Quantum Simulation of the Majorana Equation and Unphysical Operations
Open accessAbstract
Charge conjugation, complex conjugation, and time reversal of wave functions are theoretical concepts fundamental to quantum mechanics, but do not correspond to physical processes in the real world. Simulating them in a real physical setting seems then out of the question. A Spanish-Austrian team come up with a new toolbox for rendering the seemingly impossible possible.
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Written from the paper’s full text.
TL;DR
The paper proposes an analog quantum-simulation scheme for the non-Hamiltonian Majorana equation by mapping complex conjugation and other unphysical operations onto unitary operations in an enlarged Hilbert space. It outlines a trapped-ion implementation using two ions and gives measurement protocols for observables such as pseudohelicity.
Key findings
- Complex conjugation, charge conjugation, and time reversal can be implemented as unitary operations on an enlarged real Hilbert space via the mapping \( \psi \in \mathbb{C}^n \to \Xi \in \mathbb{R}^{2n} \), using an auxiliary two-level system.
- In \(1+1\) dimensions, the non-Hermitian Majorana equation for a complex spinor becomes a \(3+1\)-dimensional Dirac equation for a four-component real bispinor, Eq. (5).
- The proposed trapped-ion implementation uses two trapped ions and laser couplings to simulate the \(1+1\)-dimensional Majorana dynamics; coupled Majorana neutrinos are suggested with three ions and two vibrational modes.
- Numerical scattering examples use parameters \(m=0.5\), \(c=1\), and \(V(x)=x\) in dimensionless units, showing time reversal refocusing, charge conjugation converting a repulsive potential to attractive, and a Majorana wave packet penetrating a barrier.
Methods
- Theoretical design: introduces a complex-to-real mapping that turns antiunitary operations into local unitaries on an ancillary qubit, then applies it to the Majorana equation and discrete symmetries \(C\), \(T\), and \(K\).
- Trapped-ion scheme: derives an interaction-picture Hamiltonian for two ions with center-of-mass and stretch modes, specifying laser frequencies, phases, and Rabi frequencies in Eq. (8) to recover the simulated Majorana Hamiltonian.
- Measurement and numerics: gives a protocol to measure pseudohelicity via state-dependent displacement and spin measurements, and uses conventional numerical tools to study wave-packet scattering against a linearly growing potential.
Limitations
- The paper does not state explicit limitations.
- Evident from the text: the work is a theoretical proposal, not an experimental realization; the detailed implementation is presented for \(1+1\) dimensions with two trapped ions, while higher-dimensional and coupled-neutrino cases are only outlined.
- Evident from the text: the scattering examples are numerical, use dimensionless parameters, and are not experimental data.
Model-written summary. Check it against the paper before relying on it.
Integrity signals
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Cites
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Claims the paper makes
0 of 5 cited sources checked · 0 supported · 0 flagged
- Findingown data
A Majorana particle wave packet penetrates the barrier, showing a counterintuitive insensitivity to its presence.
“While there are no plane wave solutions in the Majorana equation, we can still see a wave packet penetrating the barrier, showing a counterintuitive insensitivity to the presence of it.”
Quote in the paperSubstantial claim - Findingreasoning
Charge conjugation is implemented in the enlarged space via the unitary operation V_C.
“we can deduce that charge conjugation is implemented in the enlarged space via the unitary operation V C ,”
Quote in the paperSubstantial claim - Findingown data
In the numerical experiment, a Dirac particle splits into a fraction that bounces back and a large antiparticle component that penetrates the barrier.
“This is shown in Fig. 2(a), where a Dirac particle splits into a fraction of a particle that bounces back, and a large antiparticle component that penetrates the barrier.”
Quote in the paperSubstantial claim - Findingreasoning
The non-Hermitian Majorana equation for a complex spinor becomes a 3+1-dimensional Dirac equation for a four-component real bispinor under the mapping.
“Surprisingly, the non-Hermitian Majorana equation for a complex spinor becomes a 3 þ 1-dimensional Dirac equation for a four-component real bispinor,”
Quote in the paperSubstantial claim - Findingcitation
The application of a state-dependent displacement operation on ion 2 followed by a measurement of σ_z^1 is equivalent to measuring the observable Σ.
“The application of a state-dependent displacement operation on ion 2, U 2 ¼ exp½Àikð1 y Þ p x =2, generated by resonant blue and red sidebands, followed by a measurement of 1 z , is equivalent to measuring the observable [13]”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [13] source
Not checked in the free run.
- Interpretationreasoning
Quantum simulations of unphysical operations can be extended to higher dimensions.
“As mentioned before, quantum simulations of unphysical operations can be straightforwardly extended to higher dimensions.”
Quote in the paperNot a substantial claim - Interpretationreasoning
A general method can be devised for implementing unphysical operations including charge conjugation, complex conjugation, and time reversal.
“Driven by the need of the simulation, we devise a general method for implementing a number of mathematical operations that are unphysical, including charge conjugation, complex conjugation, and time reversal.”
Quote in the paperSubstantial claim - Interpretationreasoning
The proposed simulator accommodates the antiunitary operations C = U_C K and T = U_T K.
“The proposed simulator also accommodates the antiunitary operations C ¼ U C K and T ¼ U T K.”
Quote in the paperSubstantial claim - Interpretationreasoning
The Majorana dynamics of Eq. (5) can be simulated with two trapped ions, with lasers coupling their internal states and motional degrees of freedom.
“We can simulate Eq. ( 5) with two trapped ions, with lasers coupling their internal states and motional degrees of freedom.”
Quote in the paperSubstantial claim - Interpretationreasoning
The second term in Eq. (10) can be measured by applying the operation U_1 and measuring the spin correlation σ_z σ_x.
“To measure the second term in Eq. ( 10), we apply the operation U 1 ¼ exp½Àikð x 1Þ p x =2, and measure the spin correlation z x .”
Quote in the paperSubstantial claim - Interpretationreasoning
Quantum dynamics that do not occur in the real space-time quantum world can be implemented.
“we show here the possibility of implementing quantum dynamics that do not occur in the real space-time quantum world.”
Quote in the paperSubstantial claim - Interpretationreasoning
Unitaries and observables can be mapped onto the real space preserving unitarity and Hermiticity.
“Furthermore, unitaries and observables can also be mapped onto the real space, O ! ¼ 1 O r À i y O i , where O r ¼ 1 2 ðO þ KOKÞ and O i ¼ À i 2 ðO À KOKÞ, preserving unitarity and Hermiticity.”
Quote in the paperSubstantial claim - Interpretationreasoning
The proposed ideas can accomplish the mid-simulation unphysical operation with a suitable implementation.
“Clearly, this task would be impossible with classical resources but could be accomplished with a suitable implementation of our proposed ideas.”
Quote in the paperNot a substantial claim - Interpretationown data
A quantum simulator for the Majorana equation can be designed.
“We design a quantum simulator for the Majorana equation, a non-Hamiltonian relativistic wave equation that might describe neutrinos and other exotic particles beyond the standard model.”
Quote in the paperSubstantial claim - Interpretationcitation
The Majorana equation can be simulated in an analog quantum simulator.
“In this paper, we show how the Majorana equation [17] can be simulated in an analog quantum simulator, having the implementation of complex conjugation of the wave function as a key requirement.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [17] source
Not checked in the free run.
- Interpretationreasoning
The first term in Eq. (10) can be measured by applying a short probe pulse to the ions and measuring the initial slope of the observable A(k).
“Therefore, the first term in (10) can be measured by applying a short probe pulse to the ions and measuring the initial slope of the observable AðkÞ.”
Quote in the paperSubstantial claim - Interpretationnone given
Implementing complex conjugation of the wave function is a key requirement for simulating the Majorana equation.
“having the implementation of complex conjugation of the wave function as a key requirement”
Quote in the paperSubstantial claim - Interpretationreasoning
The Majorana equation can be simulated in 1+1 dimensions using two trapped ions and their interactions with laser pulses.
“We show how to simulate the Majorana equation in 1 þ 1 dimensions and other unphysical operations using two trapped ions and their interactions with laser pulses.”
Quote in the paperSubstantial claim - Interpretationcitation
Complex conjugation, an unphysical operation, can be mapped to a unitary operation acting on an enlarged Hilbert space.
“Here, we introduce a mapping [25] by which complex conjugation, an unphysical operation, becomes a unitary operation acting on an enlarged Hilbert space.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [25]
Not checked in the free run.
- Interpretationreasoning
To obtain the pseudohelicity Σ, one has to measure in the enlarged simulation space.
“Therefore, to obtain the pseudohelicity Æ, we have to measure in the enlarged simulation space.”
Quote in the paperSubstantial claim - Interpretationreasoning
The paper has presented a complete toolbox for implementing unphysical operations C, T, and K.
“So far, we have presented a complete toolbox for implementing unphysical operations, C, T , and K.”
Quote in the paperSubstantial claim - Interpretationnone given
There is no apparent fundamental restriction for implementing parity, charge conjugation, and time reversal in physical systems that simulate quantum mechanics.
“However, there is no apparent fundamental restriction for implementing them in physical systems that simulate quantum mechanics.”
Quote in the paperNot a substantial claim - Interpretationreasoning
The paper introduced a general method to implement quantum simulations of unphysical operations and non-Hamiltonian dynamics, such as the Majorana equation, in a Hamiltonian system.
“In summary, we have introduced a general method to implement quantum simulations of unphysical operations and non-Hamiltonian dynamics, such as the Majorana equation, in a Hamiltonian system.”
Quote in the paperSubstantial claim - Interpretationnone given
The general method for implementing unphysical operations can be realized in a system of trapped ions.
“Furthermore, we describe how to realize the general method in a system of trapped ions.”
Quote in the paperSubstantial claim - Interpretationcitation
In general, the complex-to-real map in arbitrary space-time dimensions transforms a Majorana equation into a higher dimensional Dirac equation.
“In general, the complex-to-real map in arbitrary space-time dimensions transforms a Majorana equation into a higher dimensional Dirac equation [27].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [27]
Not checked in the free run.
- Interpretationreasoning
It is feasible to have CP-violating phases in the Dirac mass term m_D exp(iγ_5).
“It also becomes feasible to have CP-violating phases in the Dirac mass term m D expði 5 Þ.”
Quote in the paperSubstantial claim - Interpretationreasoning
Since Eq. (5) is a Hamiltonian equation, it can be simulated in a quantum system.
“Since Eq. ( 5) is a Hamiltonian equation, it can be simulated in a quantum system.”
Quote in the paperSubstantial claim - Interpretationreasoning
The proposed protocol for implementing unphysical operations within a physical setup allows dealing with situations that are otherwise intractable with conventional quantum simulations.
“The proposed protocol for implementing unphysical operations within a physical setup allows us to deal with situations that are, otherwise, intractable with conventional quantum simulations.”
Quote in the paperSubstantial claim - Interpretationreasoning
In trapped-ion experiments, laser pulses can map information about the pseudohelicity onto the internal states.
“In the type of trapped-ion experiments discussed above, we can use laser pulses to map information about the pseudohelicity onto the internal states.”
Quote in the paperSubstantial claim - Interpretationcitation
The mapping of the quantum states of an n-dimensional complex Hilbert space onto a real Hilbert space R^2n can be implemented by means of an auxiliary two-level system.
“This mapping can be implemented by means of an auxiliary two-level system, such that R 2n 2 H 2 H n .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [25]
Not checked in the free run.
- May combine two claims. 7
- Classified as probably not a substantial claim. 3
- The quote only partly states this claim. 1
May combine two claims.
Findingown dataApplying charge conjugation changes the sign of the charge and turns a repulsive electric potential into an attractive one, which can be easily penetrated by the antiparticle.
“In Fig. 2(c) we apply charge conjugation, changing the sign of the charge and turning a repulsive electric potential into an attractive one, which can be easily penetrated by the antiparticle.”
Quote in the paperSubstantial claimMay combine two claims.
Findingown dataApplying time reversal after the particle has entered the barrier reverses all momenta and refocuses the wave packet, tracing back exactly its trajectory.
“In Fig. 2(b) we apply the time-reversal operation after the particle has entered the barrier: All momenta are reversed and the wave packet is refocused, tracing back exactly its trajectory.”
Quote in the paperSubstantial claimMay combine two claims.
FindingreasoningIn the limit of large detuning, Eq. (5) is recovered with the momentum operator p_x = iħ(a†-a)/2Δ and the equivalences c = 2Δ and mc^2 = 2ħ^2 r^2/...
“In the limit of large detuning, we have ) r ffiffiffiffiffiffiffiffiffiffiffi ffi hb y bi p , jha y À aij, and we recover Eq. ( 5) with the momentum operator p x ¼ i@ða y À aÞ=2Á and the equivalences c ¼ 2Á and mc 2 ¼ 2@ 2 r 2 = with Á ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @=4m 0 p .”
Quote in the paperSubstantial claimMay combine two claims.
Findingnone givenPseudohelicity Σ = σ_x p_x is conserved in the 1+1-dimensional Majorana dynamics of Eq. (3) but not in the 1+1-dimensional Dirac equation.
“This quantity is conserved in the 1 þ 1-dimensional Majorana dynamics of Eq. ( 3) but not in the 1 þ 1-dimensional Dirac equation.”
Quote in the paperSubstantial claimMay combine two claims.
FindingreasoningThe dynamics of the mapped equation preserves the reality of the bispinor and cannot be reduced to a single 1+1 Dirac particle.
“Here, the dynamics preserves the reality of the bispinor É and cannot be reduced to a single 1 þ 1 Dirac particle.”
Quote in the paperSubstantial claimMay combine two claims.
InterpretationreasoningComplex conjugation of the simulated state becomes a local unitary V_K acting solely on the ancillary space, and is physically implementable for a wave function of arbitrary dimensions.
“In this manner, the complex conjugation of the simulated state becomes a local unitary V K acting solely on the ancillary space,”
- Also stated in: § Abstract · plain
“and thus is physically implementable for a wave function of arbitrary dimensions.”
Quote in the paperSubstantial claim- Also stated in: § Abstract · plain
Probably not a substantial claim: may not pass S4 (based).
Interpretationnone givenA situation where an unphysical operation is performed between unitary evolutions is impossible to reproduce seamlessly with classical computers.
“Such a situation is impossible to reproduce seamlessly with classical computers.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based).
Interpretationnone givenThe conventional approach to implementing an unphysical operation mid-simulation (stop, full quantum tomography, classical operation, re-encoding, and proceeding) would be impossible with classical resources.
“Clearly, this task would be impossible with classical resources but could be accomplished with a suitable implementation of our proposed ideas.”
Quote in the paperNot a substantial claimMay combine two claims.
InterpretationcitationThe mapping works in arbitrary dimensions and can be immediately applied on different experimental setups.
“The mapping works in arbitrary dimensions and can be immediately applied on different experimental setups.”
Quote in the paperNot a substantial claimThe quote only partly states this claim.
InterpretationreasoningIt is possible to tune the numerator and denominator independently to preserve the dispersive regime while exploring the range from ζ≈0 (ultrarelativistic limit) to ζ→∞ (nonrelativistic limit).
“we see that it is possible to tune the numerator and denominator independently to preserve the dispersive regime, while exploring simultaneously the range from ' 0 (ultrarelativistic limit) to ! 1 (nonrelativistic limit).”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Interpretationnone givenThe previous implementation of discrete symmetries is valid both for Majorana and Dirac equations.
“The previous implementation of discrete symmetries is valid both for Majorana and Dirac equations.”
Quote in the paperNot a substantial claim
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Direct laser-written waveguides offer the possibility of implementing complex lattice geometries including the third spatial dimension, and consequently of utilization for investigating systems beyond the standard model, in quantum biophysics, and in quantum artificial life.
“Special attention is paid to direct laser-written waveguides since they offer the possibility of implementing complex lattice geometries including the third spatial dimension [129, 130], and consequently of utilization for investigating systems beyond the standard model [131], in quantum biophysics [132-134], and in quantum artificial life [135,136].”
From Integrated photonic quantum walks (2020) · cites this paper as [131]
Split into 2 atoms — 1 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.
Experimental — accuracy not yet measuredThe spin degrees of freedom can be encoded in long-lived electronic states of the ions.
“The spin degrees of freedom can be encoded in long-lived electronic states of the ions [5, 6, 12, 14, 19, 32, 39].”
From Quantum Simulation of Interacting Fermion Lattice Models in Trapped Ions (2012) · cites this paper as [14]
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Noh, C., Rodríguez-Lara, B. M., & Angelakis, D. G. (2012) · New Journal of Physics · doi:10.1088/1367-2630/14/3/033028 · Cited by 23 · see citing works
Linear-algebraic bath transformation for simulating complex open quantum systems
Huh, J., Mostame, S., Fujita, T., Yung, M., & Aspuru‐Guzik, A. (2014) · New Journal of Physics · doi:10.1088/1367-2630/16/12/123008 · Cited by 20 · see citing works
Atomic Quantum Technologies for Quantum Matter and Fundamental Physics Applications
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Different models of gravitating Dirac fermions in optical lattices
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Efficient Measurement of Multiparticle Entanglement with Embedding Quantum Simulator
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Quantum Simulation of Noncausal Kinematic Transformations
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Measuring Entanglement in a Photonic Embedding Quantum Simulator
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Self-citationСадыков, Н. Р., Muratov, E. T., Pilipenko, I. A., & Aporoski, A. V. (2020) · Physica E Low-dimensional Systems and Nanostructures · doi:10.1016/j.physe.2020.114071 · Cited by 14 · see citing works
Mapping curved spacetimes into Dirac spinors
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Self-citationProposal for trapped-ion emulation of the electric dipole moment of neutral relativistic particles
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Quantum simulation of a quantum stochastic walk
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Finite volume effects on QCD susceptibilities with a chiral chemical potential
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Entanglement measures in ion-trap quantum simulators without full tomography
Pedernales, J. S., Candia, R. D., Schindler, P., Monz, T., Hennrich, M., Casanova, J., & Solano, E. (2014) · Physical Review A · doi:10.1103/physreva.90.012327 · Cited by 11 · see citing works
Self-citationAl-Hashimi, M. H., Shalaby, A. M., & Wiese, U. ‐. (2017) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.95.065007 · Cited by 12 · see citing works
Topological qubits with Majorana fermions in trapped ions
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Self-citationRecent progress of quantum simulation of non-Hermitian systems
Gao, X.-E., Li, D.-L., Liu, Z., & Zheng, C. (2022) · Acta Physica Sinica · doi:10.7498/aps.71.20221825 · Cited by 9 · see citing works
Quantum weak and modular values in enlarged Hilbert spaces
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Neutrino helicity reversal and fundamental symmetries
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Encoding relativistic potential dynamics into free evolution
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The nonrelativistic limit of the Majorana equation and its simulation in trapped ionsPreprint
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Self-citationNeutrino helicity reversal and fundamental symmetries
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Entanglement and Quantum Logical Gates. Part I.
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Simulating the Majorana dynamics with ultracold atomic gases in a bilayer honeycomb lattice
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Time and spatial parity operations with trapped ions
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Trapped ionic simulation of neutrino electromagnetic properties in neutrino oscillation
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Switchable particle statistics with an embedding quantum simulator
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Quantum Alchemy and Universal Orthogonality Catastrophe in One-Dimensional AnyonsPreprint
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- Authors
- Casanova, J., Sabín, C., León, J., Egusquiza, I. L., Gerritsma, R., Roos, C. F., García-Ripoll, J. J., & Solano, E.
- Affiliations
- University of the Basque Country (ES)
- Consejo Superior de Investigaciones Científicas (ES)
- Austrian Academy of Sciences (AT)
- Ikerbasque (ES)
- Venue
- Physical Review X
- Publisher
- American Physical Society (APS)
- Published
- 2011-12-09
- Volume
- 1
- Issue
- 2
- ISSN
- 2160-3308 (ISSN-L)
- OpenAlex
- W2001928329
- Cited by
- 77
- References
- 20
- Citations by year
- 2025: 1 · 2024: 3 · 2023: 1 · 2022: 3 · 2020: 5 · 2019: 4 · 2018: 7 · 2017: 6 · 2016: 6 · 2015: 10 · 2014: 11 · 2013: 8 · 2012: 10
- FWCI
- 9.13
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- Physical Review X (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- DIGITAL.CSIC (Spanish National Research Council (CSIC)) (submittedVersion)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- arXiv (Cornell University)
- Funders
- Eusko Jaurlaritza (IT-559-10, IT-472-10, IT472)
- Ministerio de Ciencia e Innovación (FIS2009-12773-C02-01)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Topological Materials and Phenomena
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- Majorana equation, quantum simulation, charge conjugation, complex conjugation, time reversal, wave function symmetry, Majorana fermions
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