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Switchable Particle Statistics with an Embedding Quantum Simulator Switchable particle statistics with an embedding quantum simulator Xiao-Hang Cheng,1, 2 I~nigoArrazola,2 Julen S. Pedernales,2 Lucas Lamata,2 Xi Chen,1 and Enrique Solano2, 3 1Department of Physics, Shanghai University, 200444 Shanghai, People's Republic of China 2Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain 3IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Spain (Dated: October 1, 2018) We propose the implementation of a switch of particle statistics with an embedding quantum sim- ulator. By encoding both Bose-Einstein and Fermi-Dirac statistics into an enlarged Hilbert space, the statistics of the simulated quantum particles may be changed in situ during the time evolution, from bosons to fermions and from fermions to bosons, as many times as desired before a measure- ment is performed. We illustrate our proposal with few-qubit examples, although the protocol is straightforwardly extendable to larger numbers of particles. This proposal can be implemented on different quantum platforms such as trapped ions, quantum photonics, and superconducting circuits, among others. The possibility to implement permutation symmetrization and antisymmetrization of quantum particles enhances the toolbox of quantum simulations, for unphysical operations as well as for symmetry transformations. PACS numbers: 03.67.Ac, 03.67.Lx, 05.30.-d, 75.10.Jm I. INTRODUCTION of a switch of particle statistics, from bosons to fermions and from fermions to bosons, reversibly, during a quan- In quantum mechanics, particles fall into two funda- tum evolution. We construct a spinor in an embedding mental categories: bosons or fermions. Bosons, with in- space which has as components the wave functions of all teger spin and a wave function that is symmetric under different permutations of the particles, such that we can the permutation of two particles, follow Bose-Einstein recover both the dynamics of Bose-Einstein and Fermi- statistics, for example photons and gluons. Fermions, Dirac statistics via a direct mapping. In this way, we with half-integer spin, have a wave function that is an- can shift from bosonic to fermionic statistics and vicev- tisymmetric under the exchange of two particles and are ersa by means of local operations. The implementation of characterized by Fermi-Dirac statistics, as is the case of this proposal can be carried out via analog [8, 9] or digi- quarks and leptons. The investigation of permutations tal [35{38] quantum simulations on a variety of quantum of quantum particles is a fundamental topic in many- platforms. body physics [1]. Superselection rules establish that ev- ery quantum particle follows a well-determined quantum II. THEORETICAL FRAME statistics during its whole lifespan. Is it possible to switch the particle statistics and to achieve the change of a quan- tum particle from a boson to a fermion or viceversa, dur- In the context of embedding quantum simulations, ing a quantum dynamics? We may answer this question one has to distinguish between the simulated or embed- in the positive, by means of a suitable encoding of the ded Hilbert space, which contains the dynamics of in- dynamics onto a quantum simulator. terest, and the simulating or embedding space, an en- Since proposed by Richard Feynman [2], the field larged Hilbert space containing the simulated model un- of quantum simulation has strived to observe complex der a suitable mapping. This embedding space will allow quantum phenomena in controllable quantum platforms. for operations that under the considered mapping corre- Some examples of the latter are trapped ions [3, 4], super- spond to a process of interest in the simulated space, in arXiv:1606.04339v2 [quant-ph] 6 Feb 2017 conducting circuits [5], ultracold quantum gases [6], and this case, a modification in the particle statistics of the photonic systems [7]. Proposals and experiments of quan- simulated model. tum simulations for relativistic quantum physics [8{14], For clarity, we will first illustrate our proposal consid- quantum chemistry [15, 16], spin systems [17{21], quan- ering the case of two particles, and later show that the tum field theories [22{26], and quantum phase transi- extension to N particles is polynomial in terms of re- tions [27], among others, have already been achieved. Be- sources. In order to encode bosonic and fermionic statis- sides the standard unitary and dissipative evolutions, the tics in the embedding space, we define an enlarged spinor paradigm of embedding quantum simulation allows for in the following way, the possibility of realizing unphysical operations [13]. En- 1 0(q1) 00(q2) coding the simulated dynamics in an enlarged embedding Ψ = ; (1) p2 0(q2) 00(q1) space, antilinear and antiunitary evolutions [13, 28{32], (j) as well as noncausal kinematic transformations [33, 34], where (qi) j=1 N denotes a set of orthonormal wave f g ··· have been mapped onto physical processes. functions for particle i, with qi representing all its de- In this work, we investigate the quantum simulation grees of freedom. We consider an orthonormal set in 2 Bose-Einstein statistics (1, 1) Ancilla 0(q ) 00(q ) 0(q ) 00(q ) 0(q1) 00(q2) 1 2 + 2 1 0(q1) 00(q2) = 0(q2) 00(q1) 0(q1) 00(q2) 0(q2) 00(q1) ( ) An (1, 1)σz − Fermi-Dirac statistics FIG. 1. (color online) Scheme of the proposed embedding quantum simulator for switchable two-particle statistics. the fermionic case without loss of generality, given that ual addressing, using ions of different species that will linearly dependent contributions will cancel out due to not interact with the lasers that rule the evolution of antisymmetrization. In the bosonic case, non-orthogonal the system, or by changing the energy splitting of the contributions could be equivalently encoded. The en- ionic qubits to bring them out of resonance [29]. In larged spinor, that we name symmetrization spinor, may circuit QED, superconducting qubits can also be taken be constructed via additional quantum levels or an aux- out of resonance with the cavity to isolate them from iliary qubit. By this encoding, Bose-Einstein and Fermi- the dynamics of the system. This is typically done by Dirac statistics can be recovered by an appropriate map- imposing a magnetic field across the superconducting ping, qubit that will change the energy splitting of its two lev- 1 els [5]. If the Hamiltonian is not permutational invari- b = (1; 1)Ψ = [ 0(q1) 00(q2) + 00(q1) 0(q2)]; (2) p2 ant, H12 = H21, then the enlarged Hamiltonian would be H~ = 6 H + H , where and are 1 j"ih"j ⊗ 12 j#ih#j ⊗ 21 j"i j#i f = (1; 1)σzΨ = [ 0(q1) 00(q2) 00(q1) 0(q2)]: the ancilla states corresponding to permutations 1; 2 p2 − f g and 2; 1 of the particles, respectively, and Hij the To physically implement such a mapping, we encode Hamiltonianf g corresponding to permutation ij. the spinor by means of an ancillary qubit such that In order to measure a given observable M in the simu- Ψ = 1 [ (q ) (q ) + (q ) (q )], where lated quantum system, one can physically implement the p2 0 1 00 2 0 2 00 1 1j"i ⊗ 0 j#i ⊗ mapping from the embedding space to the embedded one, 0 and 1 . Then, with a local operation onj"i the ≡ ancilla wej#i can ≡ generate a symmetric wavefunction as explained above, and then perform the measurement associated with the ancillary state and an antisym- of interest. However, a more direct approach would be to metric one associated with the statej"i of the ancilla , encode the expectation value of the observable in the sim- j#i ulated space into the expectation value of an observable 1 1 b in the enlarged Hilbert space. Depending on whether we (σx + σz)Ψ = : (3) are interested in the observable to be evaluated, accord- p2 p2 f ing to bosonic or fermionic statistics, we will associate Measuring the ancilla and postselecting either states our observable to a different one in the enlarged space. or , we will be left with the system in a bosonicj"i or Following the mappings in Eq. (2), we have that fermionicj#i state, respectively. We point out that in order 1 M = M = Ψ M(1; 1) Ψ to switch from bosonic to fermionic statistics of the sim- h i b h bj j bi h j 1 j i ulated particles during a quantum dynamics, all that is = Ψ (12 + σx) M Ψ ; needed is to apply a local σ operation on the enlarged h j ⊗ j i z 1 spinor in the embedding space, before implementing the M f = f M f = Ψ σz M(1; 1)σz Ψ mapping to the bosonic or fermionic state. h i h j j i h j 1 j i We assume now that the evolution in the simulated = Ψ (1 σ ) M Ψ ; h j 2 − x ⊗ j i space is ruled by a Hamiltonian H, which typically will 1 M = M = Ψ M(1; 1)σ Ψ be permutation invariant to preserve the particle statis- h i b; f h bj j f i h j 1 zj i tics. Thus, the corresponding Hamiltonian in the em- = Ψ (σ iσ ) M Ψ : (4) bedding space, which preserves the mapping between the h j z − y ⊗ j i enlarged spinor and the fermionic and bosonic wave func- In this way, one can measure not only the observable M tions at all times, is given by H~ = 12 H. This makes with the corresponding bosonic b or fermionic f wave our proposal specially suitable for a quantum⊗ simulation, function, but also correlations between these two kinds as the dynamics of any demonstrated quantum simula- of particles.
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