Infinite-fold enhancement in communications capacity using pre-shared - - PowerPoint PPT Presentation

SLIDE 1

Infinite-fold enhancement in communications capacity using pre-shared entanglement

Saikat Guha

July 21, 2020 College of Optical Sciences, Department of Electrical and Computer Engineering University of Arizona, Tucson AZ (USA)

arXiv:2001.03934 (2020)

SLIDE 2

Outline

• Background

– Quantum limit of classical communications – Joint detection receivers for superadditive capacity

• Entanglement assisted communications over the bosonic channel

– Transmitter-receiver design – Infinite-fold capacity enhancement with pre-shared entanglement

• Conclusions and ongoing work

SLIDE 3

Outline

• Background

– Quantum limit of classical communications – Joint detection receivers for superadditive capacity

• Entanglement assisted communications over the bosonic channel

– Transmitter-receiver design – Infinite-fold capacity enhancement with pre-shared entanglement

• Conclusions and ongoing work

SLIDE 4

Quantum limit of classical communications

Alice Bob

X Y

Physical (quantum) channel (e.g., loss, noise) Optical (quantum) modulation Optical (quantum) receiver

pY |X(y|x)

Shannon’s “channel”

Shannon capacity, CShannon = max

pX I(X; Y )

Shannon, 1948 Alice Bob

N A→B

A B

Holevo capacity,

Holevo, 1996 Schumacher, Westmoreland, 1997

Y n

Xn

Physical (quantum) channel (e.g., loss, noise)

ρAn(x)

ρBn(x)

(may use entangled state over n channel uses) (may use quantum joint detection over n channel uses) modulation receiver

C = sup

n

maxpXn(x),ρAn(x) S (P pXn(x)ρBn(x)) − P pXn(x)S (ρBn(x)) n

S(ρ) = −Tr(ρ log ρ)

SLIDE 5

“Superadditivity” on the transmitter and receiver side

• Superadditivite capacity

– Transmitter side: Using entangled states help achieve higher capacity – Receiver side: Quantum joint detection receiver helps achieve higher capacity

• Channel not transmitter-side superadditive, it can still be receiver-side

– Product state modulation is optimal (entanglement at transmitter doesn’t help) – Holevo Capacity, – Shown : receiver uses symbol by symbol detection; Shannon capacity – Joint detection receiver (JDR) required, unless orthogonal pure states – Calculation of does not need receiver. Finding receiver achieving is hard

ρB(x) C = max

pX S

⇣X pX(x)ρB(x) ⌘ − X pX(x)S (ρB(x))

Alice

A B

Physical (quantum) channel (e.g., loss, noise) modulation receiver

X

ρA(x)

ρB(x) Π(y) Y

Bob

pY |X(y|x) = Tr

• ρB(x)Π(y)
• C1 < C

C C

SLIDE 6

The single-mode bosonic channel

• Mean photon number per mode constraint at transmitter
• Environment mode in thermal state, mean photon number
• Any linear lossy propagation medium can be decomposed into a

collection of single mode bosonic channels

η ∈ (0, 1]

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ˆ aS

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ˆ aR = √η ˆ aS + p 1 − η ˆ aB

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ˆ aB

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hˆ a†

Sˆ

aSi  NS

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ˆ aE

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hˆ a†

Bˆ

aBi = NB

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ρB = Z 1 πNB e−|β|2/NB|βihβ|d2β

SLIDE 7

Holevo capacity of the single-mode bosonic channel

• C > Shannon capacity of all known standard optical receivers
• Product coherent state modulation achieves capacity
• Need joint detection receiver (collective measurement over codeword)

C = g(ηNS + (1 − η)NB) − g((1 − η)NB))

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bits per mode

• V. Giovannetti, SG, S. Lloyd, L. Maccone, J. H. Shapiro, and H. P. Yuen, Phys. Rev. Lett. 92, 027902 (2004)
• V. Giovannetti, SG, S. Lloyd, L. Maccone, and J. H. Shapiro, Phys. Rev. A (2004)
• V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo, Nat. Photonics 8, 796 (2014)

g(N) = (1 + N) log(1 + N) − N log(N)

|αi

p(α) = 1 πNS e−|α|2/NS

ρB = Z 1 πNB e−|β|2/NB|βihβ|d2β

ρR = Z 1 π(1 η)NB e−|β−√ηα|2/(1−η)NB|βihβ|d2β

η ∈ (0, 1]

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SLIDE 8

g(N) = (1 + N) log(1 + N) − N log(N)

1 2 log(1 + 4N)

log(1 + N)

N = ηNS

C η :

NS :

Transmissivity Mean photon number transmitted per mode

η ∼ e−αL

η ∼ 1/L2

fiber free-space

Giovannetti, SG, Lloyd, Maccone, Shapiro, & Yuen, PRL (2004)

C(N) N

C(N)

N

N

Quantum limit of classical (optical) communications over a lossy channel, with an input mean photon number constraint

Takeoka and SG, Physical Review A 89, 042309 (2014)

NB = 0

SLIDE 9

Outline

• Background

– Quantum limit of classical communications – Joint detection receivers for superadditive capacity

• Entanglement assisted communications over the bosonic channel

– Transmitter-receiver design – Infinite-fold capacity enhancement with pre-shared entanglement

• Conclusions and ongoing work

SLIDE 10

Binary hypothesis test:

|ψ1i |ψ2i

hψ1|ψ2i = σ

Inner product between the two states

= cos 2θ

|0i = ✓ 1 ◆ |1i = ✓ 0 1 ◆

θ

= ✓ cos θ sin θ ◆ = ✓ cos θ −sin θ ◆

|ψ1i ( H1) |ψ2i ( H2)

p1 = p, p2 = 1 − p

|w1i = ✓ cos φ sin φ ◆ |w2i = ✓ sin φ cos φ ◆

φ

Pe = pP(2|1) + (1 − p)P(1|2)

tan 2φ = tan 2θ 2p − 1

Optimal measurement satisfies: Helstrom bound (for minimum probability of error state discrimination)

= 1 2 h 1 − p 1 − 4p(1 − p)σ2 i

Pe,min

SLIDE 11

Binary phase shift keying (BPSK) modulation

• Consider BPSK modulation, , priors ,

{|αi, | αi}

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(p, 1 − p)

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Pe,min = 1 2  1 − q 1 − 4p(1 − p)e−4N

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T T

E −E

T T

E

2E

−E

−2E

E − u1(t) E − u2(t)

−E − u2(t) −E − u1(t)

u1(t)

u2(t)

u1(t)

u2(t)

s(t)

s(t) − ui(t) −ui(t) √1 − κ, i = 1 or 2

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z(t) ∈ {1, 2}

κ ≈ 1

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H1

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H2

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×

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Minimum error probability:

Sam Dolinar, MIT Ph.D. Thesis, 1973

hα| αi = e−2N

N = |α|2

SLIDE 12

Example of receiver-side superadditive capacity

• Maximum capacity with the optimal symbol-by-symbol measurement
• Holevo capacity (joint detection over infinite-length modulated block)
• The joint-detection gain most pronounced at low photon number

C1 = 1 − h2(p), p = 1 2[1 − p 1 − e−4N]

<latexit sha1_base64="jCIx/3LNz9K1NE0BeuShKSfivyA=">ACIHicbVDLSgMxFM3UV62vqks3wSJUsGVSC3UjFLtxJRXsAzrjkEkzbWjm0SQjlKGf4sZfceNCEd3p15g+Ftp64MLhnHu59x434kwq0/wyUiura+sb6c3M1vbO7l52/6Apw1gQ2iAhD0XbxZJyFtCGYorTdiQo9l1OW+6gNvFbD1RIFgZ3ahR28e9gHmMYKUlJ1upOQheQgQLsO+U8tHpGbSGMe7CSKuWJzBpQ4qWHIoVIK9D4plG/GY9vJ5syiOQVcJmhOcmCOupP9tLohiX0aKMKxlB1kRspOsFCMcDrOWLGkESYD3KMdTQPsU2kn0wfH8EQrXeiFQleg4FT9PZFgX8qR7+pOH6u+XPQm4n9eJ1behZ2wIoVDchskRdzqEI4SQt2maBE8ZEmAimb4Wkj3UqSmea0SGgxZeXSbNUROfF0m05V72ax5EGR+AY5AECFVAF16AOGoCAR/AMXsGb8WS8GO/Gx6w1ZcxnDsEfGN8/4wafNA=</latexit>

C∞ = S ✓|αihα| + | αihα| 2 ◆ = h2 ✓1 + e−2N 2 ◆

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We do not know, C2, C3, . . . , Cn, . . .

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bits per mode bits per mode

lim

N→0

C∞ C1 = ∞

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SLIDE 13

Joint detection receiver (JDR) for superadditive capacity

n-mode input, n-mode output (n=8 shown)             α −α α −α α −α α −α            

            √ 8α             n = 8

Mean photon number in the n-mode Hadamard codeword (input and output), n|α|2 = nN

X ∈ {1, 2, . . . , n} Y ∈ {1, 2, . . . , n, n + 1} pY |X(y|x) =    1 e−nN, y = x 0, y 6= x e−nN, y = n + 1

SG, Phys. Rev. Lett. 106, 240502 (2011)

SLIDE 14

Superadditive capacity

• Plotting photon information efficiency, C(N)/N

N

C(N) N

bits per photon mean photon number per mode,

max

n

⇢I(X; Y )/n N

• Holevo capacity

Bridging the remaining gap to the Holevo capacity requires JDRs that use quantum optical effects

Chung, SG, Zheng, PRA 96, 012320 (2017) SG, Phys. Rev. Lett. 106, 240502 (2011)

SLIDE 15

A receiver (based on quantum “belief propagation” to achieve the Holevo capacity

Narayanan, Seshadreesan, SG, Pfister, arXiv:2003.04356 (2020)

10−4 10−3 10−2 10−1 100 101 10−4 10−3 10−2 10−1 100 Mean photon number / mode Block Error Probability Sim: BP Block Error Rate Sim: Codeword ML Block Error Rate Sim: BPQM Block Error Rate Theory: Codeword Helstrom Limit

|αi|αi|αi|αi|αi

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|αi|αi|αi| αi| αi

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|αi| αi| αi|αi|αi

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|αi| αi| αi| αi| αi

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| αi|αi| αi|αi| αi

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| αi|αi| αi| αi|αi

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| αi| αi|αi|αi| αi

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| αi| αi|αi| αi|αi

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Mean photon number per mode,|α|2 = N

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Block error probability Coherent state binary-phase shift keyed (BPSK) codebook, with eight 5-mode codewords Quantum limit of minimum error probability Best achievable using ANY classical receiver

|ψi

|ψi 2 {|αi, | αi}

|ψ±i = r 1 + x 2 |0i ± r 1 x 2 |1i hψ+|ψ−i = x

U

hα| αi = e−2N ⌘ x

<latexit sha1_base64="BguqdIYvEbczysqtCHpoZP4L/fE=">ACHicbZA9SwNBEIb3/DZ+RS1tFoNgk3AXBW0E0cZKFIwJ5GKY20ySJXt75+6eGM78EBv/io2FIjYWgv/GTXKFXwMLD+87w+y8QSy4Nq76UxMTk3PzM7N5xYWl5ZX8qtrlzpKFMKi0SkagFoFxixXAjsBYrhDAQWA16x0O/eoNK80hemH6MjRA6krc5A2OlZn7HFyA7AqkPIu4CvaPFjHw1Ng4oXqXF8umA+nid8Bt628wX3JI7KvoXvAwKJKuzZv7db0UsCVEaJkDrufGpGCMpwJHOT8RGMrAcdrFuUEKJupKPjBnTLKi3ajpR90tCR+n0ihVDrfhjYzhBMV/2huJ/Xj0x7f1GymWcGJRsvKidCGoiOkyKtrhCZkTfAjDF7V8p64ICZmyeORuC9/vkv3BZLnk7pfL5buHwKItjmyQTbJNPLJHDskJOSMVwsg9eSTP5MV5cJ6cV+dt3DrhZDPr5Ec5H1+CsKBh</latexit>

SLIDE 16

Outline

• Background

– Quantum limit of classical communications – Joint detection receivers for superadditive capacity

• Entanglement assisted communications over the bosonic channel

– Transmitter-receiver design – Entanglement assisted covert communications

• Conclusions and ongoing work

SLIDE 17

Entanglement assisted classical communications

• Communication capacity, when Alice and Bob pre-share

(unlimited amount of) entanglement,

– where, is a purification of

CE = max

ρA∈HA I(N A→B, ρA)

<latexit sha1_base64="qP9s1MpoGTCMfrqZtfTIsxhy8XM=">ACKHicbVBLSwMxGMzWV62vqkcvwSJUkLJbBb2IfSDUi1SwD+jWJZumbWg2uyRZsSz7c7z4V7yIKNKrv8T0cdDWgcBk5vtIZtyAUalMc2QklpZXVteS6mNza3tnfTuXl36ocCkhn3mi6aLJGUk5qipFmIAjyXEYa7qA89huPREjq83s1DEjbQz1OuxQjpSUnfV2ruEltD305ES26PtOEdqUw8jGiMFK7BRjeJOd3m7jh6hoKx+W4hM4nT120hkzZ04AF4k1IxkwQ9VJv9sdH4ce4QozJGXLMgPVjpBQFDMSp+xQkgDhAeqRlqYceUS2o0nQGB5pQO7vtCHKzhRf29EyJNy6Ll60kOqL+e9sfif1wpV96IdUR6EinA8fagbMqjDjluDHSoIVmyoCcKC6r9C3EcCYaW7TekSrPnIi6Sez1mnufzdWaZQmtWRBAfgEGSBc5BAVRAFdQABs/gFXyAT+PFeDO+jNF0NGHMdvbBHxjfPzNupDs=</latexit>

I(N A→B, ρA) = S(ρA) + S(N A→B(ρA)) − S

• (N A→B ⊗ I)[Φ]
• <latexit sha1_base64="BqYjVRx7ousfOG7E5WVxFzUiY/A=">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</latexit>
• C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, IEEE Trans. Inform. Theory 48, 2637 (2002)

Φ

<latexit sha1_base64="SIWhw4cDOJI1oTS6WRyjqImg48=">AB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lqQY9FLx4r2A9oQ9lsN83S3U3Y3Qgl9C948aCIV/+QN/+NmzYHbX0w8Hhvhpl5QcKZNq7ZQ2Nre2d8q7lb39g8Oj6vFJV8epIrRDYh6rfoA15UzSjmG036iKBYBp71gepf7vSeqNIvlo5kl1Bd4IlnICDa5NGxHbFStuXV3AbROvILUoEB7VP0ajmOSCioN4Vjrgecmxs+wMoxwOq8MU0TKZ4QgeWSiyo9rPFrXN0YZUxCmNlSxq0UH9PZFhoPROB7RTYRHrVy8X/vEFqwhs/YzJDZVkuShMOTIxyh9HY6YoMXxmCSaK2VsRibDCxNh4KjYEb/XldJt1L2reuOhWvdFnGU4QzO4RI8uIYW3EMbOkAgmd4hTdHOC/Ou/OxbC05xcwp/IHz+QPkB4j</latexit>

ρA

<latexit sha1_base64="oaeCv8YkwU73qX5IthZL7l6vZU=">AB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHqxWME84BkCbOT2WTMPJaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqb7q4o4cxY3/2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCK92OsKGcSdqwzHLaTjTFIuK0FY1up37riWrDlHyw4SGAg8kixnB1knNrh6q3nWvVPYr/gxomQ5KUOeq/01e0rkgoqLeHYmE7gJzbMsLaMcDopdlNDE0xGeEA7jkosqAmz2bUTdOqUPoqVdiUtmqm/JzIsjBmLyHUKbIdm0ZuK/3md1MZXYcZkloqyXxRnHJkFZq+jvpMU2L52BFMNHO3IjLEGhPrAiq6EILFl5dJs1oJzivV+4ty7SaPowDHcAJnEMAl1OAO6tAo/wDK/w5invxXv3PuatK14+cwR/4H3+AGEIjv8=</latexit>

SLIDE 18

Entanglement assisted capacity of the bosonic channel

• Mean photon number constraint,
• Take,
• Purification,
• Capacity,

– –

ρA = Z 1 πNS e−|α|2|αihα|d2α

<latexit sha1_base64="OMCwZXnvDJ2gmIjREc1/uGX24k=">ACQXicbZA9b9swEIapPmo8+W0YxeiRoAsMS3QLIESJqlU+GgdWzAtIUTfbIJU5RAUgEMWX8tS/5Bt+5dOrQosnYpbWtokxA8MH73oG8N8qkMNb3v3lr6y82Nre2X9Z2dvf2D+qHr25MmuOHZ7KVPciMCiFwo4VmIv0whJLEbTa8WfvcWtRGp+mJnGQ4SGCsRCw7WSWG9x/QkDS/pOWVCWcpiDbwIyoJlgn4KP5cUh8XJnIHMJjAftsoKmQY1lkiZrO6lSud0NGytOKw3/Ka/LPoUgoapKp2WP/KRinPE1SWSzCmH/iZHRSgreASyxrLDWbApzDGvkMFCZpBsUygpEdOGdE41e64NZbqvxMFJMbMksh1JmAn5rG3EJ/z+rmNzwaFUFluUfHVQ3EuqU3pIk46Ehq5lTMHwLVwf6V8Ai5E60KvuRCxys/hZtWM3jXbF2/b1x8qOLYJm/IW3JMAnJKLshH0iYdwskd+U5+kl/evfD+09rFrXvGrmNfmvD9/AXS2sD4=</latexit>

NS

<latexit sha1_base64="H37BEz9UXM3e76sph9tsAoyLspA=">AB6nicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BL54kEvOAZAmzk95kyOzsMjMrhJBP8OJBEa9+kTf/xkmyB0saCiqunuChLBtXHdbye3tr6xuZXfLuzs7u0fFA+PmjpOFcMGi0Ws2gHVKLjEhuFGYDtRSKNAYCsY3c781hMqzWP5aMYJ+hEdSB5yRo2V6ve9eq9YcsvuHGSVeBkpQYZar/jV7csjVAaJqjWHc9NjD+hynAmcFrophoTykZ0gB1LJY1Q+5P5qVNyZpU+CWNlSxoyV39PTGik9TgKbGdEzVAvezPxP6+TmvDan3CZpAYlWywKU0FMTGZ/kz5XyIwYW0KZ4vZWwoZUWZsOgUbgrf8ipVsreRbnycFmq3mRx5OETuEcPLiCKtxBDRrAYADP8ApvjnBenHfnY9Gac7KZY/gD5/MHAIqNnA=</latexit>

|Φi =

∞

X

n=0

s N n

S

(1 + NS)1+n |niS|niI

<latexit sha1_base64="LZEcyC86/WvYgbAN0RDYaHWKT2I=">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</latexit>

CE(η, NS, NB) = g(NS) + g(N 0

S) − g(A+) − g(A)

<latexit sha1_base64="UMrEI2ytKlIGEqW3n02rzYoqtF0=">ACJ3icbZBbS0JBEMf3dDW7WT32siSBYso5FtRLYUrQUxjlBdQOe9ZRF/dc2N0TiPhteumr9BJURD32TVovQWkDu/zmPzPszt8JOJPKND+NufmFxaXlyEp0dW19YzO2tV2WfigolKjPfVF1iATOPCgpjhUAwHEdThUnG5hWK/cg5DM925VL4CGS9oeazFKlJbs2FnBvkjUQZEDfGXfDK98Ep/idkJnSZwaw109EMyFJE7r/NxO/UA6acfiZsYcBZ4FawJxNImiHXupN30auApyomUNcsMVKNPhGKUwyBaDyUEhHZJG2oaPeKCbPRHew7wvlauOULfTyFR+rviT5xpey5ju50ierI6dpQ/K9WC1XrpNFnXhAq8Oj4oVbIsfLx0DTcZAKo4j0NhAqm/4phwhClbY2qk2wplehXI2Yx1mstdH8Vx+YkcE7aI9lEAWOkY5dImKqIQoekBP6BW9GY/Gs/FufIxb54zJzA76E8bXN2lsn4E=</latexit>

D = q (NS + N 0

S + 1)2 − 4ηNS(NS + 1)

<latexit sha1_base64="qL1G23oNtB5ICgSWDsAxCbNMn7o=">ACH3icbVDLSgMxFM3Ud31VXboJFsEilpkq6kYQdeFKFG0rdGrJpHdqMPMwuSOUoX/ixl9x40IRcefmKldaOuFhJNzuXmHi+WQqNtf1m5sfGJyanpmfzs3PzCYmFpuajRHGo8khG6tpjGqQIoYoCJVzHCljgSah7d8eZXn8ApUXmE3hmbAOqHwBWdoqFZh94QeUFfK0w3zlqXdJOa+8aNlQjAPJzSTYVu0R0XkGVK5tl0Sr1WoWiX7X7RUeAMQJEM6rxV+HTbEU8CJFLpnXDsWNspkyh4BJ6eTfREDN+xzrQMDBkAehm2t+vR9cN06Z+pMwJkfbZ3x0pC7TuBp5xBgxv9bCWkf9pjQT9/WYqwjhBCPnPID+RFCOahUXbQgFH2TWAcSXMXym/ZYpxNJHmTQjO8MqjoFYpO9vlysVO8fBoEMc0WSVrZIM4ZI8cklNyTqEk0fyTF7Jm/VkvVjv1sePNWcNelbIn7K+vgG3vJ5t</latexit>

A± = (D − 1 ± (N 0

S − NS))/2

<latexit sha1_base64="oKi9HcseiSApUk/mF2jEr5O07k=">ACEXicbZC7TsMwFIadcivlVmBksaiQ0qElKUiwIJXLwISKoBepCZHjuq1VO4lsB6mK+gosvAoLAwixsrHxNrhtBij8kqXP/zlH9vn9iFGpLOvLyMzNLywuZdzK6tr6xv5za2GDGOBSR2HLBQtH0nCaEDqipGWpEgiPuMNP3B+bjevCdC0jC4VcOIuBz1AtqlGClteXnz1EuciI/gCTQvSjbUDM0r7+bOiQTlBJagvhSL+xUvX7DK1kTwL9gpFECqmpf/dDohjkJFGZIyrZtRcpNkFAUMzLKObEkEcID1CNtjQHiRLrJZKMR3NOB3ZDoU+g4MT9OZEgLuWQ+7qTI9WXs7Wx+V+tHavusZvQIoVCfD0oW7MoArhOB7YoYJgxYaEBZU/xXiPhIKx1iTodgz678FxqVsn1QrlwfFqpnaRxZsAN2gQlscASq4BLUQB1g8ACewAt4NR6NZ+PNeJ+2Zox0Zhv8kvHxDfeHme8=</latexit>

Vittorio Giovannetti, Seth Lloyd, Lorenzo Maccone, Peter W. Shor, Phys. Rev. A 68, 062323 (2003)

SLIDE 19

The entanglement assistance is most pronounced, i.e., when

Comparison of CE with C

Shi, Zhang, Zhuang, arXiv:1909.11112 (2019) SG, Zhuang, Bash, arXiv:2001.03934 (2020)

NS ⌧ 1, NB 1

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Low brightness transmitter Long center wavelength (high noise)

CE = −NS ln NS + o(NS) C = NS ln

• 1 + [(1 − η)NB]−1

+ o(NS)

lim

NS→0

CE C ln(1/NS) = 1

• 1 + (1 − η)NB
• ln
• 1 + [(1 − η)NB]−1

CE/C ∼ ln(1/NS)

Taylor series expansion at yields: NS = 0

SLIDE 20

Achieving CE for the bosonic channel

• A two-mode squeezed vacuum (TMSV) source, and phase modulation

achieves the entanglement assisted capacity CE, when

– In general, active modulation (squeezing, displacement) of TMSV maybe necessary – binary phase modulation suffices

η ∈ (0, 1]

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ˆ aS

<latexit sha1_base64="8CJV4lNc3Xzo3tZIY9IYC4eEe9Y=">AB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JUQZdFNy4r2gc0IUymk3bo5MHMjVBiFv6KGxeKuPU3Pk3TtMstPXAhcM5986de/xEcAW9W0sLa+srq1XNqbW9s7u+befkfFqaSsTWMRy5PFBM8Ym3gIFgvkYyEvmBdf3w9bsPTCoeR/cwSZgbkmHEA04JaMkzD53ijcwXKcszZ0QAk9y78yaVbcK4EVil6SGSrQ8sZxDQNWQRUEKX6tpWAmxEJnAqWV51UsYTQMRmyvqYRCZlys2J3jk+0MsBLHVFgAv190RGQqUmoa87QwIjNe9Nxf+8fgrBpZvxKEmBRXS2KEgFhPw8ADLhkFMdGEUMn1XzEdEUko6MiqOgR7/uRF0mnU7bN64/a81rwq46igI3SMTpGNLlAT3aAWaiOKHtEzekVvxpPxYrwbH7PWJaOcOUB/YHz+AIV3lmo=</latexit>

ˆ aB

<latexit sha1_base64="2zdAPgE4ZcHd6KZQblxI2SYPNfM=">AB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JUQZelblxWsA9oQriZTtqhkwczE6HELPwVNy4UcetvuPNvnKZaOuBC4dz7p079/gJZ1JZ1rexsrq2vrFZ2apu7+zu7ZsHh10Zp4LQDol5LPo+SMpZRDuKU7iaAQ+pz2/MnNzO89UCFZHN2raULdEYRCxgBpSXPHaKNzKfpzTPnDEoDLnX8syaVbcK4GVil6SGSrQ98sZxiQNaQIBykHtpUoNwOhGOE0rzqpAmQCYzoQNMIQirdrNid4zOtDHEQC12RwoX6eyKDUMp6OvOENRYLnoz8T9vkKrg2s1YlKSKRmS+KEg5VjGehYGHTFCi+FQTILpv2IyBgFE6ciqOgR78eRl0m3U7Yt64+6y1myVcVTQCTpF58hGV6iJblEbdRBj+gZvaI348l4Md6Nj3nrilHOHKE/MD5/AGuzlk=</latexit>

ˆ aE

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ˆ aI

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ˆ aR

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(NB)

<latexit sha1_base64="jyzafBPxLk+csa3JMwvmrW7MWQ=">AB7HicbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY6sWTVHBtoV1KNs2oUl2SbJCWfobvHhQxKs/yJv/xrTdg7Y+GHi8N8PMvDhTBvX/XYKa+sbm1vF7dLO7t7+Qfnw6FHqSLUJzGPVSfEmnImqW+Y4bSTKIpFyGk7HN/M/PYTVZrF8sFMEhoIPJQsYgQbK/nVu37zvF+uDV3DrRKvJxUIEerX/7qDWKSCioN4VjrucmJsiwMoxwOi31Uk0TMZ4SLuWSiyoDrL5sVN0ZpUBimJlSxo0V39PZFhoPRGh7RTYjPSyNxP/87qpia6DjMkNVSxaIo5cjEaPY5GjBFieETSzBRzN6KyAgrTIzNp2RD8JZfXiWP9Zp3UavfX1YazTyOIpzAKVTBgytowC20wAcCDJ7hFd4c6bw4787HorXg5DPH8AfO5w+s643w</latexit>

|Φi

<latexit sha1_base64="Dv9WENSYS/DXE2IyFcxOCUWmbQ=">AB83icbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ac0oWy2k3bpZhN2N0Kp/RtePCji1T/jzX/jts1BWx8MPN6bYWZemAqujet+O4W19Y3NreJ2aWd3b/+gfHjU0kmGDZIhLVCalGwSU2DTcCO6lCGocC2+Hodua3H1FpnsgHM04xiOlA8ogzaqzkP/mNIfcVlQOBvXLFrbpzkFXi5aQCORq98pfT1gWozRMUK27npuaYEKV4UzgtORnGlPKRnSAXUsljVEHk/nNU3JmlT6JEmVLGjJXf09MaKz1OA5tZ0zNUC97M/E/r5uZ6DqYcJlmBiVbLIoyQUxCZgGQPlfIjBhbQpni9lbChlRZmxMJRuCt/zyKmnVqt5FtXZ/Wanf5HEU4QRO4Rw8uI63EDmsAghWd4hTcnc16cd+dj0Vpw8plj+APn8wcq5ZHE</latexit>

Pre-shared TMSV entanglement

θ

<latexit sha1_base64="VRbFNfU2yJrhxTioHNG9u2eQ2g=">AB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2m3btZhN2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZekEh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRI3g7GtzO/cS1EbF6wEnC/YgOlQgFo2ilVg9HGm/XHGr7hxklXg5qUCORr/81RvELI24QiapMV3PTdDPqEbBJ+WeqnhCWVjOuRdSxWNuPGz+bVTcmaVAQljbUshmau/JzIaGTOJAtsZURyZW8m/ud1Uwyv/UyoJEWu2GJRmEqCMZm9TgZCc4ZyYglWthbCRtRTRnagEo2BG/5VXSqlW9i2rt/rJSv8njKMIJnMI5eHAFdbiDBjSBwSM8wyu8ObHz4rw7H4vWgpPHMfOJ8/pUWPLA=</latexit>

Shi, Zhang, Zhuang, arXiv:1909.11112 (2019)

NS ⌧ 1, NB 1

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(NS)

<latexit sha1_base64="K1pIQk/DOHtmT8xw6GKG29wpO/c=">AB7HicbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OJKrq20C4lm2b0CS7JFmhLP0NXjwo4tUf5M1/Y9ruQVsfDzem2FmXphwpo3rfjuFldW19Y3iZmlre2d3r7x/8KjVBHqk5jHqh1iTmT1DfMcNpOFMUi5LQVjq6nfuJKs1i+WDGCQ0EHkgWMYKNlfzqbe/+tFeuDV3BrRMvJxUIEezV/7q9mOSCioN4VjrjucmJsiwMoxwOil1U0TEZ4QDuWSiyoDrLZsRN0YpU+imJlSxo0U39PZFhoPRah7RTYDPWiNxX/8zqpiS6DjMkNVS+aIo5cjEaPo56jNFieFjSzBRzN6KyBArTIzNp2RD8BZfXiaP9Zp3VqvfnVcaV3kcRTiCY6iCBxfQgBtog8EGDzDK7w50nlx3p2PeWvByWcO4Q+czx/GwI4B</latexit>

(NS)

<latexit sha1_base64="K1pIQk/DOHtmT8xw6GKG29wpO/c=">AB7HicbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OJKrq20C4lm2b0CS7JFmhLP0NXjwo4tUf5M1/Y9ruQVsfDzem2FmXphwpo3rfjuFldW19Y3iZmlre2d3r7x/8KjVBHqk5jHqh1iTmT1DfMcNpOFMUi5LQVjq6nfuJKs1i+WDGCQ0EHkgWMYKNlfzqbe/+tFeuDV3BrRMvJxUIEezV/7q9mOSCioN4VjrjucmJsiwMoxwOil1U0TEZ4QDuWSiyoDrLZsRN0YpU+imJlSxo0U39PZFhoPRah7RTYDPWiNxX/8zqpiS6DjMkNVS+aIo5cjEaPo56jNFieFjSzBRzN6KyBArTIzNp2RD8BZfXiaP9Zp3VqvfnVcaV3kcRTiCY6iCBxfQgBtog8EGDzDK7w50nlx3p2PeWvByWcO4Q+czx/GwI4B</latexit>

N 0

S = ηNS + (1 − η)NB

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(NS)

<latexit sha1_base64="K1pIQk/DOHtmT8xw6GKG29wpO/c=">AB7HicbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OJKrq20C4lm2b0CS7JFmhLP0NXjwo4tUf5M1/Y9ruQVsfDzem2FmXphwpo3rfjuFldW19Y3iZmlre2d3r7x/8KjVBHqk5jHqh1iTmT1DfMcNpOFMUi5LQVjq6nfuJKs1i+WDGCQ0EHkgWMYKNlfzqbe/+tFeuDV3BrRMvJxUIEezV/7q9mOSCioN4VjrjucmJsiwMoxwOil1U0TEZ4QDuWSiyoDrLZsRN0YpU+imJlSxo0U39PZFhoPRah7RTYDPWiNxX/8zqpiS6DjMkNVS+aIo5cjEaPo56jNFieFjSzBRzN6KyBArTIzNp2RD8BZfXiaP9Zp3VqvfnVcaV3kcRTiCY6iCBxfQgBtog8EGDzDK7w50nlx3p2PeWvByWcO4Q+czx/GwI4B</latexit>

|Φi =

∞

X

n=0

s N n

S

(1 + NS)1+n |niS|niI

<latexit sha1_base64="LZEcyC86/WvYgbAN0RDYaHWKT2I=">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</latexit>

SLIDE 21

The Optical Parametric Amplifier (OPA) based receiver

η ∈ (0, 1]

<latexit sha1_base64="ds1Sc5HUGxchqQ9VKmYvHTpQkVA=">AB+HicbVDLSgNBEJyNrxgfWfXoZTAIOUjYjYIeg148RjAxkF3C7KSTDJmdXWZ6hRjyJV48KOLVT/Hm3zh5HDSxoKGo6qa7K0qlMOh5305ubX1jcyu/XdjZ3dsvugeHTZNkmkODJzLRrYgZkEJBAwVKaKUaWBxJeIiGN1P/4RG0EYm6x1EKYcz6SvQEZ2iljlsMABkNhKJl74z6YcteRVvBrpK/AUpkQXqHfcr6CY8i0Ehl8yYtu+lGI6ZRsElTApBZiBlfMj60LZUsRhMOJ4dPqGnVunSXqJtKaQz9fEmMXGjOLIdsYMB2bZm4r/e0Me1fhWKg0Q1B8vqiXSYoJnaZAu0IDRzmyhHEt7K2UD5hmHG1WBRuCv/zyKmlWK/5pXp3UapdL+LIk2NyQsrEJ5ekRm5JnTQIJxl5Jq/kzXlyXpx352PemnMWM0fkD5zPHzpEkYA=</latexit>

ˆ aS

<latexit sha1_base64="8CJV4lNc3Xzo3tZIY9IYC4eEe9Y=">AB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JUQZdFNy4r2gc0IUymk3bo5MHMjVBiFv6KGxeKuPU3Pk3TtMstPXAhcM5986de/xEcAW9W0sLa+srq1XNqbW9s7u+befkfFqaSsTWMRy5PFBM8Ym3gIFgvkYyEvmBdf3w9bsPTCoeR/cwSZgbkmHEA04JaMkzD53ijcwXKcszZ0QAk9y78yaVbcK4EVil6SGSrQ8sZxDQNWQRUEKX6tpWAmxEJnAqWV51UsYTQMRmyvqYRCZlys2J3jk+0MsBLHVFgAv190RGQqUmoa87QwIjNe9Nxf+8fgrBpZvxKEmBRXS2KEgFhPw8ADLhkFMdGEUMn1XzEdEUko6MiqOgR7/uRF0mnU7bN64/a81rwq46igI3SMTpGNLlAT3aAWaiOKHtEzekVvxpPxYrwbH7PWJaOcOUB/YHz+AIV3lmo=</latexit>

ˆ aB

<latexit sha1_base64="2zdAPgE4ZcHd6KZQblxI2SYPNfM=">AB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JUQZelblxWsA9oQriZTtqhkwczE6HELPwVNy4UcetvuPNvnKZaOuBC4dz7p079/gJZ1JZ1rexsrq2vrFZ2apu7+zu7ZsHh10Zp4LQDol5LPo+SMpZRDuKU7iaAQ+pz2/MnNzO89UCFZHN2raULdEYRCxgBpSXPHaKNzKfpzTPnDEoDLnX8syaVbcK4GVil6SGSrQ98sZxiQNaQIBykHtpUoNwOhGOE0rzqpAmQCYzoQNMIQirdrNid4zOtDHEQC12RwoX6eyKDUMp6OvOENRYLnoz8T9vkKrg2s1YlKSKRmS+KEg5VjGehYGHTFCi+FQTILpv2IyBgFE6ciqOgR78eRl0m3U7Yt64+6y1myVcVTQCTpF58hGV6iJblEbdRBj+gZvaI348l4Md6Nj3nrilHOHKE/MD5/AGuzlk=</latexit>

ˆ aE

<latexit sha1_base64="0F8L9FoJE+ujHIjawGhDzxiO8M=">AB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JUQZdFEVxWsA9oQriZTtqhkwczE6HELPwVNy4UcetvuPNvnKZaOuBC4dz7p079/gJZ1JZ1rextLyurZe2ahubm3v7Jp7+x0Zp4LQNol5LHo+SMpZRNuKU57iaAQ+px2/fH1O8+UCFZHN2rSULdEIYRCxgBpSXPHSKNzKfpzTPnBEoDLl345k1q24VwIvELkNlWh5pcziEka0kgRDlL2bStRbgZCMcJpXnVSRMgYxjSvqYRhFS6WbE7xydaGeAgFroihQv190QGoZST0NedIaiRnPem4n9eP1XBpZuxKEkVjchsUZByrGI8DQMPmKBE8YkmQATf8VkBAKI0pFVdQj2/MmLpNOo2f1xt15rXlVxlFBR+gYnSIbXaAmukUt1EYEPaJn9IrejCfjxXg3PmatS0Y5c4D+wPj8AXA/lw=</latexit>

ˆ aI

<latexit sha1_base64="s03QHc53EeNFQ824o6PEK7RNks=">AB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JUQZdFN7qrYB/QhHAznbRDJw9mJkKJWfgrblwo4tbfcOfOE2z0NYDFw7n3Dt37vETzqSyrG9jaXldW29slHd3Nre2TX39jsyTgWhbRLzWPR8kJSziLYVU5z2EkEh9Dnt+uPrqd9oEKyOLpXk4S6IQwjFjACSkuegUb2Q+T2meOSNQGHLv1jNrVt0qgBeJXZIaKtHyzC9nEJM0pJEiHKTs21ai3AyEYoTvOqkiZAxjCkfU0jCKl0s2J3jk+0MsBLHRFChfq74kMQiknoa87Q1AjOe9Nxf+8fqCSzdjUZIqGpHZoiDlWMV4GgYeMEGJ4hNgAim/4rJCAQpSOr6hDs+ZMXSadRt8/qjbvzWvOqjKOCjtAxOkU2ukBNdINaqI0IekTP6BW9GU/Gi/FufMxal4xy5gD9gfH5A3ZPlmA=</latexit>

ˆ aR

<latexit sha1_base64="Y7fd2dX/n+MNlT4QfF0LfLkJeo=">AB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JUQZdFNy6r2Ac0IUymk3bo5MHMjVBiFv6KGxeKuPU3Pk3TtMstPXAhcM5986de/xEcAW9W0sLa+srq1XNqbW9s7u+befkfFqaSsTWMRy5PFBM8Ym3gIFgvkYyEvmBdf3w9bsPTCoeR/cwSZgbkmHEA04JaMkzD53ijcwXKcszZ0QAk9y78yaVbcK4EVil6SGSrQ8sZxDQNWQRUEKX6tpWAmxEJnAqWV51UsYTQMRmyvqYRCZlys2J3jk+0MsBLHVFgAv190RGQqUmoa87QwIjNe9Nxf+8fgrBpZvxKEmBRXS2KEgFhPw8ADLhkFMdGEUMn1XzEdEUko6MiqOgR7/uRF0mnU7bN64/a81rwq46igI3SMTpGNLlAT3aAWaiOKHtEzekVvxpPxYrwbH7PWJaOcOUB/YHz+AIPzlmk=</latexit>

(NB)

<latexit sha1_base64="jyzafBPxLk+csa3JMwvmrW7MWQ=">AB7HicbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY6sWTVHBtoV1KNs2oUl2SbJCWfobvHhQxKs/yJv/xrTdg7Y+GHi8N8PMvDhTBvX/XYKa+sbm1vF7dLO7t7+Qfnw6FHqSLUJzGPVSfEmnImqW+Y4bSTKIpFyGk7HN/M/PYTVZrF8sFMEhoIPJQsYgQbK/nVu37zvF+uDV3DrRKvJxUIEerX/7qDWKSCioN4VjrucmJsiwMoxwOi31Uk0TMZ4SLuWSiyoDrL5sVN0ZpUBimJlSxo0V39PZFhoPRGh7RTYjPSyNxP/87qpia6DjMkNVSxaIo5cjEaPY5GjBFieETSzBRzN6KyAgrTIzNp2RD8JZfXiWP9Zp3UavfX1YazTyOIpzAKVTBgytowC20wAcCDJ7hFd4c6bw4787HorXg5DPH8AfO5w+s643w</latexit>

TMSV: transmitter retains the idler mode

|Φi⊗M

<latexit sha1_base64="7s9vj2UUONwXcSucm8xiiIDxpNI=">AAACAXicbVBNS8NAEJ34WetX1IvgZbEInkpSBT0WvXgRKtgPaGLZbDft0s0m7G6EEuvFv+LFgyJe/Rfe/Ddu2xy09cHA470ZZuYFCWdKO863tbC4tLyyWlgrrm9sbm3bO7sNFaeS0DqJeSxbAVaUM0HrmmlOW4mkOAo4bQaDy7HfvKdSsVjc6mFC/Qj3BAsZwdpIHXv/wav1mSex6HF6l3mxZhFV6HrUsUtO2ZkAzRM3JyXIUevYX143JmlEhSYcK9V2nUT7GZaaEU5HRS9VNMFkgHu0bajAZo+fTT4YoSOjdFEYS1NCo4n6eyLDkVLDKDCdEdZ9NeuNxf+8dqrDcz9jIkk1FWS6KEw50jEax4G6TFKi+dAQTCQztyLSxxITbUIrmhDc2ZfnSaNSdk/KlZvTUvUij6MAB3AIx+DCGVThCmpQBwKP8Ayv8GY9WS/Wu/UxbV2w8pk9+APr8wfPpZcd</latexit>

(NS)

<latexit sha1_base64="K1pIQk/DOHtmT8xw6GKG29wpO/c=">AB7HicbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OJKrq20C4lm2b0CS7JFmhLP0NXjwo4tUf5M1/Y9ruQVsfDzem2FmXphwpo3rfjuFldW19Y3iZmlre2d3r7x/8KjVBHqk5jHqh1iTmT1DfMcNpOFMUi5LQVjq6nfuJKs1i+WDGCQ0EHkgWMYKNlfzqbe/+tFeuDV3BrRMvJxUIEezV/7q9mOSCioN4VjrjucmJsiwMoxwOil1U0TEZ4QDuWSiyoDrLZsRN0YpU+imJlSxo0U39PZFhoPRah7RTYDPWiNxX/8zqpiS6DjMkNVS+aIo5cjEaPo56jNFieFjSzBRzN6KyBArTIzNp2RD8BZfXiaP9Zp3VqvfnVcaV3kcRTiCY6iCBxfQgBtog8EGDzDK7w50nlx3p2PeWvByWcO4Q+czx/GwI4B</latexit>

(NS)

<latexit sha1_base64="K1pIQk/DOHtmT8xw6GKG29wpO/c=">AB7HicbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OJKrq20C4lm2b0CS7JFmhLP0NXjwo4tUf5M1/Y9ruQVsfDzem2FmXphwpo3rfjuFldW19Y3iZmlre2d3r7x/8KjVBHqk5jHqh1iTmT1DfMcNpOFMUi5LQVjq6nfuJKs1i+WDGCQ0EHkgWMYKNlfzqbe/+tFeuDV3BrRMvJxUIEezV/7q9mOSCioN4VjrjucmJsiwMoxwOil1U0TEZ4QDuWSiyoDrLZsRN0YpU+imJlSxo0U39PZFhoPRah7RTYDPWiNxX/8zqpiS6DjMkNVS+aIo5cjEaPo56jNFieFjSzBRzN6KyBArTIzNp2RD8BZfXiaP9Zp3VqvfnVcaV3kcRTiCY6iCBxfQgBtog8EGDzDK7w50nlx3p2PeWvByWcO4Q+czx/GwI4B</latexit>

SG and B. I. Erkmen, Phys. Rev. A (2009)

OPA ˆ

c

<latexit sha1_base64="1HbXvz6CuQ28XubqD+Z8GJu2QUE=">AB/XicbVDLSgMxFL1TX7W+xsfOTbAIrspMFXRZdOygn1AZyiZNG1DM5khyQh1GPwVNy4Ucet/uPNvzLSz0NYDgcM59+YkJ4g5U9pxvq3Syura+kZ5s7K1vbO7Z+8ftFWUSEJbJOKR7AZYUc4EbWmOe3GkuIw4LQTG5yv/NApWKRuNfTmPohHgk2ZARrI/XtI292RxrwhGapN8YakaxvV52aMwNaJm5BqlCg2be/vEFEkpAKThWquc6sfZTLDUjnGYVL1E0xmSCR7RnqMAhVX46S87QqVEGaBhJc4SJz9XfGykOlZqGgZkMsR6rRS8X/N6iR5e+SkTcaKpIPOgYcKRjlBeBRowSYnmU0Mwkcy8FZExlphoU1jFlOAufnmZtOs197xWv7uoNq6LOspwDCdwBi5cQgNuoQktIPAIz/AKb9aT9WK9Wx/z0ZJV7BzCH1ifPx+dlaY=</latexit>

ˆ d

<latexit sha1_base64="gxkbZqFqVwPtRMNRxa5JnTPd5VI=">AB/XicbVDLSsNAFL2pr1pf8bFzM1gEVyWpgi6LblxWsA9oQplMJu3QyYOZiVBD8FfcuFDErf/hzr9xmahrQcuHM65d+7c4yWcSWVZ30ZlZXVtfaO6Wdva3tndM/cPujJOBaEdEvNY9D0sKWcR7SimO0nguLQ47TnTW5mfu+BCsni6F5NE+qGeBSxgBGstDQ0j5zijczjKc0zZ4wV8vOhWbcaVgG0TOyS1KFEe2h+OX5M0pBGinAs5cC2EuVmWChGOM1rTipgskEj+hA0wiHVLpZsTlHp1rxURALXZFChfp7IsOhlNPQ050hVmO56M3E/7xBqoIrN2NRkioakfmiIOVIxWgWBfKZoETxqSaYCKb/isgYC0yUDqymQ7AXT14m3WbDPm807y7qresyjiocwmcgQ2X0IJbaEMHCDzCM7zCm/FkvBjvxse8tWKUM4fwB8bnDyEilac=</latexit>

hˆ aRˆ aIi = p ηNS(NS + 1)

<latexit sha1_base64="dnLD8rGt/Z0FibiLai/QpvdPNO8=">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</latexit>

ˆ c(m) = √ Gˆ a(m)

I

+ √ G − 1ˆ a†(m)

R

<latexit sha1_base64="xBvgloPTWHEG9XxmOECC0cjXssU=">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</latexit>

ˆ d(m) = √ Gˆ a(m)

R

+ √ G − 1ˆ a†(m)

I

<latexit sha1_base64="c+EjKVjR1Y/S25JsO09tOoO2gbM=">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</latexit>

Nm

<latexit sha1_base64="0KnwSoMa7BLVbSLKaly5GUiPLjU=">AB6nicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxZNENA9IljA7mU2GzGOZmRXCk/w4kERr36RN/GSbIHTSxoKq6e6KEs6M9f1vb2V1bX1js7BV3N7Z3dsvHRw2jUo1oQ2iuNLtCBvKmaQNyn7URTLCJOW9HoZuq3nqg2TMlHO05oKPBAspgRbJ30cNcTvVLZr/gzoGUS5KQMOeq90le3r0gqLSEY2M6gZ/YMPaMsLpNhNDU0wGeEB7TgqsaAmzGanTtCpU/oVtqVtGim/p7IsDBmLCLXKbAdmkVvKv7ndVIbX4UZk0lqSTzRXHKkVo+jfqM02J5WNHMNHM3YrIEGtMrEun6EIFl9eJs1qJTivVO8vyrXrPI4CHMJnEAl1CDW6hDAwgM4Ble4c3j3ov37n3MW1e8fOYI/sD7/AEn8o2</latexit>

Nout =

M

X

m=1

Nm

<latexit sha1_base64="jb+D2nJUW3E+uUkB8Dv7ghst7o=">ACB3icbVDLSsNAFJ3UV62vqEtBovgqiRV0E2h6MaNpYJ9QBPDZDph84kYWYilJCdG3/FjQtF3PoL7vwbp20W2nrgwuGce7n3Hj9mVCrL+jYKS8srq2vF9dLG5tb2jrm715ZRIjBp4YhFousjSRgNSUtRxUg3FgRxn5GOP7qa+J0HIiSNwjs1jonL0SCkAcVIackzDxte6gOo0RlsAYdmXAv5TU7u7+BDY97ZtmqWFPARWLnpAxyND3zy+lHOEkVJghKXu2FSs3RUJRzEhWchJYoRHaEB6moaIE+m0z8yeKyVPgwioStUcKr+nkgRl3LMfd3JkRrKeW8i/uf1EhVcuCkN40SREM8WBQmDKoKTUGCfCoIVG2uCsKD6VoiHSCsdHQlHYI9/IiaVcr9mlentWrl/mcRTBATgCJ8AG56AOrkETtAGj+AZvI348l4Md6Nj1lrwchn9sEfGJ8/kO2YeA=</latexit>

θ

<latexit sha1_base64="VRbFNfU2yJrhxTioHNG9u2eQ2g=">AB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2m3btZhN2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZekEh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRI3g7GtzO/cS1EbF6wEnC/YgOlQgFo2ilVg9HGm/XHGr7hxklXg5qUCORr/81RvELI24QiapMV3PTdDPqEbBJ+WeqnhCWVjOuRdSxWNuPGz+bVTcmaVAQljbUshmau/JzIaGTOJAtsZURyZW8m/ud1Uwyv/UyoJEWu2GJRmEqCMZm9TgZCc4ZyYglWthbCRtRTRnagEo2BG/5VXSqlW9i2rt/rJSv8njKMIJnMI5eHAFdbiDBjSBwSM8wyu8ObHz4rw7H4vWgpPHMfOJ8/pUWPLA=</latexit>

CE C → 2

<latexit sha1_base64="5tGcnQYiNOwhnI/IbyC7Fcp9ye8=">AB/HicbVDLSsNAFL3xWesr2qWbwSK4KkVdFksgsK9gFNCJPpB06eTAzEUKIv+LGhSJu/RB3/o3TNgtPXDhcM693HuPn3AmlWV9G2vrG5tb25Wd6u7e/sGheXTck3EqCO2SmMdi4GNJOYtoVzHF6SARFIc+p31/2p75/UcqJIujB5Ul1A3xOGIBI1hpyTNrTiAwydvebZG3C+SoGDU9s241rDnQKrFLUocSHc/8ckYxSUMaKcKxlEPbSpSbY6EY4bSoOqmkCSZTPKZDTSMcUunm8+MLdKaVEQpioStSaK7+nshxKGUW+rozxGoil72Z+J83TFVw7eYsSlJFI7JYFKQc6RdnSaARE5QonmCiWD6VkQmWKehdF5VHYK9/PIq6TUb9kWjeX9Zb92UcVTgBE7hHGy4ghbcQe6QCDZ3iFN+PJeDHejY9F65pRztTgD4zPH8RslDA=</latexit>

E[Nm] = hˆ c†ˆ ci = GNS + (G 1)(1 + (1 η)NB + ηNS)

±2 p G(G − 1) p ηNS(NS + 1)

SLIDE 22

Receiver based on sum frequency generation (SFG)

SFG ˆ aS

<latexit sha1_base64="hP2oRf5cT0LAru9n3fY06rPj+TY=">AB8XicbVBNS8NAEJ34WetX1aOXxSJ4KkV9Fj04rGi/cA2lMl20y7dbMLuRih/8KLB0W8+m+8+W/ctjlo64OBx3szMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqyho0FrFqB6iZ4JI1DeCtRPFMAoEawWjm6nfemJK81g+mHC/AgHkoecorHSY9YdoiE46d3SmW34s5AlomXkzLkqPdKX91+TNOISUMFat3x3MT4GSrDqWCTYjfVLE6wgHrWCoxYtrPZhdPyKlV+iSMlS1pyEz9PZFhpPU4CmxnhGaoF72p+J/XSU145WdcJqlhks4XhakgJibT90mfK0aNGFuCVHF7K6FDVEiNDaloQ/AWX14mzWrFO69U7y7Ktes8jgIcwmcgQeXUINbqEMDKEh4hld4c7Tz4rw7H/PWFSefOYI/cD5/AET9kKY=</latexit>

ˆ aI

<latexit sha1_base64="1hMYOpy18FDbrlBfBIDeIO4t6Ok=">AB8XicbVBNS8NAEJ34WetX1aOXxSJ4KkV9Fj0orcK9gPbUCbTbt0swm7G6GE/gsvHhTx6r/x5r9x2+agrQ8GHu/NMDMvSATXxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqKGvQWMSqHaBmgkvWMNwI1k4UwygQrBWMbqZ+64kpzWP5YMYJ8yMcSB5yisZKj1l3iIbgpHfXK5XdijsDWSZeTsqQo94rfX7MU0jJg0VqHXHcxPjZ6gMp4JNit1UswTpCAesY6nEiGk/m108IadW6ZMwVrakITP190SGkdbjKLCdEZqhXvSm4n9eJzXhlZ9xmaSGSTpfFKaCmJhM3yd9rhg1YmwJUsXtrYQOUSE1NqSiDcFbfHmZNKsV7xSvb8o167zOApwDCdwBh5cQg1uoQ4NoCDhGV7hzdHOi/PufMxbV5x85gj+wPn8ATXVkJw=</latexit>

ˆ b

<latexit sha1_base64="ZIQ2cZVWq5mXvO5GTYO6VK2TA=">AB73icbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2m3bpZhN3J0IJ/RNePCji1b/jzX/jts1BWx8MPN6bYWZekEh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRI3g7GtzO/cS1EbF6wEnC/YgOlQgFo2ilTtYbUSTBtF+uFV3DrJKvJxUIEejX/7qDWKWRlwhk9SYrucm6GdUo2CST0u91PCEsjEd8q6likbc+Nn83ik5s8qAhLG2pZDM1d8TGY2MmUSB7YwojsyNxP/87ophtd+JlSIldsShMJcGYzJ4nA6E5QzmxhDIt7K2EjaimDG1EJRuCt/zyKmnVqt5FtXZ/Wanf5HEU4QRO4Rw8uI63EDmsBAwjO8wpvz6Lw4787HorXg5DPH8AfO5w/oM4/h</latexit>

OPA ˆ aS

<latexit sha1_base64="hP2oRf5cT0LAru9n3fY06rPj+TY=">AB8XicbVBNS8NAEJ34WetX1aOXxSJ4KkV9Fj04rGi/cA2lMl20y7dbMLuRih/8KLB0W8+m+8+W/ctjlo64OBx3szMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqyho0FrFqB6iZ4JI1DeCtRPFMAoEawWjm6nfemJK81g+mHC/AgHkoecorHSY9YdoiE46d3SmW34s5AlomXkzLkqPdKX91+TNOISUMFat3x3MT4GSrDqWCTYjfVLE6wgHrWCoxYtrPZhdPyKlV+iSMlS1pyEz9PZFhpPU4CmxnhGaoF72p+J/XSU145WdcJqlhks4XhakgJibT90mfK0aNGFuCVHF7K6FDVEiNDaloQ/AWX14mzWrFO69U7y7Ktes8jgIcwmcgQeXUINbqEMDKEh4hld4c7Tz4rw7H/PWFSefOYI/cD5/AET9kKY=</latexit>

ˆ aI

<latexit sha1_base64="1hMYOpy18FDbrlBfBIDeIO4t6Ok=">AB8XicbVBNS8NAEJ34WetX1aOXxSJ4KkV9Fj0orcK9gPbUCbTbt0swm7G6GE/gsvHhTx6r/x5r9x2+agrQ8GHu/NMDMvSATXxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqKGvQWMSqHaBmgkvWMNwI1k4UwygQrBWMbqZ+64kpzWP5YMYJ8yMcSB5yisZKj1l3iIbgpHfXK5XdijsDWSZeTsqQo94rfX7MU0jJg0VqHXHcxPjZ6gMp4JNit1UswTpCAesY6nEiGk/m108IadW6ZMwVrakITP190SGkdbjKLCdEZqhXvSm4n9eJzXhlZ9xmaSGSTpfFKaCmJhM3yd9rhg1YmwJUsXtrYQOUSE1NqSiDcFbfHmZNKsV7xSvb8o167zOApwDCdwBh5cQg1uoQ4NoCDhGV7hzdHOi/PufMxbV5x85gj+wPn8ATXVkJw=</latexit>

Zhuang, Zhang, Shapiro, Phys. Rev. Lett. 118, 040801 (2017)

κ = 1 K

<latexit sha1_base64="PkYvXvWGpikIBR7whENZe6gjc=">AB/XicbVDLSsNAFJ3UV62v+Ni5GSyCq5JUQTdC0Y3gpoJ9QFPKzXTSDp0kw8xEqCH4K25cKOLW/3Dn3zhts9DWAxcO59zLvf4gjOlHefbKiwtr6yuFdLG5tb2zv27l5TxYktEFiHsu2D4pyFtGZprTtpAUQp/Tlj+6nvitByoVi6N7PRa0G8IgYgEjoI3Usw+8EQgB+BJ7gQSul6m/XslNxpsCLxM1JGeWo9+wvrx+TJKSRJhyU6riO0N0UpGaE06zkJYoKICMY0I6hEYRUdPp9Rk+NkofB7E0FWk8VX9PpBAqNQ590xmCHqp5byL+53USHVx0UxaJRNOIzBYFCc6xpMocJ9JSjQfGwJEMnMrJkMwKWgTWMmE4M6/vEia1Yp7WqnenZVrV3kcRXSIjtAJctE5qEbVEcNRNAjekav6M16sl6sd+tj1lqw8pl9AfW5w+4UpTB</latexit>

κ

<latexit sha1_base64="bD1kN69KciN2AkFMW9xq/coE4K4=">AB7XicbVBNSwMxEM3Wr1q/qh69BIvgqexWQY9FLx4r2A9olzKbZtvYbBKSrFCW/gcvHhTx6v/x5r8xbfegrQ8GHu/NMDMvUpwZ6/vfXmFtfWNzq7hd2tnd2z8oHx61jEw1oU0iudSdCAzlTNCmZbTjtIUkojTdjS+nfntJ6oNk+LBThQNExgKFjMC1kmt3hiUgn654lf9OfAqCXJSQTka/fJXbyBJmlBhCQdjuoGvbJiBtoxwOi31UkMVkDEMadRAQk1YTa/dorPnDLAsdSuhMVz9fdEBokxkyRynQnYkVn2ZuJ/Xje18XWYMaFSwVZLIpTjq3Es9fxgGlKLJ84AkQzdysmI9BArAuo5EIl9eJa1aNbio1u4vK/WbPI4iOkGn6BwF6ArV0R1qoCYi6BE9o1f05knvxXv3PhatBS+fOUZ/4H3+AJd6jyM=</latexit>

1 − κ

<latexit sha1_base64="tVBlvAtxGqGUAjglGPN1drYNaDg=">AB73icbVBNS8NAEJ34WetX1aOXxSJ4sSRV0GPRi8cK9gPaUCbTbt0s4m7G6GE/gkvHhTx6t/x5r9x2+agrQ8GHu/NMDMvSATXxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqKGvQWMSqHaBmgkvWMNwI1k4UwygQrBWMbqd+64kpzWP5YMYJ8yMcSB5yisZKbe+8O8IkwV6p7FbcGcgy8XJShz1Xumr249pGjFpqECtO56bGD9DZTgVbFLspolSEc4YB1LJUZM+9ns3gk5tUqfhLGyJQ2Zqb8nMoy0HkeB7YzQDPWiNxX/8zqpCa/9jMskNUzS+aIwFcTEZPo86XPFqBFjS5Aqbm8ldIgKqbERFW0I3uLy6RZrXgXler9Zbl2k8dRgGM4gTPw4ApqcAd1aAFAc/wCm/Oo/PivDsf89YVJ585gj9wPn8AckSPlQ=</latexit>

κ

<latexit sha1_base64="bD1kN69KciN2AkFMW9xq/coE4K4=">AB7XicbVBNSwMxEM3Wr1q/qh69BIvgqexWQY9FLx4r2A9olzKbZtvYbBKSrFCW/gcvHhTx6v/x5r8xbfegrQ8GHu/NMDMvUpwZ6/vfXmFtfWNzq7hd2tnd2z8oHx61jEw1oU0iudSdCAzlTNCmZbTjtIUkojTdjS+nfntJ6oNk+LBThQNExgKFjMC1kmt3hiUgn654lf9OfAqCXJSQTka/fJXbyBJmlBhCQdjuoGvbJiBtoxwOi31UkMVkDEMadRAQk1YTa/dorPnDLAsdSuhMVz9fdEBokxkyRynQnYkVn2ZuJ/Xje18XWYMaFSwVZLIpTjq3Es9fxgGlKLJ84AkQzdysmI9BArAuo5EIl9eJa1aNbio1u4vK/WbPI4iOkGn6BwF6ArV0R1qoCYi6BE9o1f05knvxXv3PhatBS+fOUZ/4H3+AJd6jyM=</latexit>

1 − κ

<latexit sha1_base64="tVBlvAtxGqGUAjglGPN1drYNaDg=">AB73icbVBNS8NAEJ34WetX1aOXxSJ4sSRV0GPRi8cK9gPaUCbTbt0s4m7G6GE/gkvHhTx6t/x5r9x2+agrQ8GHu/NMDMvSATXxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqKGvQWMSqHaBmgkvWMNwI1k4UwygQrBWMbqd+64kpzWP5YMYJ8yMcSB5yisZKbe+8O8IkwV6p7FbcGcgy8XJShz1Xumr249pGjFpqECtO56bGD9DZTgVbFLspolSEc4YB1LJUZM+9ns3gk5tUqfhLGyJQ2Zqb8nMoy0HkeB7YzQDPWiNxX/8zqpCa/9jMskNUzS+aIwFcTEZPo86XPFqBFjS5Aqbm8ldIgKqbERFW0I3uLy6RZrXgXler9Zbl2k8dRgGM4gTPw4ApqcAd1aAFAc/wCm/Oo/PivDsf89YVJ585gj9wPn8AckSPlQ=</latexit>

SFG

<latexit sha1_base64="2OqbGwDbT9+tYG6sKAnjh3PFUs4=">AB83icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DEoqMeI5gHZJcxOJsmQ2QczvWJY9je8eFDEqz/jzb9xkuxBEwsaiqpurv8WAqNtv1tLS2vrK6tFzaKm1vbO7ulvf2mjhLFeINFMlJtn2ouRcgbKFDydqw4DXzJW/7oauK3HrnSIgofcBxzL6CDUPQFo2gkN3WRP2F6f32TZd1S2a7YU5BF4uSkDnq3dKX24tYEvAQmaRadxw7Ri+lCgWTPCu6ieYxZSM64B1DQxpw7aXTmzNybJQe6UfKVIhkqv6eSGmg9TjwTWdAcajnvYn4n9dJsH/hpSKME+Qhmy3qJ5JgRCYBkJ5QnKEcG0KZEuZWwoZUYmpqIJwZl/eZE0qxXntFK9OyvXLvM4CnAIR3ACDpxDW6hDg1gEMzvMKblVgv1rv1MWtdsvKZA/gD6/MHYVmR5w=</latexit>

SFG

<latexit sha1_base64="2OqbGwDbT9+tYG6sKAnjh3PFUs4=">AB83icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DEoqMeI5gHZJcxOJsmQ2QczvWJY9je8eFDEqz/jzb9xkuxBEwsaiqpurv8WAqNtv1tLS2vrK6tFzaKm1vbO7ulvf2mjhLFeINFMlJtn2ouRcgbKFDydqw4DXzJW/7oauK3HrnSIgofcBxzL6CDUPQFo2gkN3WRP2F6f32TZd1S2a7YU5BF4uSkDnq3dKX24tYEvAQmaRadxw7Ri+lCgWTPCu6ieYxZSM64B1DQxpw7aXTmzNybJQe6UfKVIhkqv6eSGmg9TjwTWdAcajnvYn4n9dJsH/hpSKME+Qhmy3qJ5JgRCYBkJ5QnKEcG0KZEuZWwoZUYmpqIJwZl/eZE0qxXntFK9OyvXLvM4CnAIR3ACDpxDW6hDg1gEMzvMKblVgv1rv1MWtdsvKZA/gD6/MHYVmR5w=</latexit>

|0i

<latexit sha1_base64="mFbyg4gJW0ZXAvu82LjFXT3NAg=">AB8HicbVBNS8NAEJ34WetX1aOXxSJ4KkV9Fj04rGC/ZA2lM120i7dbMLuRixv8KLB0W8+nO8+W/ctjlo64OBx3szMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6mfqtR1Sax/LejBP0IzqQPOSMGis9PLldReVAYK9UdivuDGSZeDkpQ456r/TV7csjVAaJqjWHc9NjJ9RZTgTOCl2U40JZSM6wI6lkao/Wx28IScWqVPwljZkobM1N8TGY20HkeB7YyoGepFbyr+53VSE175GZdJalCy+aIwFcTEZPo96XOFzIixJZQpbm8lbEgVZcZmVLQheIsvL5NmteKdV6p3F+XadR5HAY7hBM7Ag0uowS3UoQEMIniGV3hzlPivDsf89YVJ585gj9wPn8AtneQWQ=</latexit>

|0i

<latexit sha1_base64="mFbyg4gJW0ZXAvu82LjFXT3NAg=">AB8HicbVBNS8NAEJ34WetX1aOXxSJ4KkV9Fj04rGC/ZA2lM120i7dbMLuRixv8KLB0W8+nO8+W/ctjlo64OBx3szMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6mfqtR1Sax/LejBP0IzqQPOSMGis9PLldReVAYK9UdivuDGSZeDkpQ456r/TV7csjVAaJqjWHc9NjJ9RZTgTOCl2U40JZSM6wI6lkao/Wx28IScWqVPwljZkobM1N8TGY20HkeB7YyoGepFbyr+53VSE175GZdJalCy+aIwFcTEZPo96XOFzIixJZQpbm8lbEgVZcZmVLQheIsvL5NmteKdV6p3F+XadR5HAY7hBM7Ag0uowS3UoQEMIniGV3hzlPivDsf89YVJ585gj9wPn8AtneQWQ=</latexit>

. . .

<latexit sha1_base64="VBpHJEBT97tDzHG8unvL1loUmU=">AB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSRV0GPRi8cK9gPaUDabTbt2sxt2J0Ip/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZemApu0PO+ncLa+sbmVnG7tLO7t39QPjxqGZVpypUCaU7ITFMcMmayFGwTqoZSULB2uHodua3n5g2XMkHKcsSMhA8phTglZq9USk0PTLFa/qzeGuEj8nFcjR6Je/epGiWcIkUkGM6fpeisGEaORUsGmplxmWEjoiA9a1VJKEmWAyv3bqnlklcmOlbUl05+rviQlJjBknoe1MCA7NsjcT/O6GcbXwYTLNEMm6WJRnAkXlTt73Y24ZhTF2BJCNbe3unRINKFoAyrZEPzl1dJq1b1L6q1+8tK/SaPowgncArn4MV1OEOGtAECo/wDK/w5ijnxXl3PhatBSefOY/cD5/AL1tjzw=</latexit>

κ

<latexit sha1_base64="bD1kN69KciN2AkFMW9xq/coE4K4=">AB7XicbVBNSwMxEM3Wr1q/qh69BIvgqexWQY9FLx4r2A9olzKbZtvYbBKSrFCW/gcvHhTx6v/x5r8xbfegrQ8GHu/NMDMvUpwZ6/vfXmFtfWNzq7hd2tnd2z8oHx61jEw1oU0iudSdCAzlTNCmZbTjtIUkojTdjS+nfntJ6oNk+LBThQNExgKFjMC1kmt3hiUgn654lf9OfAqCXJSQTka/fJXbyBJmlBhCQdjuoGvbJiBtoxwOi31UkMVkDEMadRAQk1YTa/dorPnDLAsdSuhMVz9fdEBokxkyRynQnYkVn2ZuJ/Xje18XWYMaFSwVZLIpTjq3Es9fxgGlKLJ84AkQzdysmI9BArAuo5EIl9eJa1aNbio1u4vK/WbPI4iOkGn6BwF6ArV0R1qoCYi6BE9o1f05knvxXv3PhatBS+fOUZ/4H3+AJd6jyM=</latexit>

SFG

<latexit sha1_base64="2OqbGwDbT9+tYG6sKAnjh3PFUs4=">AB83icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DEoqMeI5gHZJcxOJsmQ2QczvWJY9je8eFDEqz/jzb9xkuxBEwsaiqpurv8WAqNtv1tLS2vrK6tFzaKm1vbO7ulvf2mjhLFeINFMlJtn2ouRcgbKFDydqw4DXzJW/7oauK3HrnSIgofcBxzL6CDUPQFo2gkN3WRP2F6f32TZd1S2a7YU5BF4uSkDnq3dKX24tYEvAQmaRadxw7Ri+lCgWTPCu6ieYxZSM64B1DQxpw7aXTmzNybJQe6UfKVIhkqv6eSGmg9TjwTWdAcajnvYn4n9dJsH/hpSKME+Qhmy3qJ5JgRCYBkJ5QnKEcG0KZEuZWwoZUYmpqIJwZl/eZE0qxXntFK9OyvXLvM4CnAIR3ACDpxDW6hDg1gEMzvMKblVgv1rv1MWtdsvKZA/gD6/MHYVmR5w=</latexit>

|0i

<latexit sha1_base64="mFbyg4gJW0ZXAvu82LjFXT3NAg=">AB8HicbVBNS8NAEJ34WetX1aOXxSJ4KkV9Fj04rGC/ZA2lM120i7dbMLuRixv8KLB0W8+nO8+W/ctjlo64OBx3szMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6mfqtR1Sax/LejBP0IzqQPOSMGis9PLldReVAYK9UdivuDGSZeDkpQ456r/TV7csjVAaJqjWHc9NjJ9RZTgTOCl2U40JZSM6wI6lkao/Wx28IScWqVPwljZkobM1N8TGY20HkeB7YyoGepFbyr+53VSE175GZdJalCy+aIwFcTEZPo96XOFzIixJZQpbm8lbEgVZcZmVLQheIsvL5NmteKdV6p3F+XadR5HAY7hBM7Ag0uowS3UoQEMIniGV3hzlPivDsf89YVJ585gj9wPn8AtneQWQ=</latexit>

1 − κ

<latexit sha1_base64="tVBlvAtxGqGUAjglGPN1drYNaDg=">AB73icbVBNS8NAEJ34WetX1aOXxSJ4sSRV0GPRi8cK9gPaUCbTbt0s4m7G6GE/gkvHhTx6t/x5r9x2+agrQ8GHu/NMDMvSATXxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqKGvQWMSqHaBmgkvWMNwI1k4UwygQrBWMbqd+64kpzWP5YMYJ8yMcSB5yisZKbe+8O8IkwV6p7FbcGcgy8XJShz1Xumr249pGjFpqECtO56bGD9DZTgVbFLspolSEc4YB1LJUZM+9ns3gk5tUqfhLGyJQ2Zqb8nMoy0HkeB7YzQDPWiNxX/8zqpCa/9jMskNUzS+aIwFcTEZPo86XPFqBFjS5Aqbm8ldIgKqbERFW0I3uLy6RZrXgXler9Zbl2k8dRgGM4gTPw4ApqcAd1aAFAc/wCm/Oo/PivDsf89YVJ585gj9wPn8AckSPlQ=</latexit>

Feed-Forward SFG

ˆ b1

<latexit sha1_base64="SZ6+Fcv8xrdlOlzQKbKeQVgOkWs=">AB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae2oWy2m3bpZhN2J0IJ/RdePCji1X/jzX/jts1BWx8MPN6bYWZekEh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRI3g7GtzO/cS1EbF6wEnC/YgOlQgFo2ilx6w3okiCad/rlytu1Z2DrBIvJxXI0eiXv3qDmKURV8gkNabruQn6GdUomOTUi81PKFsTIe8a6miETd+Nr94Ss6sMiBhrG0pJHP190RGI2MmUWA7I4ojs+zNxP+8borhtZ8JlaTIFVsClNJMCaz98lAaM5QTiyhTAt7K2EjqilDG1LJhuAtv7xKWrWqd1Gt3V9W6jd5HEU4gVM4Bw+uoA530IAmMFDwDK/w5hjnxXl3PhatBSefOY/cD5/ABL8kIU=</latexit>

ˆ b2

<latexit sha1_base64="fsgi28M0bK1XZcXTY+TFMrxg=">AB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae2oWy2m3bpZhN2J0IJ/RdePCji1X/jzX/jts1BWx8MPN6bYWZekEh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRI3g7GtzO/cS1EbF6wEnC/YgOlQgFo2ilx6w3okiCab/WL1fcqjsHWSVeTiqQo9Evf/UGMUsjrpBJakzXcxP0M6pRMmnpV5qeELZmA51JFI278bH7xlJxZUDCWNtSObq74mMRsZMosB2RhRHZtmbif953RTDaz8TKkmRK7ZYFKaSYExm75OB0JyhnFhCmRb2VsJGVFOGNqSDcFbfnmVtGpV76Jau7+s1G/yOIpwAqdwDh5cQR3uoAFNYKDgGV7hzTHOi/PufCxaC04+cwx/4Hz+ABSAkIY=</latexit>

ˆ bK

<latexit sha1_base64="fv9hNDgQyICZ8/ihlvekegU8ms=">AB8XicbVBNS8NAEJ34WetX1aOXxSJ4KkV9Fj0InipYD+wDWz3bRLN5uwOxFK6L/w4kERr/4b/4bt20O2vpg4PHeDPzgkQKg67aysrq1vbBa2its7u3v7pYPDpolTzXiDxTLW7YAaLoXiDRQoeTvRnEaB5K1gdDP1W09cGxGrBxwn3I/oQIlQMIpWesy6Q4okmPTueqWyW3FnIMvEy0kZctR7pa9uP2ZpxBUySY3peG6CfkY1Cib5pNhNDU8oG9EB71iqaMSNn80unpBTq/RJGtbCslM/T2R0ciYcRTYzoji0Cx6U/E/r5NieOVnQiUpcsXmi8JUEozJ9H3SF5ozlGNLKNPC3krYkGrK0IZUtCF4iy8vk2a14p1XqvcX5dp1HkcBjuEzsCDS6jBLdShAQwUPMrvDnGeXHenY9564qTzxzBHzifPzpkJ8=</latexit>

1

<latexit sha1_base64="TtIgPQprnJE4HS+PuM3etxya8=">AB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlptcvV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYglitbCRtRZmx2ZRsCN7y6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAcIzvMKb8+i8O/Ox6K14OQzx/AHzucPe+uMuQ=</latexit>

2

<latexit sha1_base64="jk/1fpohXujb3eq/tOFNvjxoFrw=">AB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlZq1frhVdw6ySrycVCBHo1/+6g1ilkYoDRNU67nJsbPqDKcCZyWeqnGhLIxHWLXUkj1H42P3RKzqwyIGsbElD5urviYxGWk+iwHZG1Iz0sjcT/O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaijJjsynZELzl1dJu1b1Lq15mWlfpPHUYQTOIVz8OAK6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8fW+Mug=</latexit>

K

<latexit sha1_base64="HQfadDQrozr2PpvG928HQmjcOjI=">AB6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BL4KXBMwDkiXMTnqTMbOzy8ysEK+wIsHRbz6Sd78GyfJHjSxoKGo6qa7K0gE18Z1v53c2vrG5lZ+u7Czu7d/UDw8auo4VQwbLBaxagdUo+ASG4Ybge1EIY0Cga1gdDvzW0+oNI/lgxkn6Ed0IHnIGTVWqt/3iW37M5BVomXkRJkqPWKX91+zNIpWGCat3x3MT4E6oMZwKnhW6qMaFsRAfYsVTSCLU/mR86JWdW6ZMwVrakIXP198SERlqPo8B2RtQM9bI3E/zOqkJr/0Jl0lqULFojAVxMRk9jXpc4XMiLElClubyVsSBVlxmZTsCF4y+vkmal7F2UK/XLUvUmiyMPJ3AK5+DBFVThDmrQAYIz/AKb86j8+K8Ox+L1pyTzRzDHzifP6NTjNM=</latexit>

ˆ e1

<latexit sha1_base64="gVMmAqLgehirfX4LWVPGVCPNigs=">AB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae2oWy2m3bpZhN2J0IJ/RdePCji1X/jzX/jts1BWx8MPN6bYWZekEh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRI3g7GtzO/cS1EbF6wEnC/YgOlQgFo2ilx6w3okj4tO/1yxW36s5BVomXkwrkaPTLX71BzNKIK2SGtP13AT9jGoUTPJpqZcanlA2pkPetVTRiBs/m18JWdWGZAw1rYUkrn6eyKjkTGTKLCdEcWRWfZm4n9eN8Xw2s+ESlLki0WhakGJPZ+2QgNGcoJ5ZQpoW9lbAR1ZShDalkQ/CWX14lrVrVu6jW7i8r9Zs8jiKcwCmcgwdXUIc7aEATGCh4hld4c4z4rw7H4vWgpPHMfOJ8/F5GQiA=</latexit>

ˆ e2

<latexit sha1_base64="f5jVxeon67v3DO0LB6DB13AqekY=">AB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae2oWy2m3bpZhN2J0IJ/RdePCji1X/jzX/jts1BWx8MPN6bYWZekEh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRI3g7GtzO/cS1EbF6wEnC/YgOlQgFo2ilx6w3okj4tF/rlytu1Z2DrBIvJxXI0eiXv3qDmKURV8gkNabruQn6GdUomOTUi81PKFsTIe8a6miETd+Nr94Ss6sMiBhrG0pJHP190RGI2MmUWA7I4ojs+zNxP+8borhtZ8JlaTIFVsClNJMCaz98lAaM5QTiyhTAt7K2EjqilDG1LJhuAtv7xKWrWqd1Gt3V9W6jd5HEU4gVM4Bw+uoA530IAmMFDwDK/w5hjnxXl3PhatBSefOY/cD5/ABkVkIk=</latexit>

ˆ eK

<latexit sha1_base64="6IjkcGm+9SmBJjw9qgLb4fwHbzc=">AB8XicbVBNS8NAEJ34WetX1aOXxSJ4KkV9Fj0InipYD+wDWz3bRLN5uwOxFK6L/w4kERr/4b/4bt20O2vpg4PHeDPzgkQKg67aysrq1vbBa2its7u3v7pYPDpolTzXiDxTLW7YAaLoXiDRQoeTvRnEaB5K1gdDP1W09cGxGrBxwn3I/oQIlQMIpWesy6Q4qET3p3vVLZrbgzkGXi5aQMOeq90le3H7M04gqZpMZ0PDdBP6MaBZN8UuymhieUjeiAdyxVNOLGz2YXT8ipVfokjLUthWSm/p7IaGTMOApsZ0RxaBa9qfif10kxvPIzoZIUuWLzRWEqCcZk+j7pC80ZyrElGlhbyVsSDVlaEMq2hC8xZeXSbNa8c4r1fuLcu06j6MAx3ACZ+DBJdTgFurQAYKnuEV3hzjvDjvzse8dcXJZ47gD5zPHz75kKI=</latexit>

ˆ aB

<latexit sha1_base64="2zdAPgE4ZcHd6KZQblxI2SYPNfM=">AB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JUQZelblxWsA9oQriZTtqhkwczE6HELPwVNy4UcetvuPNvnKZaOuBC4dz7p079/gJZ1JZ1rexsrq2vrFZ2apu7+zu7ZsHh10Zp4LQDol5LPo+SMpZRDuKU7iaAQ+pz2/MnNzO89UCFZHN2raULdEYRCxgBpSXPHaKNzKfpzTPnDEoDLnX8syaVbcK4GVil6SGSrQ98sZxiQNaQIBykHtpUoNwOhGOE0rzqpAmQCYzoQNMIQirdrNid4zOtDHEQC12RwoX6eyKDUMp6OvOENRYLnoz8T9vkKrg2s1YlKSKRmS+KEg5VjGehYGHTFCi+FQTILpv2IyBgFE6ciqOgR78eRl0m3U7Yt64+6y1myVcVTQCTpF58hGV6iJblEbdRBj+gZvaI348l4Md6Nj3nrilHOHKE/MD5/AGuzlk=</latexit>

ˆ aE

<latexit sha1_base64="0F8L9FoJE+ujHIjawGhDzxiO8M=">AB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JUQZdFEVxWsA9oQriZTtqhkwczE6HELPwVNy4UcetvuPNvnKZaOuBC4dz7p079/gJZ1JZ1rextLyurZe2ahubm3v7Jp7+x0Zp4LQNol5LHo+SMpZRNuKU57iaAQ+px2/fH1O8+UCFZHN2rSULdEIYRCxgBpSXPHSKNzKfpzTPnBEoDLl345k1q24VwIvELkNlWh5pcziEka0kgRDlL2bStRbgZCMcJpXnVSRMgYxjSvqYRhFS6WbE7xydaGeAgFroihQv190QGoZST0NedIaiRnPem4n9eP1XBpZuxKEkVjchsUZByrGI8DQMPmKBE8YkmQATf8VkBAKI0pFVdQj2/MmLpNOo2f1xt15rXlVxlFBR+gYnSIbXaAmukUt1EYEPaJn9IrejCfjxXg3PmatS0Y5c4D+wPj8AXA/lw=</latexit>

(NB)

<latexit sha1_base64="jyzafBPxLk+csa3JMwvmrW7MWQ=">AB7HicbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY6sWTVHBtoV1KNs2oUl2SbJCWfobvHhQxKs/yJv/xrTdg7Y+GHi8N8PMvDhTBvX/XYKa+sbm1vF7dLO7t7+Qfnw6FHqSLUJzGPVSfEmnImqW+Y4bSTKIpFyGk7HN/M/PYTVZrF8sFMEhoIPJQsYgQbK/nVu37zvF+uDV3DrRKvJxUIEerX/7qDWKSCioN4VjrucmJsiwMoxwOi31Uk0TMZ4SLuWSiyoDrL5sVN0ZpUBimJlSxo0V39PZFhoPRGh7RTYjPSyNxP/87qpia6DjMkNVSxaIo5cjEaPY5GjBFieETSzBRzN6KyAgrTIzNp2RD8JZfXiWP9Zp3UavfX1YazTyOIpzAKVTBgytowC20wAcCDJ7hFd4c6bw4787HorXg5DPH8AfO5w+s643w</latexit>

|Φi⊗M

<latexit sha1_base64="7s9vj2UUONwXcSucm8xiiIDxpNI=">AAACAXicbVBNS8NAEJ34WetX1IvgZbEInkpSBT0WvXgRKtgPaGLZbDft0s0m7G6EEuvFv+LFgyJe/Rfe/Ddu2xy09cHA470ZZuYFCWdKO863tbC4tLyyWlgrrm9sbm3bO7sNFaeS0DqJeSxbAVaUM0HrmmlOW4mkOAo4bQaDy7HfvKdSsVjc6mFC/Qj3BAsZwdpIHXv/wav1mSex6HF6l3mxZhFV6HrUsUtO2ZkAzRM3JyXIUevYX143JmlEhSYcK9V2nUT7GZaaEU5HRS9VNMFkgHu0bajAZo+fTT4YoSOjdFEYS1NCo4n6eyLDkVLDKDCdEdZ9NeuNxf+8dqrDcz9jIkk1FWS6KEw50jEax4G6TFKi+dAQTCQztyLSxxITbUIrmhDc2ZfnSaNSdk/KlZvTUvUij6MAB3AIx+DCGVThCmpQBwKP8Ayv8GY9WS/Wu/UxbV2w8pk9+APr8wfPpZcd</latexit>

(NS)

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(NS)

<latexit sha1_base64="K1pIQk/DOHtmT8xw6GKG29wpO/c=">AB7HicbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9OJKrq20C4lm2b0CS7JFmhLP0NXjwo4tUf5M1/Y9ruQVsfDzem2FmXphwpo3rfjuFldW19Y3iZmlre2d3r7x/8KjVBHqk5jHqh1iTmT1DfMcNpOFMUi5LQVjq6nfuJKs1i+WDGCQ0EHkgWMYKNlfzqbe/+tFeuDV3BrRMvJxUIEezV/7q9mOSCioN4VjrjucmJsiwMoxwOil1U0TEZ4QDuWSiyoDrLZsRN0YpU+imJlSxo0U39PZFhoPRah7RTYDPWiNxX/8zqpiS6DjMkNVS+aIo5cjEaPo56jNFieFjSzBRzN6KyBArTIzNp2RD8BZfXiaP9Zp3VqvfnVcaV3kcRTiCY6iCBxfQgBtog8EGDzDK7w50nlx3p2PeWvByWcO4Q+czx/GwI4B</latexit>

ˆ aS

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ˆ aI

<latexit sha1_base64="s03QHc53EeNFQ824o6PEK7RNks=">AB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JUQZdFN7qrYB/QhHAznbRDJw9mJkKJWfgrblwo4tbfcOfOE2z0NYDFw7n3Dt37vETzqSyrG9jaXldW29slHd3Nre2TX39jsyTgWhbRLzWPR8kJSziLYVU5z2EkEh9Dnt+uPrqd9oEKyOLpXk4S6IQwjFjACSkuegUb2Q+T2meOSNQGHLv1jNrVt0qgBeJXZIaKtHyzC9nEJM0pJEiHKTs21ai3AyEYoTvOqkiZAxjCkfU0jCKl0s2J3jk+0MsBLHRFChfq74kMQiknoa87Q1AjOe9Nxf+8fqCSzdjUZIqGpHZoiDlWMV4GgYeMEGJ4hNgAim/4rJCAQpSOr6hDs+ZMXSadRt8/qjbvzWvOqjKOCjtAxOkU2ukBNdINaqI0IekTP6BW9GU/Gi/FufMxal4xy5gD9gfH5A3ZPlmA=</latexit>

ˆ aR

<latexit sha1_base64="Y7fd2dX/n+MNlT4QfF0LfLkJeo=">AB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JUQZdFNy6r2Ac0IUymk3bo5MHMjVBiFv6KGxeKuPU3Pk3TtMstPXAhcM5986de/xEcAW9W0sLa+srq1XNqbW9s7u+befkfFqaSsTWMRy5PFBM8Ym3gIFgvkYyEvmBdf3w9bsPTCoeR/cwSZgbkmHEA04JaMkzD53ijcwXKcszZ0QAk9y78yaVbcK4EVil6SGSrQ8sZxDQNWQRUEKX6tpWAmxEJnAqWV51UsYTQMRmyvqYRCZlys2J3jk+0MsBLHVFgAv190RGQqUmoa87QwIjNe9Nxf+8fgrBpZvxKEmBRXS2KEgFhPw8ADLhkFMdGEUMn1XzEdEUko6MiqOgR7/uRF0mnU7bN64/a81rwq46igI3SMTpGNLlAT3aAWaiOKHtEzekVvxpPxYrwbH7PWJaOcOUB/YHz+AIPzlmk=</latexit>

hˆ aRˆ aIi = p ηNS(NS + 1)

<latexit sha1_base64="dnLD8rGt/Z0FibiLai/QpvdPNO8=">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</latexit>

µ =

• 1 − κ(1 + N 0

S)

2

<latexit sha1_base64="UY9I/p+5JbrbAJQvjMbIOKPg7X8=">AAACEXicbVDLSgMxFM3UV62vqks3wSK0iGWmCroRim5cSUX7gM50yKRpG5rMhCQjlKG/4MZfceNCEbfu3Pk3pu0stHrgwuGce7n3nkAwqrRtf1mZhcWl5ZXsam5tfWNzK7+901BRLDGp44hFshUgRRgNSV1TzUhLSIJ4wEgzGF5O/OY9kYpG4Z0eCeJx1A9pj2KkjeTniy6P4Tl0A9ovOkfuEAmBis7htX/bcYWknJQmVqlT8fMFu2xPAf8SJyUFkKLm5z/dboRjTkKNGVKq7dhCewmSmmJGxjk3VkQgPER90jY0RJwoL5l+NIYHRunCXiRNhRpO1Z8TCeJKjXhgOjnSAzXvTcT/vHase2deQkMRaxLi2aJezKCO4CQe2KWSYM1GhiAsqbkV4gGSCGsTYs6E4My//Jc0KmXnuFy5OSlUL9I4smAP7IMicMApqIIrUAN1gMEDeAIv4NV6tJ6tN+t91pqx0pld8AvWxzcE15tC</latexit>

NT = κNSN 0

S

<latexit sha1_base64="3WFZLVUQ8GDb/5sjOjz0pOJQbcU=">AAACBHicbVDLSsNAFJ3UV62vqMtuBovgqiRV0I1QdOOqVOwLmhgm00k7dJIMMxOhhC7c+CtuXCji1o9w5984abPQ1gMXDufcy733+JxRqSzr2yisrK6tbxQ3S1vbO7t75v5BR8aJwKSNYxaLno8kYTQibUUVIz0uCAp9Rrr++Drzuw9ESBpHLTXhxA3RMKIBxUhpyTPLDa8FL6EzRpwj2PDusrp3uKAh8cyKVbVmgMvEzkkF5Gh65pcziHESkkhhhqTs2xZXboqEopiRaclJJOEIj9GQ9DWNUEikm86emMJjrQxgEAtdkYIz9fdEikIpJ6GvO0OkRnLRy8T/vH6iggs3pRFPFInwfFGQMKhimCUCB1QQrNhEE4QF1bdCPEICYaVzK+kQ7MWXl0mnVrVPq7Xbs0r9Ko+jCMrgCJwAG5yDOrgBTdAGGDyCZ/AK3own48V4Nz7mrQUjnzkEf2B8/gB7tpa7</latexit>

The SF ( ) modes are in displaced thermal states:

ˆ bk

<latexit sha1_base64="ZMw1hLCQsVTayoENE/hMoy9Yw=">AB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae2oWy2m3bpZhN2J0IJ/RdePCji1X/jzX/jts1BWx8MPN6bYWZekEh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRI3g7GtzO/cS1EbF6wEnC/YgOlQgFo2ilx6w3okiCaX/cL1fcqjsHWSVeTiqQo9Evf/UGMUsjrpBJakzXcxP0M6pRMmnpV5qeELZmA51JFI278bH7xlJxZUDCWNtSObq74mMRsZMosB2RhRHZtmbif953RTDaz8TKkmRK7ZYFKaSYExm75OB0JyhnFhCmRb2VsJGVFOGNqSDcFbfnmVtGpV76Jau7+s1G/yOIpwAqdwDh5cQR3uoAFNYKDgGV7hzTHOi/PufCxaC04+cwx/4Hz+AGrkL8=</latexit>

ˆ ρth(α(k), NT ) = Z

C

1 πNT e−|β−α(k)|2/NT |βihβ|d2β

<latexit sha1_base64="QQZDYYNELFNnlaqArA6dpMX+AnY=">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</latexit>

α(k) = q MκηNS(1 + NS)µk−1

<latexit sha1_base64="Jwx8viOdv4VvGeeDIn6MeBJ5unU=">AAACK3icbVDLSgMxFM34tr6qLt0Ei1ARy4wKuhFEN24URatCp5Y7adqGyczE5I5QhvkfN/6KC134wK3/YVq78HUg4XDOvTc3J1BSGHTdV2doeGR0bHxisjA1PTM7V5xfuDBJqhmvskQm+ioAw6WIeRUFSn6lNIcokPwyCA96/uUt10Yk8Tl2Fa9H0I5FSzBAKzWK+35/RhbIlOcFH6TqwHVWDldzukt9c6MxO/JDUAp8jkCPG2dlb83eq36UXmfhupfnjWLJrbh90L/EG5ASGeCkUXz0mwlLIx4jk2BMzXMV1jPQKJjsbZEaroCF0OY1S2OIuKln/T1zumKVJm0l2p4YaV/93pFBZEw3CmxlBNgxv72e+J9XS7G1U89ErFLkMft6qJVKigntBUebQnOGsmsJMC3srpR1QANDG2/BhuD9/vJfcrFR8TYrG6dbpb39QRwTZIkskzLxyDbZI4fkhFQJI3fkgTyTF+feeXLenPev0iFn0LNIfsD5+AQXF6bt</latexit>

t = π/2 √ Mg

<latexit sha1_base64="Kytf83e5zL1nkxDeiEIHW4q0i2c=">AB+3icbVBNS8NAEN34WetXrEcvi0XwVJMq6EUoevEiVLAf0JSy2W7apZtN3J2IJeSvePGgiFf/iDf/jds2B219MPB4b4aZeX4suAbH+baWldW19YLG8XNre2dXuv1NRoihr0EhEqu0TzQSXrAEcBGvHipHQF6zlj64nfuRKc0jeQ/jmHVDMpA84JSAkXp2CfAl9mJ+UvX0g4L0Nhv07LJTcabAi8TNSRnlqPfsL68f0SRkEqgWndcJ4ZuShRwKlhW9BLNYkJHZMA6hkoSMt1Np7dn+MgofRxEypQEPFV/T6Qk1Hoc+qYzJDU895E/M/rJBcdFMu4wSYpLNFQSIwRHgSBO5zxSiIsSGEKm5uxXRIFKFg4iqaENz5lxdJs1pxTyvVu7Ny7SqPo4AO0CE6Ri46RzV0g+qogSh6Qs/oFb1ZmfVivVsfs9YlK5/ZR39gf4A7dCTvA=</latexit>

SLIDE 23

Transmitter and joint-detection receiver for entanglement assisted communication

µ =

• 1 − κ(1 + N 0

S)

2

<latexit sha1_base64="UY9I/p+5JbrbAJQvjMbIOKPg7X8=">ACEXicbVDLSgMxFM3UV62vqks3wSK0iGWmCroRim5cSUX7gM50yKRpG5rMhCQjlKG/4MZfceNCEbfu3Pk3pu0stHrgwuGce7n3nkAwqrRtf1mZhcWl5ZXsam5tfWNzK7+901BRLDGp4hFshUgRgNSV1TzUhLSIJ4wEgzGF5O/OY9kYpG4Z0eCeJx1A9pj2KkjeTniy6P4Tl0A9ovOkfuEAmBis7htX/bcYWknJQmVqlT8fMFu2xPAf8SJyUFkKLm5z/dboRjTkKNGVKq7dhCewmSmJGxjk3VkQgPER90jY0RJwoL5l+NIYHRunCXiRNhRpO1Z8TCeJKjXhgOjnSAzXvTcT/vHase2deQkMRaxLi2aJezKCO4CQe2KWSYM1GhiAsqbkV4gGSCGsTYs6E4My/Jc0KmXnuFy5OSlUL9I4smAP7IMicMApqIrUAN1gMEDeAIv4NV6tJ6tN+t91pqx0pld8AvWxzcE15tC</latexit>

NT = κNSN 0

S

<latexit sha1_base64="3WFZLVUQ8GDb/5sjOjz0pOJQbcU=">ACBHicbVDLSsNAFJ3UV62vqMtuBovgqiRV0I1QdOqVOwLmhgm0k7dJIMxOhC7c+CtuXCji1o9w5984abPQ1gMXDufcy73+JxRqSzr2yisrK6tbxQ3S1vbO7t75v5BR8aJwKSNYxaLno8kYTQibUVIz0uCAp9Rr+Drzuw9ESBpHLTXhxA3RMKIBxUhpyTPLDa8FL6EzRpwj2PDusrp3uKAh8cyKVbVmgMvEzkF5Gh65pcziHESkhqTs2xZXboqEopiRaclJOEIj9GQ9DWNUEikm86emMJjrQxgEAtdkYIz9fdEikIpJ6GvO0OkRnLRy8T/vH6igs3pRFPFInwfFGQMKhimCUCB1QrNhE4QF1bdCPEICYaVzK+kQ7MWXl0mnVrVPq7Xbs0r9Ko+jCMrgCJwAG5yDOrgBTdAGDyCZ/AK3own48V4Nz7mrQUjnzkEf2B8/gB7tpa7</latexit>

k-th Green Machine’s inputs: k-th Green Machine’s outputs (one of them has this mean, others have zero mean)

SG, Zhuang, Bash, arXiv:2001.03934 (2020)

SLIDE 24

Superadditivity in entanglement assisted communication rate

∼ log(1/NS)

<latexit sha1_base64="TxqUTKCDeDilaZfBIE16k2k/yOk=">AB+XicbVDLSsNAFJ34rPUVdelmsAh1U5Mq6LoxpVUtA9oQphMp+3QeYSZSaGE/okbF4q49U/c+TdO2y09cCFwzn3cu89cKoNp737aysrq1vbBa2its7u3v7sFhU8tUYdLAknVjpEmjArSMNQw0k4UQTxmpBUPb6d+a0SUplI8mXFCQo76gvYoRsZKkesGmnIYMNkv+f30eNZ5Ja8ijcDXCZ+TkogRz1yv4KuxCknwmCGtO74XmLCDClDMSOTYpBqkiA8RH3SsVQgTnSYzS6fwFOrdGFPKlvCwJn6eyJDXOsxj20nR2agF72p+J/XSU3vOsyoSFJDBJ4v6qUMGgmnMcAuVQbNrYEYUXtrRAPkELY2LCKNgR/8eVl0qxW/ItK9eGyVLvJ4yiAY3ACysAHV6AG7kAdNAGI/AMXsGbkzkvzrvzMW9dcfKZI/AHzucPsFeSYw=</latexit>

SG, Zhuang, Bash, arXiv:2001.03934 (2020)

SLIDE 25

Outline

• Background

– Quantum limit of classical communications – Joint detection receivers for superadditive capacity

• Entanglement assisted communications over the bosonic channel

– Transmitter-receiver design – Entanglement assisted covert communications

• Conclusions and ongoing work

SLIDE 26

Covert communications

• Provably covert communications needs , resulting in:
• Entanglement assisted covert communications: the log(1/NS)

enhancement in capacity results in:

• B. A. Bash, A. H. Gheorghe, M. Patel, J. L. Habif, D. Goeckel, D. Towsley, and SG, Nat. Commun. 6, 8626 (2015)

O(√n)

<latexit sha1_base64="TELHNR7/AB9bIeqPwboqfTj7TsI=">AB8nicbVBNSwMxEM3Wr1q/qh69BItQL2W3CnosevFmBVsL26Vk02wbmk3WZFYoS3+GFw+KePXePfmLZ70NYHA4/3ZpiZFyaCG3Ddb6ewsrq2vlHcLG1t7+zulfcP2kalmrIWVULpTkgME1yFnAQrJNoRuJQsIdwdD31H56YNlzJexgnLIjJQPKIUwJW8m+rXfOoIZOT01654tbcGfAy8XJSQTmavfJXt69oGjMJVBjfM9NIMiIBk4Fm5S6qWEJoSMyYL6lksTMBNns5Ak+sUofR0rbkoBn6u+JjMTGjOPQdsYEhmbRm4r/eX4K0WQcZmkwCSdL4pSgUHh6f+4zWjIMaWEKq5vRXTIdGEgk2pZEPwFl9eJu16zTur1e/OK42rPI4iOkLHqIo8dIEa6AY1UQtRpNAzekVvDjgvzrvzMW8tOPnMIfoD5/MHCg6RGA=</latexit>

n

<latexit sha1_base64="2QmInd+nHO60uhS7AYCvgpiU4a8=">AB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlpuyXK27VnYOsEi8nFcjR6Je/eoOYpRFKwTVu5ifEzqgxnAqelXqoxoWxMh9i1VNItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2oszYbEo2BG/5VXSrlW9i2qteVmp3+RxFOETuEcPLiCOtxBA1rAOEZXuHNeXRenHfnY9FacPKZY/gD5/MH2F+M9g=</latexit>

bits in modes

Gagatsos, Bullock, Bash, arXiv:2002.06733 [quant-ph] (2020)

O(√n log n)

<latexit sha1_base64="CWwE1TwSGsZQfI81Zdz1DRFCyo=">AB/HicbVDLSsNAFJ3UV62vaJduBotQUpSBV0W3bizgn1AE8pkOmHTmbizEQIof6KGxeKuPVD3Pk3TtstPXAhcM593LvPUHMqNKO820VlbX1jeKm6Wt7Z3dPXv/oK1EIjFpYcGE7AZIEUY5aWmqGenGkqAoYKQTjK+nfueRSEUFv9dpTPwIDTkNKUbaSH27fFv1IPUGZ94px4TQ8hP+nbFqTkzwGXi5qQCcjT79pc3EDiJCNeYIaV6rhNrP0NSU8zIpOQlisQIj9GQ9AzlKCLKz2bHT+CxUQYwFNIU13Cm/p7IUKRUGgWmM0J6pBa9qfif10t0eOlnlMeJhzPF4UJg1rAaRJwQCXBmqWGICypuRXiEZIa5NXyYTgLr68TNr1mntWq9+dVxpXeRxFcAiOQBW4AI0wA1oghbAIAXP4BW8WU/Wi/VufcxbC1Y+UwZ/YH3+AO9hlE0=</latexit>

n

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bits in modes

NS = O(1/√n)

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SLIDE 27

Outline

• Background

– Quantum limit of classical communications – Joint detection receivers for superadditive capacity

• Entanglement assisted communications over the bosonic channel

– Transmitter-receiver design – Entanglement assisted covert communications

• Conclusions and ongoing work

SLIDE 28

Conclusions and ongoing work

• Holevo capacity, C: Quantum limit of classical communications

– (product) coherent state, i.e., laser light modulation is capacity optimal – Attaining ultimate limit of classical communications needs quantum (joint detection) receivers

• Entanglement assisted capacity, CE: highest gain when the

transmitter is low brightness and thermal noise is high

– TMSV pre-shared entanglement and SFG receiver front end translates the problem back to receiver design for coherent states to achieve Holevo capacity – Covert communications: shared entangled breaks the “square root law”

• Find better transmitter modulation, codes, and receivers to fully

achieve CE: explore resource theories of non-Gaussian components

• Connection with quantum illumination radar

SLIDE 29

Thank you!

Quntao Zhuang Boulat Bash Jeffrey Shapiro Baris Erkmen Kaushik Seshadreesan Narayanan Rengaswamy Henry Pfister Lizhong Zheng Hye-Won Chung Christos Gagatsos