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