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T he Separability Problem and its Variants in Quantum Entanglement Theory Nathaniel Johnston Institute for Quantum Computing University of Waterloo

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Overview What is Quantum Entanglement? The Separability Problem The Bound Entanglement Problem The Separability from Spectrum Problem

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Overview What is Quantum Entanglement? Weird physical phenomenon Linear algebra works! The Separability Problem The Bound Entanglement Problem The Separability from Spectrum Problem

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Particles can be “linked” Always get correlated measurement results That’s weird! Tensor product of finite- dimensional vector spaces Tensors of rank > 1 exist That’s obvious! Physicist Mathematician What is Quantum Entanglement?

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Pure quantum state: with i.e., with Dual (row) vector: Inner product:

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What is Quantum Entanglement? Tensor product: = = =

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What is Quantum Entanglement? Outer product tensor product: Obtained via “stacking columns”:

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What is Quantum Entanglement? Definition A pure state is separable if it can be written as Otherwise, it is entangled. rank 1 rank 2 (thus entangled)

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What is Quantum Entanglement? Pure state (again): Rank 1 Trace 1 Positive semidefinite equivalent Mixed quantum state: Trace 1 Positive semidefinite equivalent

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What is Quantum Entanglement? Definition A mixed state is separable if it can be written as with each separable. Otherwise, it is entangled. Equivalent: for some positive semidefinite is the “maximally mixed” state. convex combination separable

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Overview What is Quantum Entanglement? The Separability Problem How to determine separability? Positive matrix-valued maps Funky matrix norms The Bound Entanglement Problem The Separability from Spectrum Problem

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The Separability Problem Recall: is separable if we can write for some Definition Given the separability problem is the problem of determining whether ρ is separable or entangled. This is a hard problem! This is an NP-hard problem! (Gurvits, 2003)

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The Separability Problem All states Separable states ρ

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The Separability Problem Define a linear map Γ on by Method 1: “Partial” transpose In matrices:

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The Separability Problem Apply Γ to a separable state: is positive semidefinite Not true for some entangled states: which has eigenvalues 1, 1, 1, and -1. We say that ρ has positive partial transpose (PPT).

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The Separability Problem All states Separable states ρ PPT states

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The Separability Problem Theorem (Størmer, 1963; Woronowicz, 1976; Peres, 1996) Let be a quantum state. If ρ is separable then Furthermore, the converse holds if and only if mn ≤ 6. Separability problem is completely solved when mn ≤ 6 Higher dimensions?

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The Separability Problem Given, define a linear map on by Method 1.1: Positive maps In matrices:

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The Separability Problem Transpose map: Theorem (Horodecki 3, 1996) A quantum state is separable if and only if for all positive maps Definition is positive if whenever positive semidefinite

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The Separability Problem All states Separable states ρ Transpose map

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The Separability Problem The problem: Coming up with positive maps is hard! Proving that a map is positive is NP-hard Current status: Dozens of papers Only a handful of known positive maps

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The Separability Problem Method 2: Norms Definition The operator norm and trace norm of a matrix are defined by: where are the singular values of X.

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The Separability Problem All states ≈ unit ball of Separable states ≈ unit ball of

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The Separability Problem Definition Given define the S(1)-norm via dual Separable version of

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The Separability Problem Theorem (Rudolph, 2000) Let be the dual of the S(1)-norm, defined by A quantum state ρ is separable if and only if

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The Separability Problem All states ≈ unit ball of Separable states ≈ unit ball of

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The Separability Problem The goal: derive bounds for “Swap” operator: “Realignment” map: = 1 if ρ separable because

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ρ σ The Separability Problem Theorem (Chen–Wu, 2003) If then ρ is entangled. The goal: Come up with more bounds on Lower bounds entanglement Upper bounds separability

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Overview What is Quantum Entanglement? The Separability Problem The Bound Entanglement Problem Not all entanglement is “useful” Partial transpose is awesome The Separability from Spectrum Problem

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Bound Entanglement Can we turn mixed entanglement into pure entanglement? ρ ρ ρ

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Bound Entanglement Not always! Theorem (Horodecki 3, 1998) If the quantum state has positive partial transpose then it is bound entangled (i.e., many copies of ρ can not be locally converted into an entangled pure state). Question: Are there more? Or is this “iff”?

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Bound Entanglement All states Separable states PPT states = Bound entangled states Bound entangled states

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Bound Entanglement Let’s phrase the problem mathematically! Recall: for we have Similarly, “Rank 1” and “full rank” versions of same norm

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Bound Entanglement We now want the “rank 2” version of this norm: Also need the “maximally entangled state”: standard basis of

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Bound Entanglement Theorem Define a family of projections P 1, P 2, … recursively as follows: Then there exists non-positive partial transpose bound entanglement (more or less) if and only if up to minor technical details (e.g., n ≥ 4 only)

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Bound Entanglement What do we know so far? Big gap! n = 4, k = 2: equality when k = 1

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Overview What is Quantum Entanglement? The Separability Problem The Bound Entanglement Problem The Separability from Spectrum Problem We only know eigenvalues Want to determine separable/entangled

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Separability from Spectrum Only given eigenvalues of ρ Can we prove ρ is entangled/separable? No: diagonal separable Prove entangled? arbitrary eigenvalues, but always separable

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Separability from Spectrum Sometimes: If all eigenvalues are equal then Prove separable? a separable decomposition Only given eigenvalues of ρ Can we prove ρ is entangled/separable?

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Separability from Spectrum Can also prove separability if ρ is close to Theorem (Gurvits–Barnum, 2002) Let be a mixed state. If then ρ is separable, where is the Frobenius norm. Frobenius norm: eigenvalues of ρ

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Separability from Spectrum All states Separable states Gurvits–Barnum ball

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Separability from Spectrum Definition A quantum state is called separable from spectrum if all quantum states with the same eigenvalues as ρ are separable. only depends on eigenvalues of ρ States in the Gurvits–Barnum ball are separable from spectrum: But there are more!

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Separability from Spectrum All states Separable states Gurvits–Barnum ball Separable from spectrum

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Separability from Spectrum The case of two qubits (i.e., m = n = 2) was solved long ago: Theorem (Verstraete–Audenaert–Moor, 2001) A state is separable from spectrum if and only if What about higher-dimensional systems? eigenvalues, sorted so that λ 1 ≥ λ 2 ≥ λ 3 ≥ λ 4 ≥ 0

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Separability from Spectrum Replace “separable” by “positive partial transpose”. Definition A quantum state is called positive partial transpose (PPT) from spectrum if all quantum states with the same eigenvalues as ρ are PPT.

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Separability from Spectrum All states Separable states Gurvits–Barnum ball Separable from spectrum PPT from spectrum

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Separability from Spectrum PPT from spectrum is completely solved (but complicated) Theorem (Hildebrand, 2007) A state is PPT from spectrum if and only if Recall: separability = PPT when m = 2, n ≤ 3 Thus is separable from spectrum if and only if

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Separability from Spectrum Can PPT from spectrum tell us more about separability from spectrum? Theorem (J., 2013) A state is separable from spectrum if and only if it is PPT from spectrum. Yes! obvious when n ≤ 3 weird when n ≥ 4

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Separability from Spectrum All states Separable states Gurvits–Barnum ball Separable from spectrum PPT from spectrum =

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Separability from Spectrum Sketch of proof. Lemma If then ρ is separable. Write as a block matrix: ρ becomes “more positive” as B becomes small compared to A and C

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Separability from Spectrum Want: every PPT from spectrum to satisfy hypotheses of Lemma. Lemma If then ρ is separable. Not true!

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Separability from Spectrum Instead: for every PPT from spectrum there exists a 2×2 unitary matrix U such that satisfies hypotheses of Lemma. Lemma If then ρ is separable. works for

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Separability from Spectrum Lemma If then ρ is separable. Define Then some intermediate value of t works

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Separability from Spectrum What about separability from spectrum for when m, n ≥ 3? Don’t know! All states Separable states Gurvits–Barnum ball Separable from spectrum PPT from spectrum = All states Separable states Gurvits–Barnum ball Separable from spectrum PPT from spectrum

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