Multiaccess quantum communication and product higher rank numerical - - PowerPoint PPT Presentation

Multiaccess quantum communication and product higher rank numerical ranges

Maciej Demianowicz∗ (joint work with P. Horodecki and K. Życzkowski)

support: ERC Advanced Grant QOLAPS (leader R. Horodecki)

∗Department of Atomic Physics and Luminescence

Faculty of Applied Physics and Mathematics Gdańsk University of Technology

Sep 19th, 2012

2012/09/19 Multiaccess quantum communication and product higher rank ...

Outline

Outline

introduction: quantum channels + quantum error correction (QEC) higher rank numerical range approach to QEC product higher rank numerical range: properties zero entropy codes (decoherence free subspaces) higher entropy codes conclusions + open problems

2012/09/19 Multiaccess quantum communication and product higher rank ...

Quantum channels and quantum error correction (QEC) Channels and error correction Higher rank numerical range, code entropy MACs and QEC

Quantum channels

Quantum channel = a model of noise a trace preserving (TP) completely positive (CP) map Choi–Kraus representation L(̺) =

i Ai̺A† i ,

• i A†

i Ai =

bipartite, multiple access: ̺ = ̺1 ⊗ ̺2 ⊗ · · · , broadcast, km–user Random unitary channel: L(̺) =

i piUi̺U† i ,

• i pi = 1.

Biunitary channel (BUC): L(̺) = pU1̺U1†+(1−p)U2̺U2† → p̺+(1−p)U̺U†, U = U†

1U2

2012/09/19 Multiaccess quantum communication and product higher rank ...

Quantum channels and quantum error correction (QEC) Channels and error correction Higher rank numerical range, code entropy MACs and QEC

Error correction

Error correction = a way to combat noise Definition A quantum error correction code (QECC) is a subspace C ⊆ H. Equivalently, it is the projection PC onto this subspace. |ψ = a|0 + b|1, |0 − → |000, |1 − → |111 |ψ → |φ = a|000 + b|111 C ⊂ (C2)⊗3, PC = |000000| + |111111| C is correctable: D ◦ Λ(̺) = ̺ for PC̺PC = ̺ Theorem (Knill–Laflamme conditions) [Knill&Laflamme 1997] C is correctable iff RCA†

i AjRC = βijRC for some hermitian matrix

[B]ij = βij.

2012/09/19 Multiaccess quantum communication and product higher rank ...

Quantum channels and quantum error correction (QEC) Channels and error correction Higher rank numerical range, code entropy MACs and QEC

Higher rank numerical range and QEC

BUC: L(̺) = p̺ + (1 − p)U̺U† − → KL condition: RUR = λR Definition [Choi et al. 2006] Higher rank (or rank–k) numerical range of an operator A is defined to be the following set: Λk(A) = {λ ∈ C : PAP = λP for P ∈ Pk}. λ1 ∗ ∗ ∗

• Example.

A4 =     1 2 3    , Λ2(A4) = 1; 2, P2 = |φ1φ1| + |φ2φ2|, |φ1 =

1 √ 2(|0 + |3), |φ2 = 1 √ 2(|1 + |2) −

→ P2A4P2 = 3

2P2 2012/09/19 Multiaccess quantum communication and product higher rank ...

Quantum channels and quantum error correction (QEC) Channels and error correction Higher rank numerical range, code entropy MACs and QEC

Entropy of a code

code entropy=an effort to perform recovery= ancillary qubits for D Definition [Kribs et al. 2008] The von Neumann entropy S(C) := S(B) is called the entropy of a code C. For BUC: B =

• p

±λ

• p(1 − p)

±λ

• p(1 − p)

1 − p

• S(C) = 0 iff λ = ±1 — zero–entropy codes → decoherence

free subspaces (DFS; trivial recovery) S(C) > 0 iff λ = ±1 — higher entropy codes

2012/09/19 Multiaccess quantum communication and product higher rank ...

Quantum channels and quantum error correction (QEC) Channels and error correction Higher rank numerical range, code entropy MACs and QEC

Multiple access channels and QEC

Observation Local codes Ci are correctable for a MAC with Kraus operators {Ai} with k inputs if and only if RC1⊗RC2⊗· · ·⊗RCkA†

i AjRC1⊗RC2⊗· · ·⊗RCk = βijRC1⊗RC2⊗· · ·⊗RCk

for some hermitian matrix [B]ij = βij.

2012/09/19 Multiaccess quantum communication and product higher rank ...

Quantum channels and quantum error correction (QEC) Channels and error correction Higher rank numerical range, code entropy MACs and QEC

Multiple access channels and QEC

Observation Local codes Ci are correctable for a MAC with Kraus operators {Ai} with k inputs if and only if RC1⊗RC2⊗· · ·⊗RCkA†

i AjRC1⊗RC2⊗· · ·⊗RCk = βijRC1⊗RC2⊗· · ·⊗RCk

for some hermitian matrix [B]ij = βij.

2012/09/19 Multiaccess quantum communication and product higher rank ...

Product higher rank numerical range Definitions Properties

Product higher rank numerical ranges: definitions

Definition [MD,PH&KŻ 2012] The k1⊗k2⊗· · · product higher rank numerical range of an operator A is defined to be Λk1⊗k2⊗···(A) = {λ ∈ C : R⊗R′⊗· · · AR⊗R′⊗· · · = λR⊗R′⊗· · ·} for some R ∈ Pk1, R′ ∈ Pk2, . . . k1 ⊗ k2 — bipartite, k1 ⊗ k2 ⊗ k3 ⊗ · · · — multipartite symmetric product higher rank numerical range: R = R′ = . . . locally symmetric: R = R′, R′′ = R′′′ = . . . joint: Λk1⊗k2(A1, A2) = {(λ1, λ2) ∈ C2 : R ⊗R′AiR ⊗R′ = λiR ⊗R′} common: Λcomm.

k1⊗k2(A1, A2) = {λ ∈ C : R ⊗ R′AiR ⊗ R′ = λR ⊗ R′}

2012/09/19 Multiaccess quantum communication and product higher rank ...

Product higher rank numerical range Definitions Properties

Product higher rank numerical ranges: definitions

Definition [MD,PH&KŻ 2012] The k1⊗k2⊗· · · product higher rank numerical range of an operator A is defined to be Λk1⊗k2⊗···(A) = {λ ∈ C : R⊗R′⊗· · · AR⊗R′⊗· · · = λR⊗R′⊗· · ·} for some R ∈ Pk1, R′ ∈ Pk2, . . . k1 ⊗ k2 — bipartite, k1 ⊗ k2 ⊗ k3 ⊗ · · · — multipartite symmetric product higher rank numerical range: R = R′ = . . . locally symmetric: R = R′, R′′ = R′′′ = . . . joint: Λk1⊗k2(A1, A2) = {(λ1, λ2) ∈ C2 : R ⊗R′AiR ⊗R′ = λiR ⊗R′} common: Λcomm.

k1⊗k2(A1, A2) = {λ ∈ C : R ⊗ R′AiR ⊗ R′ = λR ⊗ R′}

2012/09/19 Multiaccess quantum communication and product higher rank ...

Product higher rank numerical range Definitions Properties

Properties of the product higher rank numerical range

Λm⊗n(A) = Λm⊗n(U ⊗ VAU† ⊗ V †) Λm⊗n(A) is a compact set Λm⊗n(A) ⊆ Λmn(A) Λm⊗n(A) ⊆ Λloc(A), Λloc(A) = {λ : λ = φ ⊗ ψ|A|φ ⊗ ψ} Λcomm.

m⊗n (A, B) ⊆ Λjoint m⊗n(A, B)

Λsymm.

m⊗n⊗p(A) ⊆ Λloc. symm. m⊗n⊗p (A) ⊆ Λm⊗n⊗p(A)

Λm⊗n(A) =

U Λsymm. m⊗n

• 1 ⊗ UA 1 ⊗ U†

Λcomm.

k

(A1, A2) ⊆ Λk(αA1 + (1 − α)A2) Λk1⊗k2(A) ⊆ W loc.

R⊗R′

1 k1k2 A

• , ’optimization’ bound

W loc.

C

(A) = {trC †(U ⊗ V )†A(U ⊗ V ), U, V ∈ U}

2012/09/19 Multiaccess quantum communication and product higher rank ...

Product higher rank numerical range Definitions Properties

Properties of the product higher rank numerical range

Λm⊗n(A) = Λm⊗n(U ⊗ VAU† ⊗ V †) Λm⊗n(A) is a compact set Λm⊗n(A) ⊆ Λmn(A) Λm⊗n(A) ⊆ Λloc(A), Λloc(A) = {λ : λ = φ ⊗ ψ|A|φ ⊗ ψ} Λcomm.

m⊗n (A, B) ⊆ Λjoint m⊗n(A, B)

Λsymm.

m⊗n⊗p(A) ⊆ Λloc. symm. m⊗n⊗p (A) ⊆ Λm⊗n⊗p(A)

Λm⊗n(A) =

U Λsymm. m⊗n

• 1 ⊗ UA 1 ⊗ U†

Λcomm.

k

(A1, A2) ⊆ Λk(αA1 + (1 − α)A2) Λk1⊗k2(A) ⊆ W loc.

R⊗R′

1 k1k2 A

• , ’optimization’ bound

W loc.

C

(A) = {trC †(U ⊗ V )†A(U ⊗ V ), U, V ∈ U}

2012/09/19 Multiaccess quantum communication and product higher rank ...

Codes Model Zero entropy codes Higher entropy codes

Noise model

L : Cd ⊗ Cd → Cd2 BUC: L(̺) = p̺ + (1 − p)U̺U†, ̺ = ̺1 ⊗ ̺2 U — hermitian ⇒ U = P − Q R ⊗ R′UR ⊗ R′ = λR ⊗ R′ γ = 1 − λ 2 R ⊗ R′QR ⊗ R′ = γR ⊗ R′

• r

R ⊗R′PR ⊗R′ = (1−γ)R ⊗R′ γ = 0, 1: decoherence free subspaces 0 < γ < 1: higher entropy codes

2012/09/19 Multiaccess quantum communication and product higher rank ...

Codes Model Zero entropy codes Higher entropy codes

Decoherence free subspaces — preliminaries

Spaces of matrices of bounded rank [Flanders 1962, Westwick 1972, Atkinson&Lloyd 1980,1981, . . .]: A, B, C, . . . — spaces of matrices spanned respectively by Ai, Bi, Ci, . . ., a space of matrices of bounded (equal) rank — a space whose elements have ranks bounded by some prescribed number (or, besides the zero element, equal to it), A is equivalent to B if there exist nonsingular matrices E,F such that A = {EBF, B ∈ B} (unitary E, F suffice), a space A of a × b matrices is called (t, s)–decomposable if it is equivalent to a subspace whose all elements have the form A = [0](a−t)×(b−s) B(a−t)×s Ct×(b−s) Dt×s

• ,

A ∈ A.

2012/09/19 Multiaccess quantum communication and product higher rank ...

Codes Model Zero entropy codes Higher entropy codes

Decoherence free subspaces — preliminaries (cont’d)

Space of states vs. space of Schmidt matrices: |ψ =

ij cij|ij ⇆ C = ij cij|i

j|, H = span{|γi} and H = span{hi}, where hi is a Schmidt matrix of γi, rm(H) = rm(H), (rm– maximal rank), proposition: a subspace H is called (i, j)–decomposable if H is so.

2012/09/19 Multiaccess quantum communication and product higher rank ...

Codes Model Zero entropy codes Higher entropy codes

Decoherence free subspaces

Main results

U = P − Q, P = PH, Q = QH. P ⇄ P, Q ⇄ Q Theorem [MD 2012] A M ⊗ N DFS exists if and only if at least one of the subspaces P and Q is (d − M, d − N)–decomposable. In other words: ∃U,V ∀Ci (Ci – Schmidt matrices for a subspace): UCiV T =   0M⊗N ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗   Idea for the proof: |ψ → U ⊗ V |ψ corresponds to C → UCV T.

2012/09/19 Multiaccess quantum communication and product higher rank ...

Codes Model Zero entropy codes Higher entropy codes

Decoherence free subspaces

Main results

U = P − Q, P = PH, Q = QH. P ⇄ P, Q ⇄ Q Theorem [MD 2012] A M ⊗ N DFS exists if and only if at least one of the subspaces P and Q is (d − M, d − N)–decomposable. In other words: ∃U,V ∀Ci (Ci – Schmidt matrices for a subspace): UCiV T =   0M⊗N ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗   Idea for the proof: |ψ → U ⊗ V |ψ corresponds to C → UCV T.

2012/09/19 Multiaccess quantum communication and product higher rank ...

Codes Model Zero entropy codes Higher entropy codes

Decoherence free subspaces

Main results

U = P − Q, P = PH, Q = QH. P ⇄ P, Q ⇄ Q Theorem [MD 2012] A M ⊗ N DFS exists if and only if at least one of the subspaces P and Q is (d − M, d − N)–decomposable. In other words: ∃U,V ∀Ci (Ci – Schmidt matrices for a subspace): UCiV T =   0M⊗N ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗   Idea for the proof: |ψ → U ⊗ V |ψ corresponds to C → UCV T. Corollary If there is a M ⊗ N DFS then either of the following holds rm(Q) ≤ 2d − (M + N), rm(P) ≤ 2d − (M + N).

• r, equivalently,

rm(Q) ≤ 2d − (M + N), rm(P) ≤ 2d − (M + N).

2012/09/19 Multiaccess quantum communication and product higher rank ...

Codes Model Zero entropy codes Higher entropy codes

Decoherence free subspaces

Main results (cont’d)

d = 3 Theorem [MD 2012] If dim Q = 2 then a 2 ⊗ 2 DFS exists if and only if rm(Q) ≤ 2 and Q is not a (0, 2) or (2, 0)–decomposable 2–subspace. Q : ∃q ∈ Q, r(q) = 1, not   ∗ ∗ ∗ ∗ ∗ ∗   or   ∗ ∗ ∗ ∗ ∗ ∗   Observation [MD 2012, MD, PH&KŻ 2012] If for d = 3 there exists a 2 ⊗ 2 DFS free subspace then there exists no higher entropy code for this system.

2012/09/19 Multiaccess quantum communication and product higher rank ...

Codes Model Zero entropy codes Higher entropy codes

Examples

Assume Q =

i |φiφi|:

dim Q = 1 |φ1 = 1 √ 3 (|00 + |11 + |22) φ1 ⇄ q, r(q) = 3 ⇒ no DFS. dim Q = 2 |φ1 = 1 √ 2 (|11 + |22) |φ2 = 1 √ 2 (|10 + |21) q1 = 1 √ 2   1 1   q2 = 1 √ 2   1 1   Q is a (0, 2)–decomposable 2–subspace ⇒ no DFS.

2012/09/19 Multiaccess quantum communication and product higher rank ...

Codes Model Zero entropy codes Higher entropy codes

Examples (cont’d)

dim Q = 2 |φ1 = 1 √ 2 (|02 + |10), |φ2 = 1 √ 2 (|01 + |20) Q is a (trivialy) (1, 1)–decomposable 2–subspace ⇒ a code PC = R ⊗ R′ = P12 ⊗ P12, where P12 = |11| + |22|. dim Q = 3 |φ1 = 1 √ 2 (|01 − |10), |φ2 = 1 √ 2 (|12 − |21), |φ3 = 1 √ 2 (|02 − |20), Q is a space of skew–hermitian matrices ⇒ indecomposable ⇒ no DFS.

2012/09/19 Multiaccess quantum communication and product higher rank ...

Codes Model Zero entropy codes Higher entropy codes

Non–zero entropy codes

Fact [MD,PH&KŻ 2012] Fix a number n. Let R = k

i=1 |eiei| be a projection (orthonormal |ei) onto

a subspace of a n dimensional space H. Denote with BR = {|e1, |e2, . . . , |ek} and B⊥ its orthonormal complement, so that span{BR ⊕ B⊥} = H. If RQR = γR holds with 0 < γ < 1 then Q must have the following form in BR ⊕ B⊥              γ ... γ

• γ(1 − γ)

...

• γ(1 − γ)
• γ(1 − γ)

...

• γ(1 − γ)

1 − γ ... 1 − γ S              , where non zero blocks with off diagonal terms equal to zero are k × k and S is a (n − 2k) × (n − 2k) projector.

2012/09/19 Multiaccess quantum communication and product higher rank ...

Codes Model Zero entropy codes Higher entropy codes

Non–zero entropy codes

Fact [MD,PH&KŻ 2012] Fix a number n. Let R = k

i=1 |eiei| be a projection (orthonormal |ei) onto

a subspace of a n dimensional space H. Denote with BR = {|e1, |e2, . . . , |ek} and B⊥ its orthonormal complement, so that span{BR ⊕ B⊥} = H. If RQR = γR holds with 0 < γ < 1 then Q must have the following form in BR ⊕ B⊥              γ ... γ

• γ(1 − γ)

...

• γ(1 − γ)
• γ(1 − γ)

...

• γ(1 − γ)

1 − γ ... 1 − γ S              , where non zero blocks with off diagonal terms equal to zero are k × k and S is a (n − 2k) × (n − 2k) projector.

Corollary If RQR = γR holds then there exists the basis in which k eigenvectors |ψi of the projector Q = q

i=1 |ψiψi| can be

expressed as |ψi = √γ|ei + √1 − γ|vi, where ei|ej = δij, vi|vj = δij, and ei|vj = 0. The code is then R =

i |eiei| and

its existence is equivalent to the existence of the code ˆ R =

i |vivi| satisfying ˆ

RQ ˆ R = (1 − γ)ˆ

• R. This implies that for

any projection Q both γ and 1 − γ belong to Λk(Q).

2012/09/19 Multiaccess quantum communication and product higher rank ...

Codes Model Zero entropy codes Higher entropy codes

Non–zero entropy codes (cont’d)

Theorem [MD,PH&KŻ 2012] ’computable’ bound

Let Ql be a rank l projection. Assume RM ⊗ R′

NQlRM ⊗ R′ N = γRM ⊗ R′ N holds.

Let further x1 ≥ x2 ≥ . . . be eigenvalues of trBQl. Then

M

• i=1

|xi − Nγ| +

r(trB Ql )

• i=M+1

xi ≤ MN

• (1 − γ)(1 + 3γ) + l − MN.

Let Pd2−l be a rank d2 − l projection. Assume RM ⊗ R′

NPd2−lRM ⊗ R′ N = (1 − γ)RM ⊗ R′ N holds. Let further ˜

x1 ≥ ˜ x2 ≥ . . . be eigenvalues of trBPd2−l. Then

M

• i=1

|˜ xi − N(1 − γ)| +

r(trB Pd2−l )

• i=M+1

˜ xi ≤ MN

• γ(4 − 3γ) + d2 − l − MN.

2012/09/19 Multiaccess quantum communication and product higher rank ...

Codes Model Zero entropy codes Higher entropy codes

Non–zero entropy codes (cont’d)

Theorem [MD,PH&KŻ 2012] ’computable’ bound

Let Ql be a rank l projection. Assume RM ⊗ R′

NQlRM ⊗ R′ N = γRM ⊗ R′ N holds.

Let further x1 ≥ x2 ≥ . . . be eigenvalues of trBQl. Then

M

• i=1

|xi − Nγ| +

r(trB Ql )

• i=M+1

xi ≤ MN

• (1 − γ)(1 + 3γ) + l − MN.

Let Pd2−l be a rank d2 − l projection. Assume RM ⊗ R′

NPd2−lRM ⊗ R′ N = (1 − γ)RM ⊗ R′ N holds. Let further ˜

x1 ≥ ˜ x2 ≥ . . . be eigenvalues of trBPd2−l. Then

M

• i=1

|˜ xi − N(1 − γ)| +

r(trB Pd2−l )

• i=M+1

˜ xi ≤ MN

• γ(4 − 3γ) + d2 − l − MN.

Corollary Existence of a product code for the noise model U = P − Q for λ = 1 − 2γ does not necessarily imply the existence of a product code for λ = 2γ − 1. In other words, there are cases when γ ∈ Λ2⊗2 but 1 − γ / ∈ Λ2⊗2.

2012/09/19 Multiaccess quantum communication and product higher rank ...

Codes Model Zero entropy codes Higher entropy codes

Examples

Example 1. (d ⊗ d input , M ⊗ N codes) SWAPd↔d = Psym. − Pasym. := Vd, Vd|φ|ψ = |ψ|φ. It holds ΛM⊗N(Vd) = 0, M + N ≤ d ∅, M + N > d . Eigenvectors of the corresponding local projectors R, R′ must obey ϕi ⊥ ψj thus Λsymm.

M⊗N (Vd) = ∅.

M = N = 2: (i) d = 3: Λ2⊗2(V3) = ∅ ⇒ no faithful transmission allowed (also for many usages of the channel) (ii) d = 4: local codes R = |00| + |11| and R′ = |22| + |33|

2012/09/19 Multiaccess quantum communication and product higher rank ...

Codes Model Zero entropy codes Higher entropy codes

Examples (cont’d)

Example 2. (2 ⊗ 4 input, 2 ⊗ 2 codes) Q(γ) = |00| ⊗ Q1 + |11| ⊗ Q2, Q1 = |00| + |11|, Q2 = |η1η1| + |η2η2|, |η1 = √γ|0 + √1 − γ|2, |η2 = √γ|1 + √1 − γ|3. Λ2⊗2(Q(γ)) = Λcomm.

2

(Q1, Q2(γ)) = = 1

2(1 − √γ), 1 2(1 + √γ)

Codes: R : |00| + |11| R′ : |ξ1ξ1| + |ξ2ξ2|, |ξ1 = √ λ|0 + eiβ√ 1 − λ|2, |ξ2 = √ λ|1 + eiβ√ 1 − λ|3.

0.2 0.4 0.6 0.8 1.0 Γ 0.2 0.4 0.6 0.8 1.0 22QΓ

Λ2⊗2(Q(γ) = W loc.

R⊗R′ (Q(γ))

2012/09/19 Multiaccess quantum communication and product higher rank ...

Codes Model Zero entropy codes Higher entropy codes

Examples (cont’d)

Example 3. (4 ⊗ 4 input, 2 ⊗ 2 codes) Q(α) = 4

i=1 |ψiψi|:

|ψ1 = √α|00 + √ 1 − α|22, |ψ2 = √α|01 + √ 1 − α|23, |ψ3 = √α|10 + √ 1 − α|32, |ψ4 = √α|11 + √ 1 − α|33. Λ2⊗2(Q(α)) = 0; 1 − α, α < 1 − α, 0; α, α ≥ 1 − α Codes: R : |φ1 = √1 − β|0 + √β|2, |φ2 = √1 − β|1 + √β|3 R′ : |ψ1 = |2, |ψ2 = |3 Λ2⊗2(Q(α)) = Λsymm.

2⊗2 (Q(α))

0.2 0.4 0.6 0.8 1.0 Α 0.2 0.4 0.6 0.8 1.0 22QΑ Α1Α Α1Α

Λ2⊗2(Q(α)) = W loc.

R⊗R′ (Q(α))

2012/09/19 Multiaccess quantum communication and product higher rank ...

Summary

Conlusions: new notion of the product higher rank numerical range, zero–entropy codes – connection with spaces of matrices with bounded rank, higher entropy codes – (universal) analytical techniques for bounding the set and some methods for specific cases. Open problems: verify whether for d = 3 product higher rank numerical range is at most a one–element set, simple–connectivity of the set, efficient procedure for finding codes.

2012/09/19 Multiaccess quantum communication and product higher rank ...

References

References

1

• M. Demianowicz

Decoherence free subspaces for two–access quantum channels, arXiv: 1209.0119

2

• M. Demianowicz, P. Horodecki, and K. Życzkowski

Multiaccess quantum communication and product higher rank numerical range, arXiv: 1209.0120

2012/09/19 Multiaccess quantum communication and product higher rank ...