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Challenges in Quantum Information Theory: Erasing Correlations and Destroying Entanglement

This paper explores the new challenges in quantum information theory, focusing on reversing protocols, erasing classical correlations, and disentangling power of quantum operations. It discusses resource inequalities, super-dense coding, and the significance of coherent erasure in entanglement-assisted communication. The text delves into unitary gates, asymmetric gate capacities, and the Quantum Reverse Shannon Theorem, highlighting clean protocols and entanglement spread. The paper presents various results and poses questions on nonlocality, quantum capacities, and gate symmetries. It concludes with insights on converting resources asymptotically and the complexities of entanglement dilution.

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Challenges in Quantum Information Theory: Erasing Correlations and Destroying Entanglement

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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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