2012

Dynamics of a qubit as a classical stochastic process with time-correlated noise: minimal measurement invasiveness

So far it has been shown that the quantum dynamics cannot be described as a classical Markov process unless the number of classical states is uncountably infinite. In this paper, we present a stochastic model with time-correlated noise that exactly reproduces any unitary evolution of a qubit and requires just four classical states. The invasive updating of just one bit during a measurement accounts for the quantum violation of the Leggett-Garg inequalities. Unlike in a pilot wave theory, the stochastic forces governing the jumps among the four states do not depend on the quantum state, but only on the unitary evolution. This model is used to derive a local hidden variable model, augmented by one bit of classical communication, for simulating entangled Bell states.

quant-ph↗

arXiv2011

Communication cost of classically simulating a quantum channel with subsequent rank-1 projective measurement

A process of preparation, transmission and subsequent projective measurement of a qubit can be simulated by a classical model with only two bits of communication and some amount of shared randomness. However no model for n qubits with a finite amount of classical communication is known at present. A lower bound for the communication cost can provide useful hints for a generalization. It is known for example that the amount of communication must be greater than c 2^n, where c~0.01. The proof uses a quite elaborate theorem of communication complexity. Using a mathematical conjecture known as the "double cap conjecture", we strengthen this result by presenting a geometrical and extremely simple derivation of the lower bound 2^n-1. Only rank-1 projective measurements are involved in the derivation.

quant-ph↗

arXiv2011

Approximate simulation of entanglement with a linear cost of communication

Bell's theorem implies that the outcomes of local measurements on two maximally entangled systems cannot be simulated without classical communication between the parties. The communication cost is finite for n Bell states, but it grows exponentially in n. Three simple protocols are presented that provide approximate simulations for low-dimensional entangled systems and require a linearly growing amount of communication. We have tested them by performing some simulations for a family of measurements. The maximal error is less than 1% in three dimensions and grows sublinearly with the number of entangled bits in the range numerically tested. One protocol is the multidimensional generalization of the exact Toner-Bacon [Phys. Rev. Lett. 91, 187904 (2003)] model for a single Bell state. The other two protocols are generalizations of an alternative exact model, which we derive from the Kochen-Specker [J. Math. Mech. 17, 59 (1967)] scheme for simulating single-qubit measurements. These protocols can give some indication for finding optimal one-way communication protocols that classically simulate entanglement and quantum channels. Furthermore they can be useful for deciding if a quantum communication protocol provides an advantage on classical protocols.

quant-ph↗

arXiv2011

Impurity Scattering in a Bose-Einstein Condensate at finite temperature

We consider the effects of finite temperature on the scattering of impurity atoms in a BoseEinstein condensate, showing that the scattering rate is enhanced by the thermal atoms. Collisions can increase or decrease the impurity energy. Below the Landau velocity only the first process occurs, i.e., the collisions cool the condensate. Above the critical velocity the dissipative collisions prevail over the cooling ones for sufficiently low temperatures. These considerations are applied to a recent experiment.

cond-mat.quant-gas↗

arXiv2011

Measurement contextuality is implied by macroscopic realism

Ontological theories of quantum mechanics provide a realistic description of single systems by means of well-defined quantities conditioning the measurement outcomes. In order to be complete, they should also fulfil the minimal condition of macroscopic realism. Under the assumption of outcome determinism and for Hilbert space dimension greater than two, they were all proved to be contextual for projective measurements. In the recent years a generalized concept of non-contextuality was introduced that applies also to the case of outcome indeterminism and unsharp measurements. It was pointed out that the Beltrametti-Bugajski model is an example of measurement non-contextual indeterminist theory. Here we provide a simple proof that this model is the only one with such a feature for projective measurements and Hilbert space dimension greater than two. As a corollary, von Neumann measurement non-contextuality implies non-contextuality for unsharp measurements. By noting that the Beltrametti-Bugajski model does not satisfy the condition of macroscopic realism, we arrive at the conclusion that the only way to solve the measurement problem in the framework of an ontological theory is relaxing the hypothesis of non-contextuality in its generalized sense.

quant-ph↗

arXiv2011

Compressing the hidden variable space of a qubit

In previously exhibited hidden variable models of quantum state preparation and measurement, the number of continuous hidden variables describing the actual state of a single realization is never smaller than the quantum state manifold dimension. We introduce a simple model for a qubit whose hidden variable space is one-dimensional, i.e., smaller than the two-dimensional Bloch sphere. The hidden variable probability distributions associated with the quantum states satisfy reasonable criteria of regularity. Possible generalizations of this shrinking to a N-dimensional Hilbert space are discussed.

quant-ph↗

arXiv2010

State space dimensionality in short memory hidden variable theories

Recently we have presented a hidden variable model of measurements for a qubit where the hidden variable state space dimension is one-half the quantum state manifold dimension. The absence of a short memory (Markov) dynamics is the price paid for this dimensional reduction. The conflict between having the Markov property and achieving the dimensional reduction was proved in [A. Montina, Phys. Rev. A, 77, 022104 (2008)] using an additional hypothesis of trajectory relaxation. Here we analyze in more detail this hypothesis introducing the concept of invertible process and report a proof that makes clearer the role played by the topology of the hidden variable space. This is accomplished by requiring suitable properties of regularity of the conditional probability governing the dynamics. In the case of minimal dimension the set of continuous hidden variables is identified with an object living an N-dimensional Hilbert space, whose dynamics is described by the Schrödinger equation. A method for generating the economical non-Markovian model for the qubit is also presented.

quant-ph↗