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Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb

Moran Chen, Nicolas C. Menicucci, Olivier Pfister

arXiv:1311.2957v3quant-ph

TL;DR

Large-scale multipartite entanglement can be verified by measuring the N generators of the graph stabilizer group instead of full tomography, which scales exponentially with N in general. The experiment tests multipartite inseparability using nullifier-variance inequalities; satisfying the sufficient conditions makes each four-mode unit inseparable and extends inseparability across the whole wire by transitivity. 60 modes were measured in one wire, limited by EOM measurement ability, while two-wire checks found no squeezing connection between the wires.

Abstract

from arXiv · show

We report the experimental realization and characterization of one 60-mode copy, and of two 30-mode copies, of a dual-rail quantum-wire cluster state in the quantum optical frequency comb of a bimodally pumped optical parametric oscillator. This is the largest entangled system ever created whose subsystems are all available simultaneously. The entanglement proceeds from the coherent concatenation of a multitude of EPR pairs by a single beam splitter, a procedure which is also a building block for the realization of hypercubic-lattice cluster states for universal quantum computing.

Stabilizers and nullifiers

The paper connects qubit graph-state stabilizers to continuous-variable nullifiers, which provide experimentally accessible tests of cluster-state entanglement.

Stabilizers and nullifiers

  • For a pure entangled state, measuring the N graph-stabilizer generators offers an alternative to exponentially scaling quantum tomography.Source: Stabilizers and nullifiers · p. 6
  • Continuous-variable graph stabilizers replace the qubit Pauli group with the unitary Weyl-Heisenberg group generated by quadrature displacements.Source: Stabilizers and nullifiers · p. 6
  • A nullifier is the Hermitian operator derived from the graph adjacency matrix whose vanishing variance witnesses the cluster state's stabilization.The nullifiers coincide, up to local phase shifts, with solutions of the OPO's Heisenberg equations.Source: Stabilizers and nullifiers · p. 6
  • The same nullifiers can also implement the van Loock-Furusawa multipartite inseparability criterion.Source: Stabilizers and nullifiers · p. 6

The van Loock-Furusawa inseparability criterion

The van Loock-Furusawa criterion tests multipartite inseparability by checking necessary separability inequalities, using graph nullifiers because cluster-state correlations are local.

The van Loock-Furusawa inseparability criterion

One mode- three mode bipartitions

For one-mode-versus-three-mode bipartitions, the criterion assigns a necessary variance inequality to each possible isolated mode.

One mode- three mode bipartitions

Two-mode bipartitions

For two-mode bipartitions, the criterion tests the distinct ways of dividing the four modes into two pairs using nullifiers centered on the relevant pump frequencies.

Two-mode bipartitions

  • The two-mode bipartitions tested are {n3z,n3y}|{n4z,n4y}, {n3z,n4z}|{n3y,n4y}, and {n3z,n4y}|{n4z,n3y}.Source: Two-mode bipartitions · p. 8
  • For the third partition, P+(n3,n5) is the z-pump-centered nullifier with frequencies satisfying n3 + n5 = nzpump.Source: Two-mode bipartitions · p. 8

Sufficient conditions for inseparability

A sufficient inseparability condition is that the summed variances of appropriate P and Q nullifiers fall below one, corresponding to squeezing beyond −3 dB.

Sufficient conditions for inseparability

  • The inequalities require the relevant P- and Q-nullifier variances to sum to less than one for four-mode inseparability.Overlapping inseparable four-mode units extend inseparability to the whole wire through transitivity.Source: Sufficient conditions for inseparability · p. 8
  • When the squeezing level exceeds −3 dB, all modes are inseparable under the stated sufficient condition.The condition follows from the nullifier-variance inequalities and was demonstrated experimentally.Source: Sufficient conditions for inseparability · p. 8

EXPERIMENTAL SETUP: EQUIPMENT AND PROCEDURES

The experimental setup description is completed after detailing the procedures used in the main text and supplemental experimental work.

EXPERIMENTAL SETUP: EQUIPMENT AND PROCEDURES

  • The experimental setup and procedures described in the main text are completed by the detailed procedures reported here.The passage marks the end of the setup description.Source: EXPERIMENTAL SETUP: EQUIPMENT AND… · p. 8

Equipment

The apparatus combines temperature-controlled PPKTP crystals in a stabilized bowtie OPO with phase-locked pump and local-oscillator systems for two-tone homodyne detection.

Equipment

  • The bowtie OPO uses two temperature-controlled, orthogonally oriented PPKTP crystals placed at distinct cavity waists.The cavity has two concave and two flat mirrors, including a 5% output coupler at 1064 nm.Source: Equipment · p. 9
  • Polarization crosstalk was tested by second-harmonic generation and minimized by slightly angling the crystals to suppress resonant retroreflection.The retroreflected beam could reach about 10% of the incident seed power before mitigation.Source: Equipment · p. 9
  • Two frequency-doubled Nd:YAG pump lasers were phase-locked with frequency difference 2m∆ω and coupled separately into the two crystals.A superheterodyne setup controlled one laser through its piezotransducer.Source: Equipment · p. 9
  • Two-tone balanced homodyne detection used 95%-efficient InGaAs photodiodes, an EOM-generated local oscillator, and a filter cavity selecting measurement sidebands.The squeezing traces were recorded with a spectrum analyzer and a component-based detection network.Source: Equipment · p. 9
  • The local oscillator was phase-locked to a pump laser, with adjustable frequency offsets used to check the graph.The lock used a 70 MHz acousto-optic shift and synchronized signal generators.Source: Equipment · p. 9

Electronic noise correction

Electronic noise contaminates the raw squeezing measurements, so the experiment distinguishes measured squeezing from a corrected value using shot-noise and electronic-noise variances.

Electronic noise correction

  • Electronic noise lies about 13 dB below shot noise, and correcting for it increases the measured squeezing level by more than 0.2 dB.The correction result is summarized in Table 1.Source: Electronic noise correction · pp. 9–10, Table 1 · p. 10
  • The displayed squeezing traces are raw measurements without electronic-noise correction.The actual squeezing is higher after accounting for detector electronic noise.Source: Electronic noise correction · p. 9
  • The correction relates squeezing, shot-noise, and electronic-noise variances through Vsq, Vsn, and Ven.These variances define the measured and corrected squeezing quantities in the correction equations.Source: Electronic noise correction · pp. 9–10

Role of the EOM phase in nullifier selection

The EOM phase selects quadrature nullifiers without changing their squeezing performance, while frequency-selective checks distinguish valid wire connections from incorrect pairings. Measurements support one 60-mode wire and two independent 30-mode wires, with imbalance identified as a source of graph distortions.

Role of the EOM phase in nullifier selection

Squeezing measurements for the one-wire case

  • One-wire measurements covered 60 modes, limited by EOM measurement ability rather than the wire’s actual mode capacity; noise correction would increase the reported squeezing by 0.2 dB.Source: Squeezing measurements for the… · p. 12

Wrong-frequency checks for the one-wire case

  • Wrong-frequency one-wire checks produce phase-independent antisqueezing because the selected mode pairs satisfy neither pump phase-matching condition.The tested pairs are not symmetric about either pump, matching the observed absence of squeezing.Source: Wrong-frequency checks for the… · p. 12

Squeezing measurements for the two-wire case

  • For two wires, all tested nullifiers symmetric about either pump remain squeezed with constant levels, showing no observed loss of entanglement away from the center.Each wire was measured over a range limited by EOM capability rather than the generated state.Source: Squeezing measurements for the… · p. 12, Two-wire case · p. 12

Wrong-frequency checks for the two-wire case

  • Two-wire wrong-frequency checks show only antisqueezing or shot noise, supporting the absence of connections between the two wires.The authors note that exhaustive cross-wire testing was impractical because of the large number of modes.Source: Wrong-frequency checks for the… · p. 12

SQUEEZING IMPERFECTIONS

  • Unequal squeezing produces spurious correlations between the central z and y modes and unequal edge weights elsewhere, although tuning relative pump intensities can reduce these effects.These distortions remain relevant when evaluating future quantum-processing performance.Source: SQUEEZING IMPERFECTIONS · p. 13
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