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

Debbie Leung1 w/ Ben Toner2, John Watrous1 (0804.4118) w/ Jesse Wang3 (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

Embezzlement of entanglement Approx violation of conservation laws & Entanglement in nonlocal games

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

• 1. System (d-dim)
• 2. State

What it is Cd vector |ψi ∈ Cd

QM101 (notations)

Symbol / Concept

• 1. System (d-dim)
• 2. State
• 3. |ii

What it is Cd vector |ψi ∈ Cd ei =

: 1 : ith entry

QM101 (notations)

Symbol / Concept

• 1. System (d-dim)
• 2. State
• 3. |ii
• 4. {|ii}i=1

d

What it is Cd vector |ψi ∈ Cd ei = Basis for Cd (Computation basis)

: 1 : ith entry

QM101 (notations)

Symbol / Concept

• 1. System (d-dim)
• 2. State
• 3. |ii
• 4. {|ii}i=1

d

e.g. |ψ i = ∑i αi |ii , ∑i |αi|2 = 1 What it is Cd vector |ψi ∈ Cd ei = Basis for Cd

: 1 : ith entry

α1 α2

:

αd

QM101 (notations)

Symbol / Concept

• 5. An operation

applied to the sys What it is Isometries U applied to the vector

QM101 (notations)

Symbol / Concept

• 5. An operation

applied to the sys

• 6. A measurement

along comp basis What it is Isometries U applied to the vector f: Cd → Δd ∑i αi |ii a (|α1|2,|α2|2,...|αd|2)

QM201 (locality and correlations)

Symbol / Concept

• 1. Parties

Alice & Bob What it is CdAdB ≈ CdA ⊗ CdB

QM201 (locality and correlations)

Symbol / Concept

• 1. Parties

Alice & Bob

• 2. |iji = |ii|ji

What it is CdAdB ≈ CdA ⊗ CdB |ii ⊗ |ji

QM201 (locality and correlations)

Symbol / Concept

• 1. Parties

Alice & Bob

• 2. |iji = |ii|ji
• 3. {|iji}

What it is CdAdB ≈ CdA ⊗ CdB |ii ⊗ |ji Tensor product basis

QM201 (locality and correlations)

Symbol / Concept

• 1. Parties

Alice & Bob

• 2. |iji = |ii|ji
• 3. {|iji}
• 4. Local operation

What it is CdAdB ≈ CdA ⊗ CdB |ii ⊗ |ji Tensor product basis UA ⊗ VB

QM201 (locality and correlations)

Symbol / Concept

• 1. Parties

Alice & Bob

• 2. |iji = |ii|ji
• 3. {|iji}
• 4. Local operation
• 5. Product states

What it is CdAdB ≈ CdA ⊗ CdB |ii ⊗ |ji Tensor product basis UA ⊗ VB e.g. |ii |ji e.g. (∑i αi |ii) ⊗ (∑j βj |ji)

QM201 (locality and correlations)

Symbol / Concept

• 1. Parties

Alice & Bob

• 2. |iji = |ii|ji
• 3. {|iji}
• 4. Local operation
• 5. Product states

What it is CdAdB ≈ CdA ⊗ CdB |ii ⊗ |ji Tensor product basis UA ⊗ VB e.g. |ii |ji e.g. (∑i αi |ii) ⊗ (∑j βj |ji)

• 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

• 1. Parties

Alice & Bob

• 2. |iji = |ii|ji
• 3. {|iji}
• 4. Local operation
• 5. Product states
• 6. Entangled states

What it is CdAdB ≈ CdA ⊗ CdB |ii ⊗ |ji Tensor product basis UA ⊗ VB e.g. |ii |ji e.g. (∑i αi |ii) ⊗ (∑j βj |ji) e.g. ∑k αk |ki|ki

• completely correlated measurement outcomes

Schmidt decomposition

• Theorem. Let |ψi ∈ CdA ⊗ CdB , N = dA · dB

Then, ∃ U, V s.t. |ψi = ∑k=1

N αk (U|ki) ⊗ (V|ki).

Schmidt decomposition

• Theorem. Let |ψi ∈ CdA ⊗ CdB , N = dA · dB

Then, ∃ U, V s.t. |ψi = ∑k=1

N αk (U|ki) ⊗ (V|ki).

Pf: express |ψi as ∑ij γij |ii|ji and take the singular value decomposition of [γij] = UT D V where D is diagonal with diagonal entries {αk}.

Schmidt decomposition

• Theorem. Let |ψi ∈ CdA ⊗ CdB , N = dA · dB

Then, ∃ U, V s.t. |ψi = ∑k=1

N αk (U|ki) ⊗ (V|ki).

Pf: express |ψi as ∑ij γij |ii|ji and take the singular value decomposition of [γij] = UT D V where D is diagonal with diagonal entries {αk}. The {αk}'s are called the Schmidt coefficients of |ψi. The Schmidt rank of |ψi = # nonzero Schmidt coeffs.

Schmidt decomposition

• Theorem. Let |ψi ∈ CdA ⊗ CdB , N = dA · dB

Then, ∃ U, V s.t. |ψi = ∑k=1

N αk (U|ki) ⊗ (V|ki).

Pf: express |ψi as ∑ij γij |ii|ji and take the singular value decomposition of [γij] = UT D V where D is diagonal with diagonal entries {αk}. The {αk}'s are called the Schmidt coefficients of |ψi. The Schmidt rank of |ψi = # nonzero Schmidt coeffs. Obs 1: Local operations leave the Schmidt coeffs invariant Obs 2: Conversely, if |ψ1i, |ψ2i have the same set of Schmidt coeffs, then, |ψ1i = U ⊗ V |ψ2i for some isometries U,V.

Schmidt decomposition

• Theorem. Let |ψi ∈ CdA ⊗ CdB , N = dA · dB

Then, ∃ U, V s.t. |ψi = ∑k=1

N αk (U|ki) ⊗ (V|ki).

Relation to entanglement:

• 1. |ψi entangled iff Schmidt rank ≥ 2.
• 2. "Amount" of entanglement E(|ψi)

= entropy of {|ακ|2} = -∑k |αk|2 log |αk|2 (conserved)

Schmidt decomposition

• Theorem. Let |ψi ∈ CdA ⊗ CdB , N = dA · dB

Then, ∃ U, V s.t. |ψi = ∑k=1

N αk (U|ki) ⊗ (V|ki).

Relation to entanglement:

• 1. |ψi entangled iff Schmidt rank ≥ 2.
• 2. "Amount" of entanglement E(|ψi)

= entropy of {|ακ|2} = -∑k |αk|2 log |αk|2 (conserved) In particular, |00iA'B' ↔ |φiA'B' where |φi= a|00i+b|11i, a,b ≠ 0.

Schmidt decomposition

• Theorem. Let |ψi ∈ CdA ⊗ CdB , N = dA · dB

Then, ∃ U, V s.t. |ψi = ∑k=1

N αk (U|ki) ⊗ (V|ki).

Relation to entanglement:

• 1. |ψi entangled iff Schmidt rank ≥ 2.
• 2. "Amount" of entanglement E(|ψi)

= entropy of {|ακ|2} = -∑k |αk|2 log |αk|2 (conserved) In particular, |00iA'B' ↔ |φiA'B' where |φi= a|00i+b|11i, a,b ≠ 0. Not even with a catalyst: |ψiAB |00iA'B' ↔ |ψiAB |φiA'B'

Schmidt decomposition

• Theorem. Let |ψi ∈ CdA ⊗ CdB , N = dA · dB

Then, ∃ U, V s.t. |ψi = ∑k=1

N αk (U|ki) ⊗ (V|ki).

Relation to entanglement:

• 1. |ψi entangled iff Schmidt rank ≥ 2.
• 2. "Amount" of entanglement E(|ψi)

= entropy of {|ακ|2} = -∑k |αk|2 log |αk|2 (conserved) In particular, |00iA'B' ↔ |φiA'B' where |φi= a|00i+b|11i, a,b ≠ 0. Not even with a catalyst: |ψiAB |00iA'B' ↔ |ψiAB |φiA'B' α1 α2 : αN Schmidt coeffs aα1 aα2 : aαN bα1 bα2 : bαN

Schmidt coeffs when Alice and Bob hold multiple systems: If |ψiAB = ∑k αk |kiA |kiB then |ψiAB |00iA'B' = ∑k αk |k0iAA' |k0iBB' Schmidt coeffs: α1, α2, ... αN If |φi= a|00i+b|11i then |ψiAB |φiA'B' = ∑k aαk |k0iAA' |k0iBB' + bαk |k1iAA' |k1iBB' Schmidt coeffs: aα1, aα2, ... aαN , bα1, ... bαN

Octave demonstration with αk ∝ 1/√k. N=8; α1 through α8 : 0.607 0.429 0.350 0.303 0.271 0.248 0.229 0.214 a = 0.8; b = 0.6; a α1 through a α8 , b α1 through b α8 : 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

• verlap of above with α1 through α8 : 0.88030

Octave demonstration with αk ∝ 1/√k. N=8; α1 through α8 : 0.607 0.429 0.350 0.303 0.271 0.248 0.229 0.214 a = 0.8; b = 0.6; a α1 through a α8 , b α1 through b α8 : 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

• verlap of above with α1 through α8 : 0.88030

N=20; overlap = 0.90500 N=45; overlap = 0.92070 N=300; overlap = 0.94378

log N van Dam - Hayden L, Wang

• verlap or fidelity

Embezzlement of entanglement (I)

• Theorem. ∀ ε > 0, ∀ d, ∀ |φiA'B' ∈ Cd ⊗ Cd

∃ N, ∃ |ψiAB ∈ C

N ⊗ C N, ∃ U, V

s.t. (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' ! (while |ψiAB |00iA'B' ↔ |ψiAB |φiA'B')

Embezzlement of entanglement (I)

• Theorem. ∀ ε > 0, ∀ d, ∀ |φiA'B' ∈ Cd ⊗ Cd

∃ N, ∃ |ψiAB ∈ C

N ⊗ C N, ∃ U, V

s.t. (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' ! i.e. hψ|AB hφ|A'B' (UAA' ⊗ VBB') |ψiAB |00iA'B' ≥ 1- ε (while |ψiAB |00iA'B' ↔ |ψiAB |φiA'B')

Embezzlement of entanglement (I)

• Theorem. ∀ ε > 0, ∀ d, ∀ |φiA'B' ∈ Cd ⊗ Cd

∃ N, ∃ |ψiAB ∈ C

N ⊗ C N, ∃ U, V

s.t. (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' ! i.e. hψ|AB hφ|A'B' (UAA' ⊗ VBB') |ψiAB |00iA'B' ≥ 1- ε (while |ψiAB |00iA'B' ↔ |ψiAB |φiA'B') van Dam & Hayden (0201041)

• conceived such possibility !!
• proved the stronger, universal case where the same |ψi

works for all |φi (exchanging the red & blue quantifiers)

• relies heavily on the Schmidt decompositions for the

bipartite setting

2nd method / interpretation:

Embezzlement of entanglement (II)

Goal: (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' !

2nd method / interpretation:

• 1. Choose A = A1 ... An , B = B1 ... Bn , dim(Ai) = dim(Bi) = d

|ψiAB = C ∑r=1

n-1 |00iA1B1 |00iA2B2 ...|00iArBr |φiAr+1Br+1 ... |φiAnBn

Embezzlement of entanglement (II)

Goal: (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' !

2nd method / interpretation:

• 1. Choose A = A1 ... An , B = B1 ... Bn , dim(Ai) = dim(Bi) = d

|ψiAB = C ∑r=1

n-1 |00iA1B1 |00iA2B2 ...|00iArBr |φiAr+1Br+1 ... |φiAnBn

= C ∑r=1

n-1 |00i⊗r |φi⊗n-r

Embezzlement of entanglement (II)

Goal: (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' !

2nd method / interpretation:

• 1. Choose A = A1 ... An , B = B1 ... Bn , dim(Ai) = dim(Bi) = d

|ψiAB = C ∑r=1

n-1 |00iA1B1 |00iA2B2 ...|00iArBr |φiAr+1Br+1 ... |φiAnBn

= C ∑r=1

n-1 |00i⊗r |φi⊗n-r

• 2. Choose UAA' as:

U |i1iA1 |i2iA2 ... |iniAn |iiA' = |iiA1 |i1iA2 ... |in-1iAn |iniA' i.e. U permutes the systems cyclicly.

Embezzlement of entanglement (II)

Goal: (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' ! A1 A A2 An

2nd method / interpretation:

• 1. Choose A = A1 ... An , B = B1 ... Bn , dim(Ai) = dim(Bi) = d

|ψiAB = C ∑r=1

n-1 |00iA1B1 |00iA2B2 ...|00iArBr |φiAr+1Br+1 ... |φiAnBn

= C ∑r=1

n-1 |00i⊗r |φi⊗n-r

• 2. Choose UAA' as:

U |i1iA1 |i2iA2 ... |iniAn |iiA' = |iiA1 |i1iA2 ... |in-1iAn |iniA' i.e. U permutes the systems cyclicly.

Embezzlement of entanglement (II)

Goal: (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' ! A1 A A2 An

• 3. VBB' acts similarly.

An A1 A A2 B1 B2 B Bn

2nd method / interpretation:

• 1. |ψiAB = C ∑r=1

n-1 |00iA1B1 |00iA2B2 ...|00iArBr |φiAr+1Br+1 ... |φiAnBn

= C ∑r=1

n-1 |00i⊗r |φi⊗n-r

Embezzlement of entanglement (II)

Goal: (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' !

2nd method / interpretation:

• 1. |ψiAB = C ∑r=1

n-1 |00iA1B1 |00iA2B2 ...|00iArBr |φiAr+1Br+1 ... |φiAnBn

= C ∑r=1

n-1 |00i⊗r |φi⊗n-r

• 2. (UAA' ⊗ VBB') |ψiAB |00iA'B'

= (UAA' ⊗ VBB') C ∑r=1

n-1|00iA1B1|00iA2B2 ..|00iArBr |φiAr+1Br+1 .. |φiAnBn |00iA'B'

Embezzlement of entanglement (II)

Goal: (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' ! An A1 A A2 B1 B2 B Bn

2nd method / interpretation:

• 1. |ψiAB = C ∑r=1

n-1 |00iA1B1 |00iA2B2 ...|00iArBr |φiAr+1Br+1 ... |φiAnBn

= C ∑r=1

n-1 |00i⊗r |φi⊗n-r

• 2. (UAA' ⊗ VBB') |ψiAB |00iA'B'

= (UAA' ⊗ VBB') C ∑r=1

n-1|00iA1B1|00iA2B2 ..|00iArBr |φiAr+1Br+1 .. |φiAnBn |00iA'B'

= C ∑r=1

n-1 |00iA1B1 |00iA2B2 ...|00iArBr |00iAr+1Br+1 ... |φiAnBn |φiA'B'

Embezzlement of entanglement (II)

Goal: (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' ! An A1 A A2 B1 B2 B Bn

2nd method / interpretation:

• 1. |ψiAB = C ∑r=1

n-1 |00iA1B1 |00iA2B2 ...|00iArBr |φiAr+1Br+1 ... |φiAnBn

= C ∑r=1

n-1 |00i⊗r |φi⊗n-r

• 2. (UAA' ⊗ VBB') |ψiAB |00iA'B'

= (UAA' ⊗ VBB') C ∑r=1

n-1|00iA1B1|00iA2B2 ..|00iArBr |φiAr+1Br+1 .. |φiAnBn |00iA'B'

= C ∑r=1

n-1 |00iA1B1 |00iA2B2 ...|00iArBr |00iAr+1Br+1 ... |φiAnBn |φiA'B'

= (C ∑r=1

n-1 |00i⊗r+1 |φi⊗n-r-1 )AB |φiA'B'

Embezzlement of entanglement (II)

Goal: (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' ! An A1 A A2 B1 B2 B Bn

2nd method / interpretation:

• 1. |ψiAB = C ∑r=1

n-1 |00iA1B1 |00iA2B2 ...|00iArBr |φiAr+1Br+1 ... |φiAnBn

= C ∑r=1

n-1 |00i⊗r |φi⊗n-r

• 2. (UAA' ⊗ VBB') |ψiAB |00iA'B'

= (UAA' ⊗ VBB') C ∑r=1

n-1|00iA1B1|00iA2B2 ..|00iArBr |φiAr+1Br+1 .. |φiAnBn |00iA'B'

= C ∑r=1

n-1 |00iA1B1 |00iA2B2 ...|00iArBr |00iAr+1Br+1 ... |φiAnBn |φiA'B'

= (C ∑r=1

n-1 |00i⊗r+1 |φi⊗n-r-1 )AB |φiA'B'

Embezzlement of entanglement (II)

Goal: (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' ! An A1 A A2 B1 B2 B Bn inner product with |ψiAB is ≥ 1-1/n ∴n = 1/ε suffices.

Summary of 2nd method:

• 1. |ψiAB |00iA'B' = C ∑r=1

n-1 |00i⊗r |φi⊗n-r |00iA'B'

• 2. (UAA' ⊗ VBB') |ψiAB |00iA'B = (C ∑r=1

n-1 |00i⊗r+1 |φi⊗n-r-1 )AB |φiA'B'

Embezzlement of entanglement (II)

Achieves (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' ! Note: U, V do not depend on |φiA'B' .

Summary of 2nd method:

• 1. |ψiAB |00iA'B' = C ∑r=1

n-1 |00i⊗r |φi⊗n-r |00iA'B'

• 2. (UAA' ⊗ VBB') |ψiAB |00iA'B = (C ∑r=1

n-1 |00i⊗r+1 |φi⊗n-r-1 )AB |φiA'B'

Embezzlement of entanglement (II)

Achieves (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' ! Note: U, V do not depend on |φiA'B' . Extension 1: works for any m-party states |φiA'B'

Summary of 2nd method:

• 1. |ψiAB |00iA'B' = C ∑r=1

n-1 |00i⊗r |φi⊗n-r |00iA'B'

• 2. (UAA' ⊗ VBB') |ψiAB |00iA'B = (C ∑r=1

n-1 |00i⊗r+1 |φi⊗n-r-1 )AB |φiA'B'

Embezzlement of entanglement (II)

Achieves (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' ! Note: U, V do not depend on |φiA'B' . Extension 1: works for any m-party states |φiA'B' Extension 2: works for any initial state not just |00iA'B' So, |ψi = C ∑r=1

n-1 |ηi⊗r |φi⊗n-r enables |ψiAB |ηiA'B' ↔ |ψiAB |φiA'B'

Embezzlement of entanglement (II)

Achieves (UAA' ⊗ VBB') |ψiAB |00iA'B' ≈ ε |ψiAB |φiA'B' ! Extension 3: Coherent state exchange, i.e., embezzle / not in superposition

(a |00iAcBc |γiA'B' + b |11iAcBc |ηiA'B') |ψiAB → (a |00iAcBc |γiA'B' + b |11iAcBc |φiA'B' ) |ψiAB

systems that controls whether to embezzle or not

Extension 4 (approx violation of conservation laws)

Suppose operations are restricted and |ηi ↔ |φi . e.g., restricted to local operation, |ηi=|00i, |φi=(|00i+|11i)/√2

Extension 4 (approx violation of conservation laws)

Suppose operations are restricted and |ηi ↔ |φi . e.g., restricted to local operation, |ηi=|00i, |φi=(|00i+|11i)/√2 e.g., |ηi and |φi contain different amount of a conserved quantity

Extension 4 (approx violation of conservation laws)

Suppose operations are restricted and |ηi ↔ |φi . e.g., restricted to local operation, |ηi=|00i, |φi=(|00i+|11i)/√2 e.g., |ηi and |φi contain different amount of a conserved quantity then |ψi = C ∑r=1

n-1 |ηi⊗r |φi⊗n-r enables the approx transformation

(a|0i|γi + b|1i|ηi) |ψi ↔ ε (a|0i|γi + b|1i|φi) |ψi (Applying method 2 conditioned on the control register being 1, note that conditioned permutation respect global conservation.)

Extension 5 (macroscopically-controlled q gates)

e.g., |0iS , |1iS correspond to spin down and up respectively. |1iS is at a higher energy level than |0iS .

Extension 5 (macroscopically-controlled q gates)

e.g., |0iS , |1iS correspond to spin down and up respectively. |1iS is at a higher energy level than |0iS . Want the "X gate": a |0iS + b |1iS ↔ a |1iS + b |0iS but |0iS ↔ |1iS

Extension 5 (macroscopically-controlled q gates)

e.g., |0iS , |1iS correspond to spin down and up respectively. |1iS is at a higher energy level than |0iS . Want the "X gate": a |0iS + b |1iS ↔ a |1iS + b |0iS but |0iS ↔ |1iS Allowed: |riL |0iS ↔ |r-1iL |1iS where |kiL represents k photos in a laser beam.

Extension 5 (macroscopically-controlled q gates)

e.g., |0iS , |1iS correspond to spin down and up respectively. |1iS is at a higher energy level than |0iS . Want the "X gate": a |0iS + b |1iS ↔ a |1iS + b |0iS but |0iS ↔ |1iS Allowed: |riL |0iS ↔ |r-1iL |1iS where |kiL represents k photos in a laser beam. If one compares the photo number in the laser beam before and after the interaction, one can learn whether the spin system is in a up/down state, decohering the qubit.

Extension 5 (macroscopically-controlled q gates)

e.g., |0iS , |1iS correspond to spin down and up respectively. |1iS is at a higher energy level than |0iS . Want the "X gate": a |0iS + b |1iS ↔ a |1iS + b |0iS but |0iS ↔ |1iS Allowed: |riL |0iS ↔ |r-1iL |1iS where |kiL represents k photos in a laser beam. If one compares the photo number in the laser beam before and after the interaction, one can learn whether the spin system is in a up/down state, decohering the qubit. Idea: use |ψiL = ∑r=1

n-1 |riL so

|ψiL (a|0iS + b|1iS) ↔ ∑r=1

n-1 |r-1iL a|1iS + ∑r=1 n-1 |r+1iL b|0iS

Extension 5 (macroscopically-controlled q gates)

e.g., |0iS , |1iS correspond to spin down and up respectively. |1iS is at a higher energy level than |0iS . Want the "X gate": a |0iS + b |1iS ↔ a |1iS + b |0iS but |0iS ↔ |1iS Allowed: |riL |0iS ↔ |r-1iL |1iS where |kiL represents k photos in a laser beam. If one compares the photo number in the laser beam before and after the interaction, one can learn whether the spin system is in a up/down state, decohering the qubit. Idea: use |ψiL = ∑r=1

n-1 |riL so

|ψiL (a|0iS + b|1iS) ↔ ∑r=1

n-1 |r-1iL a|1iS + ∑r=1 n-1 |r+1iL b|0iS

both nearly indistinguishable from |ψiL so X gate nearly coherent.

Limits to embezzlement of entanglement

Qualitative no-go theorem: |ψiAB |00iA'B' ↔ |ψiAB |φiA'B' Embezzlement: ∀ ε > 0, ∀ d, ∀ |φiA'B' ∈ Cd ⊗ Cd ∃ N, ∃ |ψiAB ∈ C

N ⊗ C N, ∃ U, V

s.t. hψ|AB hφ|A'B' (UAA' ⊗ VBB') |ψiAB |00iA'B' ≥ 1- ε So No-go theorem is not robust or continuous enough. Idea: obtain lower bound on ε as a function of N by continuity

• f von Neumann entropy.

Limits to embezzlement of entanglement

Theorem: If ε > 0, |φiA'B' ∈ Cd ⊗ Cd , |ψiAB ∈ C

N ⊗ C N,

and ∃ U, V s.t. hψ|AB hφ|A'B' (UAA' ⊗ VBB') |ψiAB |00iA'B' ≥ 1- ε then ε ≥ 8 [ E(|φi) / (log N + log d) ]2

Limits to embezzlement of entanglement

Theorem: If ε > 0, |φiA'B' ∈ Cd ⊗ Cd , |ψiAB ∈ C

N ⊗ C N,

and ∃ U, V s.t. hψ|AB hφ|A'B' (UAA' ⊗ VBB') |ψiAB |00iA'B' ≥ 1- ε then ε ≥ 8 [ E(|φi) / (log N + log d) ]2 Proof: Let |ωi = (UAA' ⊗ VBB') |ψiAB |00iA'B' Then 1- ε ≤ hψ|AB hφ|A'B' |ωiAA'BB'

Limits to embezzlement of entanglement

Theorem: If ε > 0, |φiA'B' ∈ Cd ⊗ Cd , |ψiAB ∈ C

N ⊗ C N,

and ∃ U, V s.t. hψ|AB hφ|A'B' (UAA' ⊗ VBB') |ψiAB |00iA'B' ≥ 1- ε then ε ≥ 8 [ E(|φi) / (log N + log d) ]2 Proof: Let |ωi = (UAA' ⊗ VBB') |ψiAB |00iA'B' Then 1- ε ≤ hψ|AB hφ|A'B' |ωiAA'BB' ⇒ 2√2 √ε ≥ || |ψihψ|AB ⊗ |φihφ|A'B' - |ωihω|AA'BB' ||1 by relating fidelity and trace distance between pure states

Limits to embezzlement of entanglement

Theorem: If ε > 0, |φiA'B' ∈ Cd ⊗ Cd , |ψiAB ∈ C

N ⊗ C N,

and ∃ U, V s.t. hψ|AB hφ|A'B' (UAA' ⊗ VBB') |ψiAB |00iA'B' ≥ 1- ε then ε ≥ 8 [ E(|φi) / (log N + log d) ]2 Proof: Let |ωi = (UAA' ⊗ VBB') |ψiAB |00iA'B' Then 1- ε ≤ hψ|AB hφ|A'B' |ωiAA'BB' ⇒ 2√2 √ε ≥ || |ψihψ|AB ⊗ |φihφ|A'B' - |ωihω|AA'BB' ||1 ≥ || trBB' |ψihψ|AB ⊗|φihφ|A'B' - trBB' |ωihω|AA'BB' ||1 monotonicity of trace distance under quantum operations

Limits to embezzlement of entanglement

Theorem: If ε > 0, |φiA'B' ∈ Cd ⊗ Cd , |ψiAB ∈ C

N ⊗ C N,

and ∃ U, V s.t. hψ|AB hφ|A'B' (UAA' ⊗ VBB') |ψiAB |00iA'B' ≥ 1- ε then ε ≥ 8 [ E(|φi) / (log N + log d) ]2 Proof: Let |ωi = (UAA' ⊗ VBB') |ψiAB |00iA'B' Then 1- ε ≤ hψ|AB hφ|A'B' |ωiAA'BB' ⇒ 2√2 √ε ≥ || |ψihψ|AB ⊗ |φihφ|A'B' - |ωihω|AA'BB' ||1 ≥ || trBB' |ψihψ|AB ⊗|φihφ|A'B' - trBB' |ωihω|AA'BB' ||1 ≥ | S(trBB' |ψihψ|AB ⊗|φihφ|A'B') – S(trBB'|ωihω|AA'BB') | log N + log d Fannes inequality for von Neumann entropy

Limits to embezzlement of entanglement

Theorem: If ε > 0, |φiA'B' ∈ Cd ⊗ Cd , |ψiAB ∈ C

N ⊗ C N,

and ∃ U, V s.t. hψ|AB hφ|A'B' (UAA' ⊗ VBB') |ψiAB |00iA'B' ≥ 1- ε then ε ≥ 8 [ E(|φi) / (log N + log d) ]2 Proof: Let |ωi = (UAA' ⊗ VBB') |ψiAB |00iA'B' Then 1- ε ≤ hψ|AB hφ|A'B' |ωiAA'BB' ⇒ 2√2 √ε ≥ || |ψihψ|AB ⊗ |φihφ|A'B' - |ωihω|AA'BB' ||1 ≥ || trBB' |ψihψ|AB ⊗|φihφ|A'B' - trBB' |ωihω|AA'BB' ||1 ≥ | S(trBB' |ψihψ|AB ⊗|φihφ|A'B') – S(trBB'|ωihω|AA'BB') | log N + log d ≥ E(|φi) / (log N + log d)

So, embezzlement (and coherent state exchange) can be approximated better and better with larger and larger local dimensions, but never possible exactly. Applications to nonlocal games.

Nonlocal game:

Referee Alice Bob

Nonlocal game:

Referee Alice Bob x y x,y ~ pxy

Nonlocal game:

Referee Alice Bob x y x,y ~ pxy Rxy set of (a,b) that wins if x,y are inputs a b

Nonlocal game:

Referee Alice Bob x y a b Alice and Bob win if (a,b) ∈ Rxy Value of the game G = ω(G) = max win probability x,y ~ pxy Rxy set of (a,b) that wins if x,y are inputs

Nonlocal game:

Referee Alice Bob x y a b Alice and Bob win if (a,b) ∈ Rxy Value of the game G = ω(G) = max win probability Entangled value of G = ω*(G) = sup win probability if Alice and Bob share entanglement x,y ~ pxy Rxy set of (a,b) that wins if x,y are inputs

Nonlocal game:

Referee Alice Bob x y a b Alice and Bob win if (a,b) ∈ Rxy Value of the game G = ω(G) = max win probability Entangled value of G = ω*(G) = sup win probability if Alice and Bob share entanglement x,y ~ pxy Rxy set of (a,b) that wins if x,y are inputs If ω*(G) > ω(G) , the game corresponds to a Bell's inequality where x,y are measurement settings and a,b are outcomes. e.g., x,y,a,b ∈ {0,1}, (a,b) ∈ Rxy iff ab = x⊕y corr to CHSH ineq

Nonlocal game:

Referee Alice Bob x y a b Alice and Bob win if (a,b) ∈ Rxy Value of the game G = ω(G) = max win probability Entangled value of G = ω*(G) = sup win probability if Alice and Bob share entanglement Qn: how much and what type of entanglement is needed to attain the supremum? x,y ~ pxy Rxy set of (a,b) that wins if x,y are inputs

Nonlocal game:

Referee Alice Bob x y a b Alice and Bob win if (a,b) ∈ Rxy Value of the game G = ω(G) = max win probability Entangled value of G = ω*(G) = sup win probability if Alice and Bob share entanglement Qn: how much and what type of entanglement is needed to attain the supremum? Open if sup attained with finite dim if x,y ∈ {0,1,2}, a,b ∈ {0,1} x,y ~ pxy Rxy set of (a,b) that wins if x,y are inputs

Quantum cooperative game:

Strategy: |Ψi, U, V Game: |ξi, Mw , Ml acc/rej

Meas

X Y R

|ξi

EA EB |Ψi U V Referee prepares a quantum state |ξiXYR , sends X to Alice and Y to Bob receives A from Alice and B from Bob measures ABR according to POVM {Mw , Ml} Alice and Bob win if outcome is w. Qn: does sharing entangled state |Ψi increases the winning prob? how much and what entangled state are needed?

Game that cannot be won with finite entanglement:

Strategy: |Ψi, U, V Game: |ξi, Mw , Ml acc/rej

Meas

X Y R

|ξi

EA EB |Ψi U V |ξiXYR = (|0i|00i+|1i|Φi)RXY where |Φi:=(|11i+|22i)/√2 Let |γi = (|000i+|111i) RAB , POVM: Mw = |γihγ|, Ml = I-Macc 1 √2 1 √2

Game that cannot be won with finite entanglement:

Strategy: |Ψi, U, V Game: |ξi, Mw , Ml acc/rej

Meas

X Y R

|ξi

EA EB |Ψi U V |ξiXYR = (|0i|00i+|1i|Φi)RXY where |Φi:=(|11i+|22i)/√2 Let |γi = (|000i+|111i) RAB , POVM: Mw = |γihγ|, Ml = I-Macc 1 √2 1 √2 Then, with coherent state exchange, prob(win) increases with dim(EA,B) but never reaches 1.

Open problem 1 Now that we know there is no bound on the entanglement needed in the optimal prover strategy in general for quantum multi-prover interactive proof system .... if we allow a small deviation from optimal, is there a bound on the amount of entanglement? Simpler question: for cooperative games with fixed small (constant) system dimensions and ², is there a universal (indep of game) upper bound on amt of entanglement that is sufficient to achieve accepting probability ²-close to optimal?

Open problem 2 The coherent state exchange protocol for 3 or more parties can be made universal (just like embezzlement of entanglement) but it is very

• inefficient. Is there a more efficient universal

protocol?