The computational complexity of entanglement detection

Based on 1211.6120, 1301.4504 and 1308.5788With Gus Gutoski, Daniel Harlow, Kevin Milner and Mark Wilde

Patrick HaydenStanford 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 mixed– State versus channel– Product versus separable– Choice of distance measure (equivalently, nature of promise)

Why ask?• Provides a natural set of complete

problems for many widely studied classes in quantum complexity

• Personal motivation:– Quantum gravity!• Personal frustration at inability to find a “fast

scrambler”• Possible implications for the black hole

firewall problem

Entanglement detection: The platonic ideal

αYES

NOα

β

Some complexity classes…

P / BPP / BQP NP / 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 distance

Mixed state circuitProduct output?Trace distance

Mixed state circuitSeparable output?1-LOCC distance (1/poly)

BQP-complete

QSZK-complete

NP-hard QSZK-hard

In QIP(2)

Results: Channels

Isometric channelSeparable output?1-LOCC distance

Isometric channelSeparable output?Trace distance

Noisy channelSeparable output?1-LOCC distance

QMA-complete

QMA(2)-complete

QIP-complete

The computational universe through the entanglement lens

Results: States

Pure state circuitProduct output?Trace distance

Mixed state circuitProduct output?Trace distance

Mixed state circuitSeparable output?1-LOCC distance

BQP-complete

QSZK-complete

NP-hard QSZK-hard

In QIP(2)

Baby steps: Detecting pure product states

Baby steps:Detecting pure product states

1. QPROD-PURE-STATE is in BQP

2. QPROD-PURE-STATE is BQP-hard

2. QPROD-PURE-STATE is BQP-hard

Results: States

Pure state circuitProduct output?Trace distance

Mixed state circuitProduct output?Trace distance

Mixed state circuitSeparable output?1-LOCC distance

BQP-complete

QSZK-complete

NP-hard QSZK-hard

In QIP(2)

Jaunty stroll:Detecting mixed product states

Jaunty stroll:Detecting mixed product states

Jaunty stroll: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

QPROD-STATE and Quantum Error Correction

QPROD-STATE:

QEC:

These are the SAME problem!

A: “Reference”

B: “Environment”

R: “System”

Cloning, Black Holes and Firewalls

Radial light rays:

In Out

SingularityU V

HawkingRadiation

Msg

Horizon

[Page, Preskill, Susskind 93][Susskind, Thorlacius, Uglum 93]

Quantum information appears to be cloned

Spacetime structure prevents comparison of the clones (?)

Is unitarity safe?

2007: H & Preskill study old black holes.(Only just) safe

2012: Almheiri et al. consider φ to be entanglement with late time Hawking photon

Firewalls!

Cloning, Black Holes and Firewalls

Radial light rays:

In Out

SingularityU V

EarlyHawkingRadiation

Horizon

[Page, Preskill, Susskind 93][Susskind, Thorlacius, Uglum 93]

2012: Almheiri et al. consider φ to be entanglement with late time Hawking photonFirewalls!

If black hole entropy is to decrease, φ must be present in early Hawking radiation.

If infalling Bob is to experience thevacuum as he crosses the horizon, φmust be in infalling Hawking partner.

But has cloning really occurred?Do two copies of φ exist?

To test, Bob would need to decode (QEC)the early Hawking radiation: QSZK-hardbut BH lifetime is poly(# qubits).

φφ

Results: States

Pure state circuitProduct output?Trace distance

Mixed state circuitProduct output?Trace distance

Mixed state circuitSeparable output?1-LOCC distance

BQP-complete

QSZK-complete

NP-hard QSZK-hard

In QIP(2)

Jogging:Detecting mixed separable states

ρAB 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.

Summary• Entanglement detection provides a unifying

paradigm for parametrizing quantum complexity classes

• Tunable knobs:– State versus channel– Pure versus mixed– Trace norm versus 1-LOCC norm– Product versus separable

• Implications for the (worst case) complexity of decoding quantum error correcting codes

• Provides challenge to the black hole firewall argument

Entanglement detection: The platonic ideal

αYES

NOα

β

Complexity of QSEP-STATE?

Who knows?

Soundness: NO instances