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Entanglement by Path Identity
Mario Krenn, Armin Hochrainer, Mayukh Lahiri, Anton Zeilinger
TL;DR
The paper addresses the lack of investigation of path identity as a route to entanglement generation. It combines coherent superposition of photon pairs from different crystals with overlapping paths, producing multiphoton and high-dimensional entangled states, including arbitrary two-photon high-dimensional states.
Problem
Path identity had not been investigated as a method for generating quantum entanglement, despite prior uses in other photonic contexts.
Source: p. 1Method
The method coherently superposes photon pairs from different crystals and overlaps paths to generate multiphoton and high-dimensional entanglement from separable pairs.
Source: p. 1Results
The schemes produce multiphoton polarization and high-dimensional entangled states, including arbitrary high-dimensional two-photon states.
Source: pp. 1–3Takeaways & Limitations
The approach may suit photonic quantum-computation schemes, particularly in miniaturized compact devices.
Source: p. 4General treatment of temporal coherence in multiphoton experiments remains a topic for future research.
Source: p. 4
Abstract
from arXiv · showhide
Quantum entanglement is one of the most prominent features of quantum mechanics and forms the basis of quantum information technologies. Here we present a novel method for the creation of quantum entanglement in multipartite and high-dimensional systems. The two ingredients are 1) superposition of photon pairs with different origins and 2) aligning photons such that their paths are identical. We explain the experimentally feasible creation of various classes of multiphoton entanglement encoded in polarization as well as in high-dimensional Hilbert spaces -- starting only from non-entangled photon pairs. For two photons, arbitrary high-dimensional entanglement can be created. The idea of generating entanglement by path identity could also apply to other quantum entities than photons. We discovered the technique by analyzing the output of a computer algorithm. This shows that computer designed quantum experiments can be inspirations for new techniques.
ACKNOWLEGDEMENTS
The acknowledgements credit a quantum-optics investigation, helpful discussions, computation resources, and institutional and grant support.
ACKNOWLEGDEMENTS
- The authors thank Melvin for investigating 100 million quantum optical experiments and acknowledge colleagues, computation resources, and Austrian and European funding.Support included the Austrian Academy of Sciences, ERC SIQS Grant No. 600645 EU-FP7-ICT, and Austrian Science Fund projects.Source: ACKNOWLEGDEMENTS · p. 4
APPENDIX I. REALISTIC DIAGRAM OF THE POLARIZATION GHZ SETUP AND COHERENCE TIME REQUIREMENTS
The appendix describes a four-photon polarization GHZ setup and its coherence requirements. Path misalignment changes relative amplitudes without reducing coherence between state terms.
APPENDIX I. REALISTIC DIAGRAM OF THE POLARIZATION GHZ SETUP AND COHERENCE TIME REQUIREMENTS
- The depicted 4-photon GHZ apparatus splits the pump toward crystals 1 and 2, then routes pump and down-conversion paths toward crystals 3 and 4.Type-I SPDC with a diagonally polarized pump can produce the initial horizontal and vertical photon pairs in non-collinear phase-matching conditions.Source: Figure 5 · p. 6, Appendix I · p. 6
- Pump and down-conversion paths must be matched within the relevant coherence time or length so the crystal-pair alternatives remain indistinguishable.The pump paths to crystals 1 and 2, then 3 and 4, are matched pairwise; down-conversion paths ℓ1–ℓ4 are matched within the photon coherence length, with ℓp3 = ℓ1 within the laser coherence time.Source: Appendix I · p. 6
- Misaligned overlapping paths change the relative state amplitudes but do not reduce coherence; undetected photons consequently do not produce four-fold coincidences.Similar conditions apply to the other experimental setups in the main text.Source: Appendix I · p. 7
APPENDIX II. DETAILED CALCULATION OF THE QUANTUM STATES AND THE EFFECT OF MISALIGNMENT
The appendix calculates multiphoton and high-dimensional states while modeling imperfect path overlap. It shows how misalignment changes state weights and how pump-power adjustments can compensate in several schemes.
APPENDIX II. DETAILED CALCULATION OF THE QUANTUM STATES AND THE EFFECT OF MISALIGNMENT
- The appendix models intermediate path misalignment as a variable beam splitter with transmissivity T, between perfect overlap and complete loss at a beam stop.Its detailed four-photon calculation starts from a series expansion of the down-conversion process.Source: APPENDIX II. DETAILED CALCULATION OF… · p. 7
- Misalignment leaves coherence intact but makes state terms unequally weighted, so the state is not maximally entangled; adjusting pump power can compensate.The four-fold calculation discards terms that do not yield one photon in each of paths a, b, c, d, and neglects higher-order terms when g ≪ 1.Source: APPENDIX II. DETAILED CALCULATION OF… · p. 7
- The calculated six-photon polarization state is a GHZ state with horizontal and vertical components whose relative amplitude depends on T.The supplied calculation identifies the result as a six-photon GHZ state and gives its state expression.Source: APPENDIX II. DETAILED CALCULATION OF… · p. 7
- In the W-state example, induced emission is expected to raise noise slightly because the input states match those produced by the crystals.Pump power can compensate for the different coefficients in the state terms.Source: APPENDIX II. DETAILED CALCULATION OF… · pp. 7–8
- The high-dimensional examples use mode shifters with down-conversion crystals to calculate a four-photon, three-dimensional entangled state.The appendix also describes a single-mode analysis using creation and annihilation operators.Source: APPENDIX II. DETAILED CALCULATION OF… · pp. 7–8
- For two-photon high-dimensional states, misalignment unweights the state, but adjusting pump power before each crystal can restore maximal entanglement.The scheme allows adjustable phases and supports arbitrary high-dimensional entangled states.Source: APPENDIX II. DETAILED CALCULATION OF… · p. 8
APPENDIX III. CONSTRUCTION OF GENERAL EXPERIMENTS
The appendix constructs two-photon and multiphoton entangled states by combining crystal-produced photon pairs, with extensions to arbitrary high-dimensional states and several multipartite forms.
Two-Photon Case
- For two photons, coherently pumped crystals and adjustments to mode number and phase allow arbitrary high-dimensional quantum states.The mode number and phase can be adjusted for each term of the resulting superposition.Source: APPENDIX III. CONSTRUCTION OF GENERAL… · p. 8
Multi-Photon Case
- Figure 6 shows a 6-particle, 5-dimensional GHZ configuration and an asymmetric three-photon state with Schmidt-rank vector (4,2,2), using a trigger photon.The trigger photon is detected in T.Source: Figure 6 · p. 8, APPENDIX III. CONSTRUCTION OF GENERAL… · p. 9
- An n-photon GHZ construction uses n/2 crystals firing simultaneously; for n=6, three crystals have five arrangements that yield sixfold coincidences.The construction yields arbitrary (n−1)-dimensional entangled n-photon GHZ states.Source: APPENDIX III. CONSTRUCTION OF GENERAL… · pp. 8–9
- Replacing each crystal with a row producing arbitrary high-dimensional two-photon states generalizes the GHZ construction; the W state is a special case.Figure 7 marks arbitrary mode shifters, such as OAM holograms, on the paths.Source: APPENDIX III. CONSTRUCTION OF GENERAL… · p. 9, Figure 7 · p. 9
APPENDIX IV. EFFICIENCY OF STATE CREATION
State creation is probabilistic, and the appendix defines efficiency by the fraction of possible n-photon events that produce a valid n-fold coincidence.
Efficiency
- A generated n-photon state does not always produce a valid n-fold event with one photon in each path.The efficiency E is the number of valid n-fold events divided by the number of possible n-photon events.Source: APPENDIX IV. EFFICIENCY OF STATE… · p. 9
- For d-dimensional n-particle GHZ states, d combinations yield valid n-fold events, while c crystals allow c^(n/2) possible n-photon combinations.Source: APPENDIX IV. EFFICIENCY OF STATE… · p. 9
- For GHZ states, c = n·d/2, giving E = d/(n·d/2)^(n/2).Source: APPENDIX IV. EFFICIENCY OF STATE… · p. 9
APPENDIX V. HOW WE FOUND THE TECHNIQUE
The authors traced the technique to unusual solutions from a computer algorithm, then tested increasingly restricted setups and extracted the underlying principle.
How We Found the Technique
- The algorithm produced experimentally feasible high-dimensional multiphoton setups, including an unusual three-photon state that prompted closer analysis.The unusual solution used superpositions of multiple SPDC processes.Source: Appendix V · p. 9, Figure 8 · p. 10
- The analyzed solution coherently combined cases with one SPDC event in each crystal and cases with two events in each.Each crystal produced a three-dimensional two-photon entangled state.Source: Figure 8 · p. 10
- After allowing crystal superpositions and overlapping paths, the program still found high-dimensional states using crystals and mode shifters; a further polarization-only restriction also yielded a solution.The authors extracted the principle from that restricted solution and generalized it.Source: Appendix V · pp. 9–10, Figure 8 · p. 10