Coherent quantum channel discrimination Mark M. Wilde Hearne - - PowerPoint PPT Presentation

Coherent quantum channel discrimination

Mark M. Wilde

Hearne Institute for Theoretical Phyiscs Department of Physics and Astronomy Center for Computation and Technology Louisiana State University Baton Rouge, Louisiana 70803, USA

Available as arXiv:2001.02668 ISIT 2020 (virtual)

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Motivation

Diamond distance is a fundamental metric used for characterizing the distinguishability of quantum channels One reason is that it appears in the expression for the success probability in distinguishing two quantum channels in quantum channel discrimination Here I propose a fully coherent or fully quantum version of quantum channel discrimination Motivation: Ideal quantum computers operate in a fully coherent manner, and we often take a circuit and construct a controlled version

• f it. We may not implement a perfect controlled version and so we

want to understand how the unideal case deviates from the ideal one, even when the channel is called in controlled form

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Review of diamond distance

Diamond distance quantifies the distance between two quantum channels N 0

A→B and N 1 A→B, and there are two good reasons to use it

(discussed later) It is defined as follows:

• N 0 − N 1
• 1 := sup

ρRA

• N 0

A→B(ρRA) − N 1 A→B(ρRA)

• 1

where · 1 is the trace norm and the optimization is over all bipartite states ρRA with unbounded reference system R. It suffices to restrict the optimization to be over pure bipartite states ψRA with system R isomorphic to system A.

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Reason 1: Diamond distance as “observational error”

Diamond distance characterizes “observational error” between quantum channels N 0

A→B and N 1 A→B.

Most general way that we can process a channel to obtain classical data is to 1) prepare a state ρRA, 2) send A through the channel, and 3) perform a measurement on systems RB Probability for a particular measurement outcome to occur when using first channel is Tr[ΛRBN 0

A→B(ρRA)].

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Reason 1 (ctd.): Diamond dist. as “observational error”

Observational error between the different probabilities obtained by using the same procedure on different channels is

• Tr[ΛRBN 0

A→B(ρRA)] − Tr[ΛRBN 1 A→B(ρRA)]

• Maximum observational error = normalized diamond distance:

1 2

• N 0 − N 1
• 1 :=

sup

ρRA,ΛRB

• Tr[ΛRBN 0

A→B(ρRA)] − Tr[ΛRBN 1 A→B(ρRA)]

• Mark M. Wilde

Coherent quantum channel discrimination 5 / 17

Reason 2: Diamond distance from channel discrimination

Diamond distance arises in the expression for the optimal success probability in distinguishing two quantum channels. A quantum channel discrimination protocol:

R A

Alice Bob

Channel

B

Alice sends one share of a state ψRA. Bob flips a fair coin and applies N 0

A→B if heads and N 1 A→B if tails.

Optimal success probability in distinguishing channels is equal to 1 2

• 1 + 1

2

• N 0 − N 1
• 1
• Mark M. Wilde

Coherent quantum channel discrimination 6 / 17

Purifying quantum channel discrimination

Every quantum channel can be purified, such that it is realized by attaching an auxiliary system in the state |00|G, applying an isometric channel WAG→BE, and then tracing over E: N i

A→B(ωA) = TrE[WAG→BE(ρA ⊗ |00|G)].

Can depict conventional quantum channel discrimination as

R A

Verifier Prover

G

|0〉 π

R1

Wi

B E

V

F

|0〉

E’

|0〉

R F B E’ B

where π = (|00| + |11|)/2 is the maximally mixed qubit state, representing random choice of channel selection

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Coherent quantum channel discrimination

Main idea behind coherent quantum channel discrimination is to “coherify” each step of a channel discrimination protocol

R A

Verifier Prover

G

|0〉

R1

Wi

B E

V

F

|0〉

E’

|0〉

R B B E’ F

Wi

†

B E A G

|+〉

F

Replaces initial maximally mixed state with coherent version |++|, includes an uncomputing step, and final measurement is a Bell measurement

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Details of coherent channel discrimination

Step 1 → |+R1|ψRA Step 2 → 1 √ 2

• i∈{0,1}

|iR1W i

AG→BE|ψRA|0G

Step 3 → 1 √ 2

• i∈{0,1}

|iR1VRBE ′FW i

AG→BE|ψRA|000GE ′F.

Step 4 → 1 √ 2

• i∈{0,1}

|iR1W i†VW i|ψRA|000GE ′F, Step 5 → Measure {ΦR1F ⊗ |00|G, IR1FG − ΦR1F ⊗ |00|G} Success if first outcome occurs. Probability of success: 1 2

• Φ|R1F0|G
• i∈{0,1}

|iR1W i†VW i|ψRA|000GE ′F

• 2

2

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Optimal success probability of coh. channel discrimination

Optimizing over all input states |ψψ|RA and unitaries V of the prover, the optimal success probability is pcoh

s

(N 0, N 1) =      1 2 sup {Pi}i∈{0,1}:

• i Pi†Pi=IRB
• i∈{0,1}

(N i

A→B)†(Pi RB→RBE ′)

• ∞

    

2

This can be compared with an alternate expression for success probability of conventional channel discrimination: pinc

s (N 0, N 1) = 1

2 sup {Λi

RB}i∈{0,1}

• i∈{0,1}

(N i

A→B)†(Λi RB)

• ∞

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Bounding the optimal success probability

Optimal success probability is bounded as 1/2 ≤ pcoh

s

(N 0, N 1) ≤ 1 Upper bound is saturated if and only if the channels are orthogonal. Lower bound is saturated if the channels are identical. Bounds relating optimal success probability in coherent versus conventional channel discrimination pcoh

s

(N 0, N 1) ≤ pinc

s (N 0, N 1) ≤

• pcoh

s

(N 0, N 1)

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Quantum superchannels

Most general physical transformation of a quantum channel is a superchannel, which accepts as input a quantum channel and

• utputs a quantum channel

The superchannel Θ(A→B)→(C→D) takes as input a quantum channel NA→B and outputs a quantum channel KC→D, which we denote by Θ(A→B)→(C→D)(NA→B) = KC→D.

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Physical realizations of quantum superchannels

Superchannel has a physical realization in terms of pre- and post-processing quantum channels: Θ(A→B)→(C→D)(NA→B) = DBM→D ◦ NA→B ◦ EC→AM, where EC→AM and DBM→D are pre- and post-processing channels

C D B

N

A

D

M

E

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Optimal success probability and superchannels

Fundamental property: Optimal success probability of coherent channel discrimination does not increase under a superchannel Expected property because superchannel can be viewed as a particular kind of discrimination strategy; this is main proof idea:

C L

|0〉 G’

R C

Verifier Prover

G

|0〉

R1

Wi

B E

V

F

|0〉

E’

|0〉

R D D E’ F

Wi

†

B E A G

|+〉

F

WE |0〉

L A M1 M

WD

D M2

WD

M2 D

†

G’ B M M1

WE†

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Optimal success probability as SDP

Optimal success probability can be computed by a semi-definite program (SDP): sup

σR1FBE , ρA

• Tr[YR1FBEσR1FBE] : TrBF[σR1FBE] = Z ρ

R1E

• ,

where σR1FBE and ρA are density operators and YR1FBE := 1 2

• i,j∈{0,1},k,ℓ

|iijj|R1F ⊗ Ni

kNj† ℓ ⊗ |kℓ|E,

Z ρ

R1E := 1

2

• i,j∈{0,1},k,ℓ

Tr[Nj†

ℓ Ni kρA]|ij|R1 ⊗ |kℓ|E,

with {Ni

k}k a set of Kraus operators for the channel N i A→B for

i ∈ {0, 1}.

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Comparing conventional to coherent channel discrimination

SDP allows for comparing conventional (incoherent) to coherent quantum channel discrimination for simple examples.

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Noise parameter N 0.5 0.505 0.51 0.515 0.52 0.525 0.53 0.535 0.54 0.545 0.55 Success probability Incoherent Channel Discrimination Coherent Channel Discrimination

Figure: Comparison of the success probabilities of coherent and incoherent channel discrimination for a generalized amplitude damping channel with damping parameter γ = 0.2 and another with damping parameter γ = 0.3. The channels have the same value of the noise parameter N, which is varied in the plot.

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Conclusion and future directions

Defined the task of coherent quantum channel discrimination and motivated it use in the context of discriminating quantum channels called in superposition Found expression for the optimal success probability, compared it to that for conventional channel discrimination, showed that it does not increase under superchannels, and that it can be computed by SDP For future work, try to show that the optimal success probability is faithful (i.e., equal to 1/2 if and only if the channels are identical)

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