G Toy model

, (92)

where

¯λAB =













(F_A¹? F_B¹)(ω_AB) (F_A¹? F_B²)(ω_AB)

...

(F_A^KdA ? F_B^KdB)(ωAB)













, (93)

is the locally tomographic part and

ηAB=









F_AB¹ (ωAB) ...

F_AB^{KdAdB−KdAKdB}(ωAB)









, (94)

is the holistic part. The probability for a joint outcomeF_A? F_B is given by:

(FA? FB)(ωAB) = ( ¯EFA ⊗E¯FB)·λ¯AB (95) Joint local effects are computed using the locally tomographic part of the state space only.

F.4 The reduced state space

The reduced states for Alice and Bob can be computed as follows:

ω¯A = (¯Γ^dA(IA)⊗u_B)¯λAB (96) ω¯_B = (u_A⊗Γ¯^dB(IB))¯λ_AB (97) Reduced states are determined using only the locally tomographic part of the global state.

G Toy model

In this appendix we show that the toy model introduced in section 4 meets consistency constraints C1 -C5(apart from associativity of the ?product).

G.1 Constraints C1 and C2

It is immediate that consistency constraintsC1and C2 are met by the toy model.

G.2 Constraint C3

We prove(FA? FB)(ψA⊗φB) =FA(ψA)FB(φB): (F_A? F_B)(ψ_A⊗φ_B)

= tr|ψihψ|^⊗2_A |φihφ|^⊗2_B ( ˆFAFˆB+tr( ˆFAFˆB)

tr(S_AS_B)AAAB)

=tr|ψihψ|^⊗2_A |φihφ|^⊗2_B Fˆ_AFˆ_B

=FA(ψA)FB(φB) . (98)

In the penultimate line we have used the fact that the overlap of product states|ψihψ|^⊗2_A |φihφ|^⊗2_B andAAAB

is0.

G.3 Constraint C4

In the following it will occasionally useful to label the two copies of C^dA with 1 and 3 and to label the two copies ofC^dB with 2 and 4. We write S˜_A for the normalised projector onto the symmetric subspace of (C^dA)^⊗2. We make use of the identityS= ¹₂(I+ SWAP) throughout this section.

In this section we show that normalised conditional states for Alice are valid states of a C^dA system. We first show this for the specific case where the state is conditioned on the unit effect, i.e. is a reduced state.

A reduced state of Alice for a bi-partite system in pure state|ψ_ABihψ_AB|^⊗2 is:

ω¯_A = tr_BS_B|ψ_ABihψ_AB|^⊗2+ S_A

trS_Atr_ABA_AA_B|ψ_ABihψ_AB|^⊗2 . (99) We show that these reduced states lie in the convex hull of|ψ_Aihψ_A|^⊗2.

Lemma 9. S_B|ψ_ABi^⊗2 =S_AS_B|ψ_ABi^⊗2 Proof.

|ψ_ABi^⊗2 =αi1i2αi3i4|i₁i₂i₃i₄i . (100) S_B|ψ_ABi^⊗2 = 1

2αi1i2αi3i4(|i₁i₂i₃i₄i+|i₁i₄i₃i₂i). (101) Let us relabeli₁ ↔i₃ in the last term:

S_B|ψ_ABi^⊗2 = 1

2(α_i1i2α_i3i4|i₁i₂i₃i₄i+α_i3i2α_i1i4|i₃i₄i₁i₂i). (102) S_AS_B|ψ_ABi^⊗2 = 1

4α_i1i2α_i3i4(|i₁i₂i₃i₄i+|i₁i₄i₃i₂i+|i₃i₂i₁i₄i+|i₃i₄i₁i₂i) . (103) Let us relabel i1 ↔i3 in the second term , i2 ↔i4 in the penultimate term andi1 ↔i3 and i2 ↔i4 in the last term :

SASB|ψ_ABi^⊗2= 1

4(αi1i2αi3i4|i₁i2i3i4i+αi3i2αi1i4|i₃i4i1i2i (104) +α_i1i4α_i3i2|i₃i₄i₁i₂i+α_i1i2α_i3i4|i₁i₂i₃i₄i) = 1

2(α_i1i2α_i3i4|i₁i₂i₃i₄i+α_i3i2α_i1i4|i₃i₄i₁i₂i). (105)

From the above lemma we can write a reduced state ω:¯ ω¯A = tr_BSASB|ψ_ABihψ_AB|^⊗2+ S_A

trSA

tr_ABAAAB|ψ_ABihψ_AB|^⊗2 . (106) Lemma 10. The reduced state ¯ω_A can be written as:

ω¯A=SA(ρA⊗ρA)SA+ (1−tr(SA(ρA⊗ρA)SA)) ˜SA , (107) whereρ_A= tr_B(|ψihψ|_AB).

Proof. We first show that:

tr_B(S_AS_B|ψihψ|^⊗2_AB) =S_A(ρ_A⊗ρ_A)S_A . (108) From the proof of Lemma 9:

S_AS_B|ψ_ABi^⊗2 = 1

4α_i1i2α_i3i4(|i₁i₂i₃i₄i+|i₁i₄i₃i₂i+|i₃i₂i₁i₄i+|i₃i₄i₁i₂i) . (109) Hence:

S_AS_B|ψ_ABihψ_AB|^⊗2 =1

4α_i1i2α_i3i4α¯_j1b1α¯_j3b2(|i₁i₂i₃i₄ihj₁j₂j₃j₄| (110) +|i₁i4i3i2ihj₁j2j3j4|+|i₃i2i1i4ihj₁j2j3j4|+|i₃i4i1i2ihj₁j2j3j4|). (111)

which implies:

tr_B(S_AS_B|ψihψ|^⊗2_AB) = 1

2α_i1b1α_i3b2α¯_j1b1α¯_j3b2(|i₁i₃ihj₁j₃|+|i₃i₁ihj₁j₃|) . (112) We now computeS_A(ρ_A⊗ρ_A)S_A.

|ψihψ|_AB=αi1i2α¯j1j2|i₁i₂ihj₁j₂|. (113) ρA= tr_B(|ψihψ|_AB) =αi1b1α¯j1b1|i₁ihj₁|. (114) ρ_A⊗ρ_A =α_i1b1α_i3b2α¯_j1b1α¯_j3b2|i₁i₃ihj₁j₃|. (115) S_A(ρ_A⊗ρ_A) = 1

2α_i1b1α_i3b2α¯_j1b1α¯_j3b2(|i₁i₃ihj₁j₃|+|i₃i₁ihj₁j₃|) . (116) S_A(ρ_A⊗ρ_A)S_A =1

4α_i1b1α_i3b2α¯_j1b1α¯_j3b2(|i₁i3ihj₁j3|+|i₃i1ihj₁j3|) (117) +1

4α_i1b1α_i3b2α¯_j1b1α¯_j3b2(|i₁i₃ihj₃j₁|+|i₃i₁ihj₃j₁|) . (118) In the second term we relabel j1 ↔j3,i1 ↔i3 and b1 ↔b2 to obtain:

S_A(ρ_A⊗ρ_A)S_A = 1

2(α_i1b1α_i3b2α¯_j1b1α¯_j3b2(|i₁i₃ihj₁j₃|+|i₃i₁ihj₁j₃|)) . (119) Hence,

ω¯_A=S_A(ρ_A⊗ρ_A)S_A+ (1−tr(|ψihψ|^⊗2_ABS_AS_B)) ˜S_A

=S_A(ρ_A⊗ρ_A)S_A+ (1−tr(S_A(ρ_A⊗ρ_A)S_A)) ˜S_A . (120)

Lemma 11. The reduced states ¯ωA belong to the convex hull of the local pure states |ψihψ|^⊗2. Proof. By Lemma10 the reduced state can be written as:

ω¯_A=S_A(ρ_A⊗ρ_A)S_A+ (1−tr(S_A(ρ_A⊗ρ_A)S_A)) ˜S_A , (121) whereρ_A= tr_B(|ψihψ|_AB). In the following we drop the Alabel.

ρ=^X

i

αi|iihi|, (122)

Here the |ii are not necessarily orthogonal. The trace of ρ is ^P_iαi = 1, where αi > 0. Let us write Φ_ij = ^√1

2(|i, ji+|j, ii) and observe:

X

i6=j

|Φ_ijihΦ_ij|= 2^X

i<j

|Φ_ijihΦ_ij|=^X

i6=j

(|i, jihi, j|+|i, jihj, i|) . (123) Consider the (not necessarily normalised) matrix

S(ρ⊗ρ)S=S(^X

ij

α_iα_j|i, jihi, j|)S =^X

i

α²_i|i, iihi, i|+^X

i<j

α_iα_j|Φ_ijihΦ_ij|, (124) The trace of this matrix is 1−^P_i<jαiαj; hence:

ω¯ =^X

i

α²_i|i, iihi, i|+^X

i<j

αiαj|Φ_ijihΦ_ij|+^X

i<j

αiαjS .˜ (125)

We now show that this arbitrary mixed state ¯ω can be expressed as a convex combination of local pure states|ψihψ|^⊗2. Consider the general vector

|ψi=^X

i

e^iθj√

α_j|ji , (126)

whereαj ≥0 and for allj. Normalisation implies^P_jαj = 1. Now, let us write the pure product state

|ψihψ|^⊗2 = ^X

j,k,j⁰,k⁰

e^iθje^iθke^−iθj0e^−iθk0√

α_jα_kα_j⁰α_k⁰|j, kihj⁰, k⁰|. (127) Let us make the following observations. When j6=j⁰:

Z π

−π

e^−iθje^iθj0dθ_jdθ_j⁰ = 0 . (128) When j=j⁰:

Z π

−π

e^−iθje^iθj0dθ_jdθ_j⁰ = (2π)² . (129) Now consider:

Eθi|ψihψ|^⊗2 = 1 (2π)⁴

Z π

−π

X

j,k,j⁰,k⁰

e^iθje^iθke^−iθj0e^−iθk0√

αjαkαj⁰αk⁰|j, kihj⁰, k⁰|dθ_jdθkdθj⁰dθk⁰ . (130) The non zero contributions will arise from the following terms:

j=j⁰ =k=k⁰:

Z π

−π

|e^iθj|⁴dθjdθkdθ_j⁰dθ_k⁰ = (2π)⁴ . (131) j=j⁰ 6=k=k⁰:

Z π

−π

|e^iθj|²|e^iθk|²dθjdθ_kdθ_j⁰dθ_k⁰ = (2π)⁴ . (132) j=k⁰ 6=k=j⁰:

Z π

−π

|e^iθj|²|e^iθk|²dθ_jdθ_kdθ_j⁰dθ_k⁰ = (2π)⁴ . (133) All other contributions will be zero.

Now, we write the mixed state corresponding to the uniform average over all values of the phases θi, ω¯1 =Eθi|ψihψ|^⊗2=^X

j,k

αjαk|j, kihj, k|+^X

j6=k

αjαk|j, kihk, j|, (134) where the first term arises from contributions (131) and (132) and the second term arises from contribu-tion (133). Then we can write

ω¯1 =^X

i

α²_i|i, iihi, i|+ 2^X

i<j

αiαj|Φ_ijihΦ_ij|, (135) Let us take the state:

ω¯2=^X

i

α²_i|i, iihi, i|+ 2^X

i<j

αiαjS .˜ (136)

This is a mixture of states of the form|ψihψ|^⊗2 since ˜S =^R |ψihψ|^⊗2dψ and^P_iα²_i + 2^P_i<jα_iα_j = 1. If we take the mixture ¹₂(¯ω₁+ ¯ω₂) we obtain (125).

We now consider the more general case where Alice’s state is conditioned on an arbitrary effectF_B. The conditional state for Alice given one of Bob’s effectsFB is:

ω¯_A|FB = Tr_B(S_AFˆ_B|ψihψ|^⊗2_AB) + Tr(|ψihψ|^⊗2_ABA_AA_B)Tr( ˆF_B) Tr(SB)

S˜_A . (137)

Although effects of the formFˆ_B =|φihφ|^⊗2 are not valid (since the complement effects would not be of the required form), we calculate the conditional state for such effects, as this will allow us to later determine conditional states for general effectsFˆ_B =^P_iαi|φ_iihφ_i|^⊗2.

Lemma 12. For ˆF_B =|φihφ|^⊗2:

Tr_B(S_AFˆ_B|ψihψ|^⊗2_AB) =S_A(ρ^ψ_A|φ⊗ρ^ψ_A|φ)S_A =ρ^ψ_A|φ⊗ρ^ψ_A|φ, (138) whereρ^ψ_A|φ= Tr((IA⊗ |φihφ|_B)|ψihψ|_AB) .

Proof.

Tr_B(S_AFˆ_B|ψihψ|^⊗2_AB) =S_ATr_B( ˆF_B|ψihψ|^⊗2_AB) =S_A(ρ^ψ_A|φ⊗ρ^ψ_A|φ) . (139) Let

ρ^ψ_A|φ= Tr (IA⊗ |φihφ|_B)|ψihψ|_AB= ^X

i1,j1

α_i1,φα¯_j1,φ|i₁ihj₁|. (140) We assume without loss of generality that|φi is one of the basis vectors|ii.

ρ^ψ_A|φ⊗ρ^ψ_A|φ= ^X

i1,i3,j1,j3

α_i1,φα¯_j1,φα_i3,φα¯_j3,φ|i₁i₃ihj₁j₃|. (141)

SA(ρA|φ⊗ρA|φ) =1 2( ^X

i1,i3,j1,j3

αi1,φα¯j1,φαi3,φα¯j3,φ|i₁i3ihj₁j3|

+ ^X

i1,i3,j1,j3

αi1,φα¯j1,φαi3,φα¯j3,φ|i₃i1ihj₁j3|) . (142)

We relabeli₁↔i₃ on the last line to obtainS_A(ρ_A|φ⊗ρ_A|φ) = (ρ_A|φ⊗ρ_A|φ).

S_A(ρ_A|φ⊗ρ_A|φ)S_A= 1 4( ^X

i1,i3,j1,j3

α_i1,φα¯_j1,φα_i3,φα¯_j3,φ|i₁i₃ihj₁j₃|

+ ^X

i1,i3,j1,j3

α_i1,φα¯_j1,φα_i3,φα¯_j3,φ|i₃i₁ihj₁j₃|+ ^X

i1,i3,j1,j3

α_i1,φα¯_j1,φα_i3,φα¯_j3,φ|i₁i₃ihj₃j₁|

+ ^X

i1,i3,j1,j3

α_i1,φα¯_j1,φα_i3,φα¯_j3,φ|i₃i₁ihj₃j₁|) . (143) We relabelj1 ↔j3 on the last two lines to obtainS_A(ρ_A|φ⊗ρ_A|φ) =S_A(ρ_A|φ⊗ρ_A|φ)S_A.

We need one more lemma before proving that normalised conditional states belong to the convex hull of the local pure states|ψihψ|^⊗2_A .

Lemma 13. If S(ρ⊗ρ)S =ρ⊗ρ with Tr(ρ) = 1 thenρ⊗ρ=^P_ipi|ψ_iihψ_i|^⊗2. Proof. By Lemma11 this is a valid reduced state and belongs to conv(|ψihψ|^⊗2).

We observe that since all pure states are such thatS(|ψihψ|^⊗2)S=|ψihψ|^⊗2 andTr(|ψihψ|^⊗2) = 1, we can characterise the state space of the systems of the toy model as being given byconv(ρ⊗ρ)for all normalised density operators such thatS(ρ⊗ρ)S=ρ⊗ρ.

Lemma 14. The normalised conditional states ˜ω_A|FB belong to the convex hull of the local pure states

|ψihψ|^⊗2.

Proof. We first show that the conditional state is a valid local state for effects ˆF_B =|φihφ|^⊗2. As a conditional state, this state can be subnormalised.

ω¯A|F_B = Tr_B(SAFˆB|ψihψ|^⊗2_AB) +cS˜A , (144)

wherec= Tr(|ψihψ|^⊗2_ABAAAB)^{Tr( ˆ}_Tr(S^FB)

B). By Lemma12 we have the equivalence:

Tr_B(S_AFˆ_B|ψihψ|^⊗2_AB) = (ρ^ψ_A|φ⊗ρ^ψ_A|φ). (145) The normalised conditional state is:

ω˜_A|FB = ω¯_A|FB

Tr(¯ωA|F_B) . (146)

Let us set e= Tr(¯ωA|F_B) and d= Tr(SAFˆB|ψihψ|^⊗2_AB) = Tr(ρ^ψ_A|φ⊗ρ^ψ_A|φ);e=c+d.

ω˜_A|FB = 1

e(ρ_A|φ⊗ρ_A|φ) +c e

S˜_A . (147)

We use the equalityρ^ψ_A|φ⊗ρ^ψ_A|φ=d( ˜ρ^ψ_A|φ⊗ρ˜^ψ_A|φ) where ˜ρ^ψ_A|F is a standard normalised quantum conditional state.

ω˜_A|FB =d

e( ˜ρ^ψ_A|φ⊗ρ˜^ψ_A|φ) +c e

S˜_A . (148)

By Lemma 12 ρ˜^ψ_A|φ⊗ρ˜^ψ_A|φ = S_A( ˜ρ^ψ_A|φ⊗ρ˜^ψ_A|φ)S_A hence by Lemma 13 it is a valid normalised state (i.e. of the form ^P_ipi|ψ_iihψ_i|^⊗2). Since 0 < ^d_e <1, 0 < ^c_e <1 and ^d+c_e = 1 the above is a convex combination of ρ˜^ψ_A|φ⊗ρ˜^ψ_A|φ and ˜S_A which are both valid states. Hence the state ˜ω_A|FB is a valid local state.

LetF_B =^Pα_i|φ_iihφ_i|^⊗2, withα_i>0.

ω¯_A|FB =^X

i

α_i(Tr_B(S_AFˆ_B|φ_iihφ_i|^⊗2|ψihψ|^⊗2_AB) + Tr(|ψihψ|^⊗2_ABA_AA_B) 1 Tr(S_B)

S˜_A)

=^X

i

α_i (ρ^ψ_A|φ

i⊗ρ^ψ_A|φ

i) +c_iS˜_A) . (149)

Lete= tr(ω_A|FB) andd_i = tr(ρ^ψ_A|φ

i⊗ρ^ψ_A|φ

i). We havee=^P_iα_i(c_i+d_i). From above:

ω˜_A|FB =^X

i

αi(di

e( ˜ρ^ψ_A|φ

i⊗ρ˜^ψ_A|φ

i) +ci

e

S˜_A) . (150)

Since 0 < ^αi_e^di < 1 and 0 < ^P_i ^αi_e^ci < 1 and ^P_i^αi(ci_e^+di) = 1 the above is a convex combination of ( ˜ρ^ψ_A|φ

i⊗ρ˜^ψ_A|φ

i) with coefficients ^αi_e^di and the state ˜S_A with coefficient ^P_i ^αi_e^ci. G.4 Constraint C5

In this section we show that every OPFs in F_d^G

AdB applied to a product state ψ_A⊗φ_B has a corresponding OPFF_A⁰ inF_d^L

A. LetF_AB be an arbitrary effect inF_d^L

AdB. The corresponding operator is Fˆ_AB=^X

i

α_i|x_iihx_i|₁₂⊗ |x_iihx_i|₃₄ . (151) We evaluate it on product states:

F_AB(ψ_A⊗φ_B) =^X

i

αiTr |ψihψ|^⊗2_A |φihφ|^⊗2_B (|x_iihx_i|^⊗2_AB)

=^X

i

αiTr_A(|ψihψ|^⊗2_A Tr_B(|φihφ|^⊗2_B (|x_iihx_i|^⊗2_AB)))

=^X

i

α_iTr_A(|ψihψ|^⊗2_A S_ATr_B(IA|φihφ|^⊗2_B |x_iihx_i|^⊗2_AB))

= Tr_A(|ψihψ|^⊗2_A (^X

i

α_i(S_A(ρ^x_A|φⁱ ⊗ρ^x_A|φⁱ )S_A))). (152)

By Lemma13(S_A(ρ^x_A|φ⊗ρ^x_A|φ)S_Ais of the form^P_jβj|φ_jihφ_j|^⊗2withβj ≥0. HenceFˆ_A⁰ = (^P_iαi(S_A(ρ^x_A|φⁱ ⊗ ρ^x_A|φⁱ )SA) is a valid effect on A as long as its complement is also of the form ^P_iγi|φ_iihφ_i|^⊗2. Since the complement ofFˆ_AB is of the form^P_iα_i|x_iihx_i|₁₂⊗ |x_iihx_i|₃₄it follows that the associated effect on A(which is the complement of Fˆ_A⁰) is of the required form. From this it follows that Fˆ_A⁰ is a valid effect. The set F_d^G

AdB also contains effects F_A? F_B which are not necessarily of the form given above. However since these are product effects they trivially are consistent with C5.

Dans le document Any modification of the Born rule leads to a violation of the purification and local tomography principles (Page 25-31)