Bell’s Inequalities - University of Rochester
He derived an inequality based on Bohm-type quantum systems which showed that any local realistic theory and quantum mechanics pre-dicted two di erent probabilistic outcomes His work was further elaborated on by Clauser, Horne, Shimony and Holt The derivation of Bell’s inequalities start with the following consideration If a local
CHSH Inequality - Stony Brook University
Bell’s theorem stood the test of time and was repeatedly veri ed through a variety of experimental setups One of the most popular forms of Bell’s inequality is the CHSH (John Clauser, Michael Horne, Abner Shimony, and Richard Holt) in-equality, which is what will be tested in our experiment The CHSH inequality
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1 1 1 Bell Inequality - Spin correlation Consider the nal spin singlet state of the decay 0 + We suppose that the 0 decays and the muon and + travel in opposite directions, with equal but opposite momenta, and equal but opposite spin Now we can measure the component of the muon and/or + spin along any axis we like
Classical probability model for Bell inequality
Bell’s inequality is typically coupled to notions of realism, locality, and free will [1], [3]{[11] In mathematically oriented literature, violation of Bell’s inequality is considered as exhibition of nonclassicality of quantum probability - impossibility to use the Kolmogorov model of probability theory [36]
Bell inequality, separability and entanglement distillation
Bell inequalities has been demonstrated experimentally, e g [28] 1 2 Bell inequalities for multipartite qubit systems The MABK inequality is a kind of Bell inequality for multi-partite qubits [7–9] whereas the WWZB inequality [10, 11] is a kind of generalization of the MABK inequality Here we introduce the WWZB inequality and consider the MABK
2251 Lecture slides: EPR paradox, Bell inequalities
J S Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, 195-200 (1964) • Alain Aspect, Philippe Grangier, and Gerard Roger, Phys Rev Lett 47, 460 - 463 (1981) Experimental Tests of Realistic Local Theories via Bell's Theorem 37
Violating Bell’s inequality with remotely-connected
The Bell inequality (10) is an im-portant benchmark for entanglement, providing a straightforward test of whether a local and deterministic theory can explain measured correlations To date, however, only local violations of the Bell or Leggett-Garg (11) inequalities have been demonstrated using superconducting
news and views Bell’s inequality test: more ideal than ever
Bell’s theorem changed the nature of the debate In a simple and illuminating paper1, Bell proved that Einstein’s point of view (local realism) leads to algebraic predictions (the celebrated Bell’s inequality) that are contradicted by the quantum-mechanical predictions for an EPR gedanken experiment
Entangled Photons and Bell’s Inequality
Entangled Photons and Bell’s Inequality Graham Jensen, Christopher Marsh and Samantha To University of Rochester, Rochester, NY 14627, U S November 20, 2013 Abstract – Christopher Marsh Entanglement is a phenomenon where two particles are linked by some sort of characteristic A particle such as an electron can be entangled by its spin
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Violating Bell"s inequality with remotely-connected superconducting qubits
Y. P. Zhong,
1H.-S. Chang,1K. J. Satzinger,1;2M.-H. Chou,1;3
A. Bienfait,
1C. R. Conner,1´E. Dumur,1;4J. Grebel,1
G. A. Peairs,
1;2R. G. Povey,1;3D. I. Schuster,3A. N. Cleland1;4
1 Institute for Molecular Engineering, University of Chicago, Chicago IL 60637, USA2Department of Physics, University of California, Santa Barbara CA 93106, USA
3Department of Physics, University of Chicago, Chicago IL 60637, USA
4Institute for Molecular Engineering and Materials Science Division,
Argonne National Laboratory, Argonne IL 60439, USA To whom correspondence should be addressed; E-mail: anc@uchicago.edu. Quantum communication relies on the efficient generation of entanglement between remote quantum nodes, due to entanglement"s key role in achieving and verifying secure communications. Remote entanglement has been real- ized using a number of different probabilistic schemes, but deterministic re- mote entanglement has only recently been demonstrated, using a variety of superconducting circuit approaches. However, the deterministic violation of a Bell inequality, a strong measure of quantum correlation, has not to date been demonstrated in a superconducting quantum communication architecture, in part because achieving sufficiently strong correlation requires fast and accu- rate control of the emission and capture of the entangling photons. Here we present a simple and scalable architecture for achieving this benchmark result in a superconducting system.1arXiv:1808.03000v1 [quant-ph] 9 Aug 2018
Superconducting quantum circuits have made significant progress over the past few years, demonstrating improved qubit lifetimes, higher gate fidelities, and increasing circuit complex- ity (1,2). Superconducting qubits also offer highly flexible quantum control over other systems, includingelectromagnetic(3,4)andmechanicalresonators (5,6). These devicesarethusappeal- ing for testing quantum communication protocols, with recent demonstrations of deterministic remote state transfer and entanglement generation (7-9). The Bell inequality (10) is an im- portant benchmark for entanglement, providing a straightforward test of whether a local and deterministic theory can explain measured correlations. To date, however, only local violations of the Bell or Leggett-Garg (11) inequalities have been demonstrated using superconducting qubits (12,13), as remote state transfer and entanglement generation with sufficiently high fi- delity is still an experimental challenge. (14) form of the Bell inequality, using a pair of superconducting qubits coupled through a 78 cm-long transmission line, with the photon emission and capture rates controlled by a pair of electrically-tunable couplers (15). In one experiment, we use a single standing mode of the transmission line to relay quantum states between the qubits, achieving a transfer fidelity of0:9520:009. This enables the deterministic generation of a Bell state with a fidelity of0:9570:005. Measurements on this remotely-entangled Bell state achieve a CHSH correla-
tionS= 2:2370:036, exceeding the classical correlation limit ofjSj 2by 6.6 standard deviations. In the second experiment, we control the time-dependent emission and capture rates of itinerant photons through the transmission line, a method independent of transmission dis- tance. These shaped photons enable quantum state transfer with a fidelity of0:9400:008, and deterministic generation of a Bell state with a fidelity of0:9360:006. Measurements on this Bell state demonstrate a CHSH correlation ofS= 2:2230:023, exceeding the clas- sical limit by 9.7 standard deviations. We note that the Bell state fidelities for both methods 2 are close to the threshold fidelity of 0.96 for surface code quantum communication (16). This simple yet efficient circuit architecture thus provides a powerful tool to explore complex quan- tum communication protocols and network designs, and can serve as a testbed for distributed implementations of the surface code. The device layout is shown in Fig. 1A, comprising two transmon qubits (17,18),Q1and Q2, connected via two tunable couplers (15),G1andG2, to a coplanar waveguide (CPW)
transmission line of length`= 0:78m. The device is fabricated on a single sapphire substrate, with the serpentine transmission line covering most of the area of a615mm2chip. A circuit diagram is shown in Fig. 1B, with more details in the Supplementary Information (SI). Ignoring the couplers, the transmission line is shorted to ground on both sides, supporting a sequence of standing modes with frequencies equally-spaced by!FSR=2= 1=2T`= 79MHz, whereT`= 6:3ns is the photon travel time along the line. The coupling strengthgibetween qubitQiand thenth standing mode is determined by the control signals sent to the coupler G i, and can be set dynamically between zero and about 45 MHz. The coupling strength is proportional to pn, but for the experiments herenis large (70) and the variation ofnsmall ( 5), so this dependence can be safely ignored. More details can be found in the SI. When one coupler is set to a small non-zero coupling, withjgij !FSR, and the other coupler is turned off, the coupled qubit can selectively address each standing mode of the trans- mission line. This is observed by performing qubit spectroscopy, which reveals a sequence of avoided-level crossings with the standing mode resonances (Fig. 1C). In the time domain, we observe vacuum Rabi swaps with each mode by first preparingQ1in its excited statejeiusing apulse, then setting the qubit frequency (Fig. 1D). The weak coupling allows the qubit to interact with each mode separately, with weak interference fringes visible only near frequencies halfway between each mode. By weakly coupling both qubits to a single mode, we can relay qubit states through that 3Qubit Frequency (GHz)
5.6 5.7 5.8 5.9 6.0Drive Frequency (GHz)
n = 71n= 72n= 73n= 74n= 75n= 765.65.75.85.96.0
Qubit Frequency (GHz)
0 250500
Interaction Time (ns)
A B Q 1 Q 278 cm transmission line
1 mm G 1 G 2 Q 1 Z 1 XY 1 G 1 readout resonator Q 2 Z 2 XY 2 G 2 readout resonator C q C qL J L J L g L g L g L g L T L T L w L w DC 01 P eFigure 1:Device description. APhotograph of device, showing two qubitsQ1andQ2(blue) connected via tunable couplersG1andG2(green) to a 78 cm-long coplanar transmission line (cyan).BCircuit schematic, with parameters listed in Table S2.CSpectroscopy of qubitQ1 interacting with six transmission line standing modes. Black dashed lines: Numerical simula- tions.DVacuum Rabi swaps betweenQ1and the six standing modes. The coupling is set to g1=2= 5MHz!FSR=2.
4 mode (Fig. 2A). We prepareQ1in its excited statejei, then turn on theG1coupling for a time, while simultaneously adjustingQ1"sZbias to match its frequency to the selected mode, swapping the excitationto the mode. Wethen turn on theG2coupling and adjustQ2"s frequency to swap the excitation toQ2. Atswap= 52ns, one photon is completely transferred fromQ1to Q2, with a transfer probability of0:9360:008. We perform quantum process tomography (19)
to characterize this transfer process, yielding the process matrix1shown in Fig. 2B, with a processfidelityFp1= Tr(1ideal) = 0:9520:009. Hereideal=Iistheidealprocessmatrix.
This experimental result agrees well with the numerically-simulated fidelityFp1= 0:955. Note
a related experiment (20) has demonstrated quantum state transfer through a 1 m-long normal- metal coaxial cable using an innovative "dark" relay mode, achieving a transfer fidelity of 61% with a significantly lossier channel. We also use the relay mode to generate a Bell statej Belli= (jgei+jegi)=p2between the two qubits, by terminating theQ1swap process at the half-swap timehalf= 26ns. We perform quantum state tomography (21), with the reconstructed density matrix1displayed in Fig. 2C, from which we calculate a state fidelityFs1=h Bellj1j Belli= 0:9500:005and a concurrenceC1= 0:9270:013. Numerical simulations using the master equation give a state fidelityFs1= 0:947and a concurrenceC1= 0:914, in good agreement with experiment. We next perform the CHSH Bell inequality test (12) on this remotely entangled Bell state (see SI). We measureQ1along directiona=xora0=y, and simultaneously measureQ2 alongborb0?b, varying the anglebetweenaandb(Fig. 2D inset). We then calculate the CHSH correlationS, as shown in Fig. 2D. We find thatSis maximized at= 5:5rad, where S= 2:2370:036with no measurement correction, exceeding the maximum classical value of2 by 6.6 standard deviations. If we correct for readout error (12), we findS= 2:6650:044,
approaching the quantum limit of2p22:828. The entanglement is deterministic and the measurement is single-shot (see SI), so the detection loophole (22) is closed in this experiment. 5 BA C D 03060(ns) 0 1 P e Q 1 Q 2 Q 1 G 1 Q 2 G 2 02 (rad) -2 0 2 S -22 22
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