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Detecting Incapacity Graeme Smith IBM Research Joint Work with John Smolin QECC 2011 December Detecting Incapacity Graeme Smith IBM Research Joint Work with John Smolin QECC 2011 December 6, 2011

Noisy Channel Capacity X N Y p(y|x) Capacity: bits per channel use in the Noisy Channel Capacity X N Y p(y|x) Capacity: bits per channel use in the limit of many channels C = max. X I(X; Y) is the mutual information

Zero-quantum-capacity channels Sym. PPT Q=0 ? Q>0 Zero-quantum-capacity channels Sym. PPT Q=0 ? Q>0

Outline • Zero-capacity channels • Anti-degradable • PPT • Proof that PPT has no Outline • Zero-capacity channels • Anti-degradable • PPT • Proof that PPT has no capacity • General Incapacity criterion • Linear Maps Motivation: unify thinking about incapacity, appeal to physical principles, apply to generalized prob. Models or mobits

Quantum Capacity N. . . E D N N • Want to encode qubits Quantum Capacity N. . . E D N N • Want to encode qubits so they can be recovered after experiencing noise. • Quantum capacity is the maximum rate, in qubits per channel use, at which this can be done. • We’d like to know when Q(N ) > 0.

Quantum Capacity 0 Ã U E B • Coherent Information: Q 1 (N ) Quantum Capacity 0 Ã U E B • Coherent Information: Q 1 (N ) = max S(B)-S(E) (cf Shannon formula) • Q(N ) ¸ Q 1(N ) (Lloyd-Shor-Devetak) • Q(N ) = limn ! 1(1/n) Q 1(N N ) … • Q(N ) Q 1(N ) (Di. Vincenzo-Shor-Smolin ‘ 98)

Zero Quantum Capacity Channels: Symmetric Channels Output symmetric in B and E 0 Ã Zero Quantum Capacity Channels: Symmetric Channels Output symmetric in B and E 0 Ã U E B = U 0 Ã Example: 50% attenuation channel vacuum Input mode 50: 50 environment Output mode B E

Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 0 Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 0 Ã U E B

Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 0 Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 0 Ã Un En Bn

Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 0 Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 0 Ã E Un En D Ã

Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 D Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 D 0 Ã E Ã D Ã Un

Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 D Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 D 0 Ã E Ã D Ã Un So, symmetric channels must have zero quantum capacity. Specifically, the 50% attenuation channel has zero capacity. It will be one of our two zero quantum capacity channels. IMPOSSIBLE!

Zero Quantum Capacity Channels: Positive Partial Transpose • Partial transpose: (|iihj|A B) = |iihj|A Zero Quantum Capacity Channels: Positive Partial Transpose • Partial transpose: (|iihj|A B) = |iihj|A B |kihl| |lihk| • If AB is not positive, then the state is entangled • If AB ¸ 0, it may be entangled, but then it is very noisy. Bound entanglement---can’t get any pure entanglement from it. • A PPT-channel enforces PPT between output and purification of the input: is PPT • Implies Q(N ) = 0, but can have P(N ) > 0

Outline • Zero-capacity channels • Anti-degradable • PPT • Proof that PPT has no Outline • Zero-capacity channels • Anti-degradable • PPT • Proof that PPT has no capacity • General Incapacity criterion • Linear Maps Motivation: unify thinking about incapacity, appeal to physical principles, apply to generalized prob. Models or mobits

PPT has no quantum capacity • Let T(½) = ½T. N is PPT iff PPT has no quantum capacity • Let T(½) = ½T. N is PPT iff is CP

PPT has no quantum capacity • Let T(½) = ½T. N is PPT iff PPT has no quantum capacity • Let T(½) = ½T. N is PPT iff is CP • Say such N could send quantum info

PPT has no quantum capacity • Let T(½) = ½T. N is PPT iff PPT has no quantum capacity • Let T(½) = ½T. N is PPT iff is CP • Say such N could send quantum info • Then

PPT has no quantum capacity • • Let T(½) = ½T. N is PPT PPT has no quantum capacity • • Let T(½) = ½T. N is PPT iff is CP Say such N could send quantum info Then Acting on both sides with T, we get

PPT has no quantum capacity • • Let T(½) = ½T. N is PPT PPT has no quantum capacity • • Let T(½) = ½T. N is PPT iff is CP Say such N could send quantum info Then Acting on both sides with T, we get

PPT has no quantum capacity • • Let T(½) = ½T. N is PPT PPT has no quantum capacity • • Let T(½) = ½T. N is PPT iff is CP Say such N could send quantum info Then Acting on both sides with T, we get • LHS is transpose. RHS is physical. Can’t be!

PPT has no quantum capacity • • Let T(½) = ½T. N is PPT PPT has no quantum capacity • • Let T(½) = ½T. N is PPT iff is CP Say such N could send quantum info Then Acting on both sides with T, we get • LHS is transpose. RHS is physical. Can’t be! • T is continuous, and is PPT when is, so also for capacity.

Outline • Zero-capacity channels • Anti-degradable • PPT • Proof that PPT has no Outline • Zero-capacity channels • Anti-degradable • PPT • Proof that PPT has no capacity • General Incapacity criterion • Linear Maps Motivation: unify thinking about incapacity, appeal to physical principles, apply to generalized prob. Models or mobits

P-commutation • Let R be unphysical on a set S • Say for any P-commutation • Let R be unphysical on a set S • Say for any physical map there’s a physical map with • Then, if is physical, can’t transmit S ,

Is this really more general? Lemma: If R is linear, invertible, preserves system dimension Is this really more general? Lemma: If R is linear, invertible, preserves system dimension and trace, and is pcommutative, it is either of the form R(½) = (1 -p)½T + p I/d or R(½) = (1 -p)½ + p I/d Proof: unitaries Consider conj by Has to be faithful rep. of proj. unitary gp. We know what these are.

Improved P-commutation • We want to move to non-linear maps, R, but it gets Improved P-commutation • We want to move to non-linear maps, R, but it gets very hard to make sure they Pcommute • So, we can generalize the notion of Pcommutation: for a family of unphysical maps • If then can’t send quantum info

Anti-degradable channels • Recall that a channel is antidegradable if there’s an with • Anti-degradable channels • Recall that a channel is antidegradable if there’s an with • Roughly speaking, let R = • This map will clone. • For unitary decoder, let and • Gives

Teleportation P M Teleportation P M

Teleportation P Classical information Unitary rotation recovers the state M other information goes back Teleportation P Classical information Unitary rotation recovers the state M other information goes back in time, wraps around

Teleportation R Classical information Unitary rotation recovers the state M other information goes back Teleportation R Classical information Unitary rotation recovers the state M other information goes back in time, wraps around

Time-traveling information gets confused R Classical information Unitary rotation recovers the state M Now Time-traveling information gets confused R Classical information Unitary rotation recovers the state M Now suppose state is PT invariant---PT on A leaves state alone

Time-traveling information gets confused R Classical information Unitary rotation recovers the state M Other Time-traveling information gets confused R Classical information Unitary rotation recovers the state M Other information gets stuck here because it doesn’t know which direction in time to go---can’t get around the bend!!! Now suppose state is PT invariant---PT on A leaves state alone

Summary • Channels with zero classical capacity are trivial, but there’s lots of structure Summary • Channels with zero classical capacity are trivial, but there’s lots of structure in zero quantum capacity channels • Two known tests for incapacity---symmetric extension and PPT • Both can be understood as specal cases of the House diagram, P-commutation, etc. • Gives operational proof the PPT channels have zero quantum capacity---otherwise we could implement the unphysical time-reversal operation. • To go beyond PPT, need nonlinear R.

Questions • Are there other non-linear forbidden operations that give interesting new channels with Questions • Are there other non-linear forbidden operations that give interesting new channels with no capacity? • Can we apply this to generalized probabilistic theories, mobits, etc. ? Should be yes for mobits. • Given a zero-capacity channel, can we find a “reason” for its incapacity? • Sensible classification of unphysical maps? • Can we make the time-travel story more rigorous?