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Aram Harrow, Bristol Peter Shor, MIT

Erasing correlations, destroying entanglement and other new challenges for quantum information theory. quant-ph/0511219. Aram Harrow, Bristol Peter Shor, MIT. QIP, 19 Jan 2006. outline. General rules for reversing protocols Coherent erasure of classical correlations

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Aram Harrow, Bristol Peter Shor, MIT

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  1. Erasing correlations, destroying entanglement and other new challenges for quantum information theory quant-ph/0511219 Aram Harrow, Bristol Peter Shor, MIT QIP, 19 Jan 2006

  2. outline • General rules forreversing protocols • Coherent erasureof classical correlations • Disentangling powerof quantum operations

  3. Everything is a resource resource inequalities super-dense coding: [q!q] + [qq] > 2[c!c] 2 [q!qq] In fact, [q!q] + [qq] = 2 [q!qq]

  4. Undoing things is also a resource [q!q]y = [qÃq] (relation between time-reversal and exchange symmetry) [qq]y = -[qq] (disentangling power) [q!qq]y = [qÃqq] (?) |0iA |0iB! |0iA and |1iA |1iB!|1iA (coherent erasure??) reversal meaning

  5. [qq!q] > [q!q] - [q!qq] = [q!qq] - [qq] = ([q!q] - [qq]) / 2 What good is coherent erasure? = = entanglement-assisted communication only In fact, these are all equalities! (Proof: reverse SDC.) |xi E Alice (I ­ XxZy)|Fi |Fi |yi X Z |xi |xi Bob |yi |yi a|0iA + b|1iA! a|0iA|0iB + b|1iA|1iB (using [q!qq]) !a|0iB + b|1iB (using [qq!q]) [q!qq] + [qq!q] > [q!q]

  6. application to unitary gates U is a bipartite unitary gate (e.g. CNOT) Known: U > C[c! c] implies U > C[q!qq] Time reversal means: Uy> C [qÃqq] = C [qqÃq] - C [qq] Corollary: If entanglement is free then C!E(U) = CÃE(Uy).

  7. The quest for asymmetric unitary gate capacities the construction: (Um acts on 2m £ 2m dimensions) Um|xiA|0iB = |xiA|xiB for 0 6 x < 2m Um|xiA|yiB = |xiA|y-1iB for 0 <y 6 x < 2m Um|xiA|yiB = |xiA|yiB for 0 6 x < y < 2m Problem: If U is nonlocal, it has nonzero quantum capacities in both directions. Are they equal? Yes, if U is 2£2. No, in general, but for a dramatic separation we will need a gate that violates time-reversal symmetry.

  8. et voilà l’asymétrie! Um|xiA|0iB = |xiA|xiB for 0 6 x < 2m Um|xiA|yiB = |xiA|y-1iB for 0 <y 6 x < 2m Um|xiA|yiB = |xiA|yiB for 0 6 x < y < 2m Um> m[q!qq] Upper bound by simulation: m[q!qq] + O((log m)(log m/e)) ([q!q] + [qÃq]) & Um Similarly, Umy> m[qÃqq] and m[qÃqq] + O((log m)(log m/e)) ([q!q] + [qÃq]) & Umy Meaning: Um¼ m [q!qq] and Umy¼ m [qÃqq] (almost worthless w/o ent. assistance!)

  9. disentanglement clean clean Example: [q!q] > [qq] and [q!q] > -[qq] clean Example: Um> m[qq], but can only destroy O(log2m) [qq] Umy> -m[qq], but can only create O(log2m) [qq] clean clean resource inequalities: means that a­n can be asymptotically converted to b­n while discarding only o(n) entanglement. (equivalently: while generating a sublinear amount of local entropy.)

  10. You can’t just throw it away Q: Why not? A: Given unlimited EPR pairs, try creating the state Hayden & Winter [quant-ph/0204092] proved that this requires ¼n bits of communication.

  11. more relevant examples [Bennett, Devetak, Harrow, Shor, Winter] Quantum Reverse Shannon Theorem for general inputs Input r­n requires I(A;B)r [c!c] + H(N(r)) [qq]. Superpositions of different r­n mean consuming superpositions of different amounts of entanglement: we need either extra cbits, embezzling, or another source of entanglement spread. Entanglement dilution: |yiAB is partially entangled. E = S(yA). |Fi­nE+o(n)!|yi­n Even |Fi­1!|yi­n requires W(n½) cbits (in either direction). OR a size O(n½/e) embezzling state [q-ph/0205100, Hayden-van Dam]

  12. summary • new ideas • coherent erasure • clean protocols • entanglement spread • new results • asymmetric unitary gate capacities • QRST and other converses • new directions • formalizing entanglement spread • clean protocols involving noisy resources (cbits?) READ ALL ABOUT IT! quant-ph/0511219

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