Embezzlement of entanglement Approx violation of conservation laws - PowerPoint PPT Presentation

Embezzlement of entanglement Approx violation of conservation laws & Entanglement in nonlocal games Debbie Leung 1 w/ Ben Toner 2 , John Watrous 1 (0804.4118) w/ Jesse Wang 3 (1311.6842 + ongoing work) Built on initial results by van Dam & Hayden (0201041) QMATH13, Georgia Tech, Oct 09, 2016. 1 U Waterloo 2 CWI/BQP Sol'n 3 Cambridge $ NSERC, CRC, CIFAR

Plan: - Quantum mechanics notations - Locality and correlations - Schmidt decomposition and entanglement - Embezzling of entanglement by reordering Schmidt coeffs - Embezzling of entanglement by superposing different # of entangled states - Violating conservation law by superposing different # of conserved quantities - Limitations to embezzlement - Nonlocal games that cannot be won with finite amount of entanglement

QM101 (notations) Symbol / Concept What it is 1. System (d-dim) C d vector | ψ i ∈ C d 2. State

QM101 (notations) Symbol / Concept What it is 1. System (d-dim) C d vector | ψ i ∈ C d 2. State 3. |i i e i = 0 : i th entry 1 : 0

QM101 (notations) Symbol / Concept What it is 1. System (d-dim) C d vector | ψ i ∈ C d 2. State 3. |i i e i = 0 : i th entry 1 : 0 4. {|i i } i=1 d Basis for C d (Computation basis)

QM101 (notations) Symbol / Concept What it is 1. System (d-dim) C d vector | ψ i ∈ C d 2. State 3. |i i e i = 0 : i th entry 1 : 0 4. {|i i } i=1 d Basis for C d α 1 α 2 : e.g. | ψ i = ∑ i α i |i i , ∑ i | α i | 2 = 1 α d

QM101 (notations) Symbol / Concept What it is 5. An operation Isometries U applied to the sys applied to the vector

QM101 (notations) Symbol / Concept What it is 5. An operation Isometries U applied to the sys applied to the vector f: C d → Δ d 6. A measurement along comp basis ∑ i α i |i i a (| α 1 | 2 ,| α 2 | 2 ,...| α d | 2 )

QM201 (locality and correlations) Symbol / Concept What it is 1. Parties C dAdB ≈ C dA ⊗ C dB Alice & Bob

QM201 (locality and correlations) Symbol / Concept What it is 1. Parties C dAdB ≈ C dA ⊗ C dB Alice & Bob 2. |ij i = |i i |j i |i i ⊗ |j i

QM201 (locality and correlations) Symbol / Concept What it is 1. Parties C dAdB ≈ C dA ⊗ C dB Alice & Bob 2. |ij i = |i i |j i |i i ⊗ |j i 3. {|ij i } Tensor product basis

QM201 (locality and correlations) Symbol / Concept What it is 1. Parties C dAdB ≈ C dA ⊗ C dB Alice & Bob 2. |ij i = |i i |j i |i i ⊗ |j i 3. {|ij i } Tensor product basis 4. Local operation U A ⊗ V B

QM201 (locality and correlations) Symbol / Concept What it is 1. Parties C dAdB ≈ C dA ⊗ C dB Alice & Bob 2. |ij i = |i i |j i |i i ⊗ |j i 3. {|ij i } Tensor product basis 4. Local operation U A ⊗ V B 5. Product states e.g. |i i |j i e.g. ( ∑ i α i |i i ) ⊗ ( ∑ j β j |j i )

QM201 (locality and correlations) Symbol / Concept What it is 1. Parties C dAdB ≈ C dA ⊗ C dB Alice & Bob 2. |ij i = |i i |j i |i i ⊗ |j i 3. {|ij i } Tensor product basis 4. Local operation U A ⊗ V B 5. Product states e.g. |i i |j i e.g. ( ∑ i α i |i i ) ⊗ ( ∑ j β j |j i ) - 2 measurements applied separately to the two sys result in independent outcomes (no mutual information) - holds with any local operation applied before the meas

QM201 (locality and correlations) Symbol / Concept What it is 1. Parties C dAdB ≈ C dA ⊗ C dB Alice & Bob 2. |ij i = |i i |j i |i i ⊗ |j i 3. {|ij i } Tensor product basis 4. Local operation U A ⊗ V B 5. Product states e.g. |i i |j i e.g. ( ∑ i α i |i i ) ⊗ ( ∑ j β j |j i ) e.g. ∑ k α k |k i |k i 6. Entangled states - completely correlated measurement outcomes

Schmidt decomposition Theorem. Let | ψ i ∈ C dA ⊗ C dB , N = d A · d B N α k (U|k i ) ⊗ (V|k i ). Then, ∃ U, V s.t. | ψ i = ∑ k=1

Schmidt decomposition Theorem. Let | ψ i ∈ C dA ⊗ C dB , N = d A · d B N α k (U|k i ) ⊗ (V|k i ). Then, ∃ U, V s.t. | ψ i = ∑ k=1 Pf: express | ψ i as ∑ ij γ ij |i i |j i and take the singular value decomposition of [ γ ij ] = U T D V where D is diagonal with diagonal entries { α k }.

Schmidt decomposition Theorem. Let | ψ i ∈ C dA ⊗ C dB , N = d A · d B N α k (U|k i ) ⊗ (V|k i ). Then, ∃ U, V s.t. | ψ i = ∑ k=1 Pf: express | ψ i as ∑ ij γ ij |i i |j i and take the singular value decomposition of [ γ ij ] = U T D V where D is diagonal with diagonal entries { α k }. The { α k }'s are called the Schmidt coefficients of | ψ i . The Schmidt rank of | ψ i = # nonzero Schmidt coeffs.

Schmidt decomposition Theorem. Let | ψ i ∈ C dA ⊗ C dB , N = d A · d B N α k (U|k i ) ⊗ (V|k i ). Then, ∃ U, V s.t. | ψ i = ∑ k=1 Pf: express | ψ i as ∑ ij γ ij |i i |j i and take the singular value decomposition of [ γ ij ] = U T D V where D is diagonal with diagonal entries { α k }. The { α k }'s are called the Schmidt coefficients of | ψ i . The Schmidt rank of | ψ i = # nonzero Schmidt coeffs. Obs 1: Local operations leave the Schmidt coeffs invariant Obs 2: Conversely, if | ψ 1 i , | ψ 2 i have the same set of Schmidt coeffs, then, | ψ 1 i = U ⊗ V | ψ 2 i for some isometries U,V.

Schmidt decomposition Theorem. Let | ψ i ∈ C dA ⊗ C dB , N = d A · d B N α k (U|k i ) ⊗ (V|k i ). Then, ∃ U, V s.t. | ψ i = ∑ k=1 Relation to entanglement: 1. | ψ i entangled iff Schmidt rank ≥ 2. 2. "Amount" of entanglement E(| ψ i ) = entropy of {| α κ | 2 } = - ∑ k | α k | 2 log | α k | 2 (conserved)

Schmidt decomposition Theorem. Let | ψ i ∈ C dA ⊗ C dB , N = d A · d B N α k (U|k i ) ⊗ (V|k i ). Then, ∃ U, V s.t. | ψ i = ∑ k=1 Relation to entanglement: 1. | ψ i entangled iff Schmidt rank ≥ 2. 2. "Amount" of entanglement E(| ψ i ) = entropy of {| α κ | 2 } = - ∑ k | α k | 2 log | α k | 2 (conserved) In particular, |00 i A'B' ↔ | φ i A'B' where | φ i = a|00 i +b|11 i , a,b ≠ 0.

Schmidt decomposition Theorem. Let | ψ i ∈ C dA ⊗ C dB , N = d A · d B N α k (U|k i ) ⊗ (V|k i ). Then, ∃ U, V s.t. | ψ i = ∑ k=1 Relation to entanglement: 1. | ψ i entangled iff Schmidt rank ≥ 2. 2. "Amount" of entanglement E(| ψ i ) = entropy of {| α κ | 2 } = - ∑ k | α k | 2 log | α k | 2 (conserved) In particular, |00 i A'B' ↔ | φ i A'B' where | φ i = a|00 i +b|11 i , a,b ≠ 0. Not even with a catalyst: | ψ i AB |00 i A'B' ↔ | ψ i AB | φ i A'B'

Schmidt decomposition Theorem. Let | ψ i ∈ C dA ⊗ C dB , N = d A · d B N α k (U|k i ) ⊗ (V|k i ). Then, ∃ U, V s.t. | ψ i = ∑ k=1 Relation to entanglement: 1. | ψ i entangled iff Schmidt rank ≥ 2. 2. "Amount" of entanglement E(| ψ i ) = entropy of {| α κ | 2 } = - ∑ k | α k | 2 log | α k | 2 (conserved) In particular, |00 i A'B' ↔ | φ i A'B' where | φ i = a|00 i +b|11 i , a,b ≠ 0. Not even with a catalyst: | ψ i AB |00 i A'B' ↔ | ψ i AB | φ i A'B' α 1 a α 1 b α 1 Schmidt α 2 a α 2 b α 2 coeffs : : : α N a α N b α N

Schmidt coeffs when Alice and Bob hold multiple systems: If | ψ i AB = ∑ k α k |k i A |k i B then | ψ i AB |00 i A'B' = ∑ k α k |k0 i AA' |k0 i BB' Schmidt coeffs: α 1 , α 2 , ... α N If | φ i = a|00 i +b|11 i then | ψ i AB | φ i A'B' = ∑ k a α k |k0 i AA' |k0 i BB' + b α k |k1 i AA' |k1 i BB' Schmidt coeffs: a α 1 , a α 2 , ... a α N , b α 1 , ... b α N

Octave demonstration with α k ∝ 1/ √ k. N=8; α 1 through α 8 : 0.607 0.429 0.350 0.303 0.271 0.248 0.229 0.214 a = 0.8; b = 0.6; a α 1 through a α 8 , b α 1 through b α 8 : 0.485 0.343 0.280 0.243 0.217 0.198 0.183 0.172 0.364 0.257 0.210 0.182 0.163 0.149 0.138 0.129 sorting the above: 0.485 0.364 0.343 0.280 0.257 0.243 0.217 0.210 0.198 0.183 0.182 0.172 0.163 0.149 0.138 0.129 overlap of above with α 1 through α 8 : 0.88030

Octave demonstration with α k ∝ 1/ √ k. N=8; α 1 through α 8 : 0.607 0.429 0.350 0.303 0.271 0.248 0.229 0.214 a = 0.8; b = 0.6; a α 1 through a α 8 , b α 1 through b α 8 : 0.485 0.343 0.280 0.243 0.217 0.198 0.183 0.172 0.364 0.257 0.210 0.182 0.163 0.149 0.138 0.129 sorting the above: 0.485 0.364 0.343 0.280 0.257 0.243 0.217 0.210 0.198 0.183 0.182 0.172 0.163 0.149 0.138 0.129 overlap of above with α 1 through α 8 : 0.88030 N=20; overlap = 0.90500 N=45; overlap = 0.92070 N=300; overlap = 0.94378

overlap or fidelity L, Wang van Dam - Hayden log N

Embezzlement of entanglement (I) Theorem. ∀ ε > 0, ∀ d, ∀ | φ i A'B' ∈ C d ⊗ C d N ⊗ C N , ∃ U, V ∃ N, ∃ | ψ i AB ∈ C s.t. (U AA' ⊗ V BB' ) | ψ i AB |00 i A'B' ≈ ε | ψ i AB | φ i A'B' ! (while | ψ i AB |00 i A'B' ↔ | ψ i AB | φ i A'B' )

Embezzlement of entanglement (I) Theorem. ∀ ε > 0, ∀ d, ∀ | φ i A'B' ∈ C d ⊗ C d N ⊗ C N , ∃ U, V ∃ N, ∃ | ψ i AB ∈ C s.t. (U AA' ⊗ V BB' ) | ψ i AB |00 i A'B' ≈ ε | ψ i AB | φ i A'B' ! i.e. h ψ | AB h φ | A'B' (U AA' ⊗ V BB' ) | ψ i AB |00 i A'B' ≥ 1- ε (while | ψ i AB |00 i A'B' ↔ | ψ i AB | φ i A'B' )