# The computational complexity of entanglement detection

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The computational complexity of entanglement detection. Mark M. Wilde Louisiana State University. Based on 1211.6120 and 1308.5788 With Gus Gutoski , Patrick Hayden, and Kevin Milner. How hard is entanglement detection?. - PowerPoint PPT Presentation

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The computational complexity of entanglement detectionBased on 1211.6120 and 1308.5788With Gus Gutoski, Patrick Hayden, and Kevin MilnerMark M. WildeLouisiana State University

How hard is entanglement detection?Given a matrix describing a bipartite state, is the state separable or entangled? NP-hard for d x d, promise gap 1/poly(d) [Gurvits 04 + Gharibian 10]Quasipolynomial time for constant gap [Brandao et al. 10]Probably not the right question for large systems.Given a description of a physical process for preparing a quantum state (i.e. quantum circuit), is the state separable or entangled?Variants:Pure versus mixedState versus channelProduct versus separableChoice of distance measure (equivalently, nature of promise)Entanglement detection: The platonic ideal

YESNOSome complexity classes

P / BPP / BQPNP / MA / QMA AM / QIP(2)QIP = QIP(3)NP / MA / QMA = QIP(1) P / BPP / BQP = QIP(0)QIP = QIP(3) = PSPACE [Jain et al. 09]Cryptographic variant: Zero-knowledgeVerifier, in YES instances, can simulate proverZK / SZK / QSZK = QSZK(2)

QMA(2)Results: States

Pure state circuitProduct output?Trace distanceMixed state circuitProduct output?Trace distanceMixed state circuitSeparable output?1-LOCC distance (1/poly)BQP-completeQSZK-completeNP-hardQSZK-hardIn QIP(2)Results: ChannelsIsometric channelSeparable output?1-LOCC distanceIsometric channelSeparable output?Trace distanceNoisy channelSeparable output?1-LOCC distanceQMA-completeQMA(2)-completeQIP-complete

The computational universe through the entanglement lens

Results: States

Pure state circuitProduct output?Trace distanceMixed state circuitProduct output?Trace distanceMixed state circuitSeparable output?1-LOCC distanceBQP-completeQSZK-completeNP-hardQSZK-hardIn QIP(2)Detecting mixed product states

Detecting mixed product states

Detecting mixed product states

Completeness: YES instances

Soundness: NO instances

Zero-knowledge (YES instances):Verifier can simulate prover output

QPROD-STATE is QSZK-hard

Reduction from co-QSD to QPROD-STATE

Results: States

Pure state circuitProduct output?Trace distanceMixed state circuitProduct output?Trace distanceMixed state circuitSeparable output?1-LOCC distanceBQP-completeQSZK-completeNP-hardQSZK-hardIn QIP(2)Detecting mixed separable states

AB close to separable iff it has a suitable k-extension for sufficiently large k. [BCY 10]Send R to the prover, who will try to produce the k-extension.Use phase estimation to verify that the resulting state is a k-extension.

SummaryEntanglement detection provides a unifying paradigm for parametrizing quantum complexity classesTunable knobs:State versus channelPure versus mixedTrace norm versus 1-LOCC normProduct versus separable

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