Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484 (Dated: November 25, 2016)
Entanglement entropy has proven invaluable to our understanding of quantum criticality. It is natural to try to extend the concept to “non-unitaryquantum mechanics”, which has seen growing interest from areas as diverse as open quantum systems, non-interacting electronic disordered sys- tems, or non-unitary conformal field theory (CFT). We propose and investigate such an extension here, by focussing on the case of one-dimensional quantum group symmetric or supergroup sym- metric spinchains. We show that the consideration of left and right eigenstates combined with appropriate definitions of the trace leads to a natural definition of Rényi entropies in a large variety of models. We interpret this definition geometrically in terms of related loop models and calculate the corresponding scaling in the conformal case. This allows us to distinguish the role of the central charge and effective central charge in rational minimal models of CFT, and to define an effective central charge in other, less well understood cases. The example of the sl(2|1) alternating spin chain for percolation is discussed in detail.
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
4 Research Center for Quantum Information, Slovak Academy of Sciences, Bratislava, Slovakia
(Received 24 August 2012; published 16 November 2012)
Frustration-free (FF) spinchains have a property that their ground state minimizes all individual terms in the chain Hamiltonian. We ask how entangled the ground state of a FF quantumspin-s chain with nearest-neighbor interactions can be for small values of s. While FF spin-1=2 chains are known to have unentangled ground states, the case s ¼ 1 remains less explored. We propose the first example of a FF translation-invariant spin-1 chain that has a unique highly entangled ground state and exhibits some signatures of a critical behavior. The ground state can be viewed as the uniform superposition of balanced strings of left and right brackets separated by empty spaces. Entanglement entropy of one half of the chain scales as 1 2 logn þ Oð1Þ, where n is the number of spins. We prove that the energy gap above the ground state is polynomial in 1=n. The proof relies on a new result concerning statistics of Dyck paths which might be of independent interest.
by spin-1’s on each site. We may group a number of sites into an effective site. When the effective site is large enough, the direct entanglement between two spin- 1’s (represented by the blue curves) can only appear be- tween the neighboring effective sites. Then we can use an unitary transformation acting within an effective site to simplify the entanglement within each effective site (see Fig. 5 c). After removing the degrees of freedom that are entangled only within each effective site, we obtain a sim- plified coarse-grained wave function (see Fig. 5 d) which corresponds to the corner-double-line structure. We note that, in the coarse-grained wave function, each effective site has four states: spin-0 ⊕ spin-1, which can be viewed as two spin-1/2’s: spin-1/2 ⊗ spin-1/2. We also see that the coarse-grained wave function is a product state of spin-singlets (see Fig. 5 a,d). This seems confirm that the Haldane phase is a trivial product state formed by spin-0’s.
on an infinite lattice. It will be an exact description since our finite-depth quantum circuits U,V are exact. In this section, we will refer to local degrees of freedom as qudits.
Let us review multiscale entanglement renormalization ansatz (MERA) states [ 3 , 23 ]. The MERA state is a many-qudit state that is obtained by reversing the entanglement RG transformations as follows. One starts with a qudit system on some lattice. (Step 1) Apply a finite-depth quantum circuit with some ancillary qudits in a fixed state |↑. Due to the insertion of the ancillary qudits, the number density of qudits is increased. In order to retain the number density (Step 2) one expands the lattice. Then, (Step 3) iterate Steps 1 and 2. In a scale-invariant system, one expects that the quantum circuit in Step 1 is the same for every level of the iterations. The class of states that can be written as a MERA is proposed to describe ground states of some critical systems, and is shown to admit efficient classical algorithms.
correlation exponent ν av = 1/2 exactly as for the Fernandez-Pacheco self-dual procedure for the time independent
random quantum Ising chain [43–46].
B. Strong Disorder RG procedure
Since the Block Self-dual RG rules discussed above points towards an Infinite Disorder Fixed Point, the critical properties are expected to be described exactly in the asymptotic regime by the appropriate Strong Disorder RG rules [26, 27]. Here one does not need to do new computations, since one can derive them as a special limit from the block self-dual RG rules given above. The idea is that one wishes to eliminate only one degree of freedom at each step (instead of the N
Furthermore, it becomes possible to use known results of pure-state transport to devise protocols for perfect spin transfer, even using highly mixed states. Specifically, we have shown that combining a simple encoding of the transmitted state into one or more spin pairs with engineered couplings in the chain allows for the perfect transfer of quantum information and potentially of entanglement. An additional advantage of mixed-state chains is that they enable trans- port of relevant states via a non-spin-excitation conserving Hamiltonian, the DQ Hamiltonian, which can be obtained by coherent averaging from the naturally occurring magnetic dipolar interaction. These results have been combined to obtain a proposal for scalable quantum-computation architecture using electronic spin defects in diamond [15,21,23], which may be experimentally viable with existing or near-term capabilities.
tal physical constants  – and the result is astonishingly independent of the disorder in the sample or of its geometry 3 . Although the plateaux themselves are well understood, the ques-
tion of the transition between them and their associated critical exponents is a famous still unsolved problem. It is a localization-delocalization transition of electrons in a disordered po- tential moving from one Landau level to another, that is both a geometrical percolation-like and disordered problem. One of the major attempts to solve it is the Chalker-Coddington model , that describes the evolution of an electron in a disordered potential landscape with possible tunnel effect and whose numerical simulation has yielded exponents in good agreement with experiment ; however, there are still unresolved problems in a purely numerical approach, which would require analytical work to fill the gaps; but it is of con- siderable difficulty to tackle analytically, because of its supersymmetric formulation, of its non-unitarity, namely the non-hermiticity of its Hamiltonian, and of its non-compactness, true even in finite-size since each site of the lattice is described by an infinite-dimensional vector space. On the field theory side, it is expected to be described by a sigma model with target space U (1,1)×U (1,1) U (1,1|2) [42, 43], but out of reach of analytical study. All these particularities call for the development of new concepts and techniques, and this manuscript takes place among this wider project.
JHEP02(2021)130 among these is the nature of fields whose conformal weights are in the extended Kac table,
h = h r,s with r, s ∈ N ∗ . While inunitary CFTs the resulting degenerate behaviour implies the existence of certain differential equations satisfied by the correlators of these fields, such result does not necessarily hold in the non-unitary case, where Virasoro “norm-squares” are not positive definite any longer (see more discussion of this below). The second purpose of this paper is to find out specifically what kind of Virasoro modules occur in the XXZ chain when the Virasoro representations are degenerate — that is, (some) fields belong to the extended Kac table. We will do this straightforwardly, by exploring the action of the lattice Virasoro generators, and checking directly whether the relevant combinations vanish or not — in technical parlance, whether “null states” or “singular vectors” are zero indeed. This is of course of utmost importance in practice, as this criterion determines the applicability of the BPZ formalism [ 26 ] to the determination of correlation functions, such as the four-point functions currently under investigation [ 19 – 21 ]. We shall find some unex- pected results, that we hope to complete in a subsequent paper [ 27 ] by studying the cases when the central charge is rational (e.g., the case c = 0 with applications to percolation).
numerical tensor network methods using the Matrix Product States (MPS) formalism, we can follow the non-trivial crossover occurring incriticalchains between the high-temperature diffusive classical regime and the low-temperature response described by the Tomonaga-Luttinger liquid (TLL) theory, for which analytical expressions are known. In order to compare analytics and numerics, we focus on a generic spin-1/2 XXZ chain which is a paradigm of gapless TLL, as well as a more realistic spin-1 anisotropic chain, modelling the DTN material, which can be either in a trivial gapped phase or in a TLL regime induced by an external magnetic field. Thus, by monitoring the finite temperature crossover, we provide quantitative limits on the range of validity of TLL theory, that will be useful when interpreting experiments on quasi one-dimensional materials.
Outlook.—Our work supports a very general principle that one can obtain an accurate representation of inherently exponential problems dealing with QMBS by approximat- ing them with far less complexity. This realization is at the heart of other recent developments in QMBS research such as matrix product states [ 8 , 9 ], and density matrix renor- malization group [ 10 ], where the state (usually the ground state of one-dimensional chains) can be adequately repre- sented by a matrix product state ansatz whose parameters grow linearly with the number of quantum particles. Future work includes explicit treatment of fermionic systems and numerical exploration of higher dimensional systems.
Non-equilibrium dynamics, classical and quantum, is one of the main current focuses of both theoretical and experimental condensed matter physics. In the classical theory, important theoretical progresses were recently achieved by solving simple paradigmatic models, such as the exclusion processes [1, 2]. This collection of results culminated in the formulation of the macroscropic fluctuation theory (MFT)  which provides a framework to study, and to understand, a large class of out-of-equilibrium classical systems. In the quantum theory, recent progresses arose through studies of simple, often integrable, out-of-equilibrium systems [4, 5]. Those deal for instance with quantum quenches [6, 7, 8], with boundary driven integrable spinchains [9, 10], or with transport phenomena incritical one dimensional systems either from a conformal field theory perspective [11, 12, 13] or from a hydrodynamic point of view [14, 15]. However, these simple systems generally exhibit a ballistic behaviour while the MFT deals with locally diffusive systems satisfying Fick’s law. Therefore, to decipher what the quantum analogue of the macroscopic fluctuation theory could be –a framework that we may call the mesoscopic fluctuation theory–, we need, on the one hand, to quantize its set-up and, on the other hand, to add some degree of diffusiveness in the quantum systems under study.
cation C-band (1530-1565 nm) and with an SPDC pro- cess efficiency of 4 · 10 −6 photon pairs per pump pho- ton. Such a bandwidth would allow entanglement dis- tribution in up to 31 standard channel pairs using off- the-shelves, and high-performance multi-channel DWDM components . The emitted spectrum is directly col- lected thanks to a bare fiber with 55 % efficiency and then, ab initio, deterministically demultiplexed so as to provide Alice and Bob with short and long wavelength photons apart from degeneracy, respectively, by means of standard broadband fiber Bragg gratings (AOS GmbH) and associated circulators. This strategy allows avoiding the 50% loss that would arise when separating the photon pairs using a beam-splitter. Then, to reveal energy-time entanglement, we employ a set of unbalanced Michel- son interferometers (UMI) in the “Franson configura- tion” . They are made of a fiber optics beam-splitter connected to two Faraday mirrors allowing to automati- cally compensate polarization rotations such that excel- lent long term stability is guaranteed. To further exploit the potential of the broadband photon pair generator, Al- ice and Bob analyze entanglement, for a proof-of-concept demonstration, in 2×8 complementary channels simulta- neously, by demultiplexing them with standard DWDMs (AC Photonics). As shown in Fig. 1, Alice is supplied with channels 39 to 46 and Bob with 48 to 55, according to the ITU grid. The total optical loss from the photon pair generator to after the DWDMs is about 5 − 6 dB. In the end, the photons are detected using free-running indium-gallium-arsenide (InGaAs) single photon detec- tors. The detector at Alice’s location features 440 Hz of dark counts at 28% detection efficiency (ID Quantique id230), while the detector at Bob’s place shows 1400 Hz at 20% (ID Quantique id220). Both detectors are set to a dead-time of 9 µs in order to keep the probability of afterpulses low. Coincidence measurements between cor- related pairs of detectors are performed using a time to amplitude converter (ORTEC 567) and related electron- ics. The timing jitter of the full detection system was measured to be 155 ps.
a good description of our results by using a three-band model that is an extension of the model
found to be useful for a quantitative analysis of the optical properties of bulk ZnO [2, 3]. Due to an
efficient Quantum Confined Stark Effect (QCSE), the short-range electron-hole exchange
interaction redistributes oscillator strengths among exciton states built from different valence bands
B. Related Works
Despite its original motivations,  showed that Lineal can handle the l 1 -norm. This can be used for example to
represent probabilistic distributions of terms. Also, a simpli- fication of Lineal, without scalars, can serve as a model for non-deterministic computations . And, in general, if we consider the standard values of the lambda calculus as the basis, then linear combinations of those form a vector space, which can be characterized using types . In  a similar distinction between classical bits (B) and qbits (]B) has been also studied. However, without unitarity, it is impossible to obtain a calculus that could be compiled onto a quantum machine. Finally, a concrete categorical semantics for such a calculus has been recently given in .
Semiconductor quantum dot arrays provide a fully tunable platform for manipulating the coherent coupling of quantum states. Great control has already been demon- strated in the double quantum dot system with the obser- vation of molecularlike superpositions via clear resonances in the current flowing through the system . The spin degree of freedom plays a critical role and has led to various proposals utilizing quantum dots as spin or coded spin qubits . An extension to fully coherent triple quantum dot circuits has recently been achieved [10–16]. In addition to being a first step towards more complex quantum simulation architectures , such devices make it possible to investigate phenomena which rely on quantum super- positions of distant states mediated by tunneling [18,19]. Long-range tunneling involves the transfer of states from one side of the three-dot array to the other without the occupation of the center site. A recent experiment reported the observation of such an effect as a transport resonance : If the two edge dots of the triple quantum dot array are coupled to source and drain electron reservoirs, left-right superpositions provide a direct channel for the current. The relevant resonant transitions can be measured by
from an experimental point of view, due to the several and complex measurements that are required for determining it. In order to address the challenge of a complete entan- glement characterization, we developed a multipartite d-level entanglement witness that is capable of detecting the presence of any arbitrarily complex pure quantum state, as well as its eventual entanglement. A main goal in the derivation of this operator was to make it as feasible as possible to measure in practice, which is significantly in con- trast with established theoretically optimal witness operators. We developed a general approach allowing us to construct experimentally optimal entanglement witnesses that provide a good trade-off between (white) noise tolerance and experimental complexity. The detection of the here derived witnesses demands indeed measurements that are re- duced in number and complexity, while still having a good noise robustness. We used this approach to derive a witness capable of detecting the genuine multipartite entanglement of d-level cluster states. To this end, we made use of high dimensional stabilizers that, since formed by the generalized Pauli matrices, can be measured by means of single-qudit projections, thus resulting relatively easy to measure in practice. We further showed how to customize a witness towards experimental restrictions by considering the explicit ex- ample of a four-partite three-level optical cluster state. It is always possible to measure a witness by making use of just a specific set of stabilizers, which are chosen according to the measurement capabilities. Finally, we showed that it is possible to construct a witness by making use of partial operators instead of stabilizers. It can happen indeed that stabi- lizers cannot be measured, since the available measurement settings allow us to perform projections on a number of levels that is lower with respect to the state dimensionality. We also exploited the novel witness to test the robustness of cluster states towards white noise. We demonstrated that the noise robustness increases as the single-state dimen- sionality d increases, while decreasing when the number of parties increases. We finally found that different experimental restrictions lead to different optimal witnesses as well as to different noise sensitivities for the same given quantum state.
PACS numbers: 78.67.Hc, 78.55.Cr, 78.66.Fd
Spin memory effects in semiconductor quantum dots (QDs) attract presently much attention in the physics of nanostructures. The discrete energy spectrum of zero- dimensional carriers in QDs is expected to lead to an inhi- bition of the main spin relaxation mechanisms which are known in bulk semiconductors and planar heterostruc- tures [1, 2, 3]. In some novel QD devices, the preser- vation of the exciton spin coherence is a central issue, for instance for the generation of polarization-entangled photon-pairs inquantum information processing [4, 5]. Recent studies of epitaxially grown InGaAs/GaAs QDs have shown that the longitudinal exciton spin relaxation may be quenched over tens of ns at low temperature [6, 7, 8]. However, it was also suggested in Ref.  that some QDs could undergo a rapid spin relaxation, because the longitudinal spin dynamics exhibits, for QD arrays, a fast decay component (40 ps). This fact highlights the need for experiments probing spin relaxation dynamics on the single QD level. However, the implementation of the standard time-resolved techniques used for QDs en- sembles [6, 7, 8] still remains an experimental challenge in the field of single QD spectroscopy.
exchange gas. The photoluminescence (PL) of the QDs was excited either above the gap of the ZnTe barrier (at 532 nm) or using a tunable dye laser in the range 570–610 nm. Both the exciting and the collected light were transmitted through a monomode fiber coupled di- rectly to the microscope objective. The overall spatial resolution of the setup was better then 1 m which assured the possibility of selecting different single quantum dots containing a single Mn 2 þ ion. The dots without Mn 2 þ ion were observed in the same samples. The PL analysis was done for the dots having emission lines in the low-energy tail of the broad PL emission band [ 9 , 25 ], which assured good separation from the lines related to the other dots. The
x Mn x Te quantum well. Under magnetic field, the conduction band splits into two spin subbands with, in the absence of carriers, the splitting described by a modified Brillouin
function . The single SPE spectrum observed at 0T is thus expected to split at non vanishing field into four new structures associated with spin conserving ( and ) and spin flip ( and ) transitions, which were degenerate at 0T. Spin conserving transitions are expected to vary smoothly due to the increase in Fermi velocity v
In one direction, these investments relax the assumption of standard oblivious trans- fers, Rabin OT and one-out-of-two OT. This weakening action may cover a larger class of possible noisy models [CK88, Cr´e97, DKS99, KM01, SW02, CMW04, Mor05].
In the other direction, one would find out practically physical channels that match the theoretical assumptions. Since the introduction of quantum mechanics into the field of communication and cryptography [Wie83], the successful implementation of key exchange schemes [BB84, Eke91, Ben92] with provable unconditional security [LC99, SP00] has encour- aged researchers to seek for quantum unconditionally secure bit commitment and oblivious transfer [CK88, BBCS92, BCJL93]. Much interest aimed to exploit the uncertainty princi- ple and the non-cloning property to implement wanted noisy channels for oblivious trans- fer [CK88, BBCS92]. However, this intention was rejected by a no-go theorem of Mayers and Lo & Chau, which was first discovered for quantum bit commitment protocols [May97, LC97] and then for quantum oblivious transfer protocol [Lo97].