The computational complexity of entanglement detection

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The computational complexity of entanglement detection. Patrick Hayden Stanford University. Based on 1211.6120, 1301.4504 and 1308.5788 With Gus Gutoski , Daniel Harlow, Kevin Milner and Mark Wilde. How hard is entanglement detection?. - PowerPoint PPT Presentation

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<p>Slide 1</p> <p>The computational complexity of entanglement detectionBased on 1211.6120, 1301.4504 and 1308.5788With Gus Gutoski, Daniel Harlow, Kevin Milner and Mark WildePatrick HaydenStanford University</p> <p>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)Why ask?Provides a natural set of complete problems for many widely studied classes in quantum complexityPersonal motivation:Quantum gravity!Personal frustration at inability to find a fast scramblerPossible implications for the black hole firewall problemEntanglement detection: The platonic ideal</p> <p>YESNOSome complexity classes</p> <p>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)</p> <p>QMA(2)Results: States</p> <p>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</p> <p>The computational universe through the entanglement lens</p> <p>Results: States</p> <p>Pure state circuitProduct output?Trace distanceMixed state circuitProduct output?Trace distanceMixed state circuitSeparable output?1-LOCC distanceBQP-completeQSZK-completeNP-hardQSZK-hardIn QIP(2)Baby steps: Detecting pure product states</p> <p>Baby steps:Detecting pure product states</p> <p>1. QPROD-PURE-STATE is in BQP</p> <p>2. QPROD-PURE-STATE is BQP-hard</p> <p>2. QPROD-PURE-STATE is BQP-hard</p> <p>Results: States</p> <p>Pure state circuitProduct output?Trace distanceMixed state circuitProduct output?Trace distanceMixed state circuitSeparable output?1-LOCC distanceBQP-completeQSZK-completeNP-hardQSZK-hardIn QIP(2)Jaunty stroll:Detecting mixed product states</p> <p>Jaunty stroll:Detecting mixed product states</p> <p>Jaunty stroll:Detecting mixed product states</p> <p>Completeness: YES instances</p> <p>Soundness: NO instances</p> <p>Zero-knowledge (YES instances):Verifier can simulate prover output</p> <p>QPROD-STATE is QSZK-hard</p> <p>Reduction from co-QSD to QPROD-STATE</p> <p>QPROD-STATE and Quantum Error Correction</p> <p>QPROD-STATE:</p> <p>QEC:These are the SAME problem!</p> <p>A: ReferenceB: EnvironmentR: SystemCloning, Black Holes and FirewallsRadial light rays:InOutSingularityUVHawkingRadiationMsgHorizon</p> <p>[Page, Preskill, Susskind 93][Susskind, Thorlacius, Uglum 93]Quantum information appears to be clonedSpacetime structure prevents comparison of the clones (?)Is unitarity safe?2007: H &amp; Preskill study old black holes.(Only just) safe</p> <p>2012: Almheiri et al. consider to be entanglement with late time Hawking photonFirewalls!Cloning, Black Holes and FirewallsRadial light rays:InOutSingularityUVEarlyHawkingRadiationHorizon[Page, Preskill, Susskind 93][Susskind, Thorlacius, Uglum 93]2012: Almheiri et al. consider to be entanglement with late time Hawking photonFirewalls!If black hole entropy is to decrease, must be present in early Hawking radiation.If infalling Bob is to experience thevacuum as he crosses the horizon, must be in infalling Hawking partner. But has cloning really occurred?Do two copies of exist? To test, Bob would need to decode (QEC)the early Hawking radiation: QSZK-hardbut BH lifetime is poly(# qubits).Results: States</p> <p>Pure state circuitProduct output?Trace distanceMixed state circuitProduct output?Trace distanceMixed state circuitSeparable output?1-LOCC distanceBQP-completeQSZK-completeNP-hardQSZK-hardIn QIP(2)Jogging:Detecting mixed separable states</p> <p>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.</p> <p>SummaryEntanglement detection provides a unifying paradigm for parametrizing quantum complexity classesTunable knobs:State versus channelPure versus mixedTrace norm versus 1-LOCC normProduct versus separableImplications for the (worst case) complexity of decoding quantum error correcting codesProvides challenge to the black hole firewall argumentEntanglement detection: The platonic ideal</p> <p>YESNOComplexity of QSEP-STATE?Who knows?</p> <p>Soundness: NO instances</p>

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