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) Mark M. Wilde Coherent quantum channel discrimination 1 / 17
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 of 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 Mark M. Wilde Coherent quantum channel discrimination 2 / 17
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 � � � � N 0 A → B ( ρ RA ) − N 1 � 1 := sup A → B ( ρ RA ) � � 1 ρ RA where � · � 1 is the trace norm and the optimization is over all bipartite states ρ RA with unbounded reference system R . It suffices to restrict the optimization to be over pure bipartite states ψ RA with system R isomorphic to system A . Mark M. Wilde Coherent quantum channel discrimination 3 / 17
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 ρ RA , 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[Λ RB N 0 A → B ( ρ RA )]. Mark M. Wilde Coherent quantum channel discrimination 4 / 17
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[Λ RB N 0 A → B ( ρ RA )] − Tr[Λ RB N 1 � � A → B ( ρ RA )] � Maximum observational error = normalized diamond distance: � N 0 − N 1 � � Tr[Λ RB N 0 A → B ( ρ RA )] − Tr[Λ RB N 1 1 � � � 1 := sup A → B ( ρ RA )] � � 2 ρ RA , Λ RB 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 B Bob Channel Alice sends one share of a state ψ RA . 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 � 1 + 1 � � N 0 − N 1 � � � 1 2 2 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 | 0 �� 0 | G , applying an isometric channel W AG → BE , and then tracing over E : N i A → B ( ω A ) = Tr E [ W AG → BE ( ρ A ⊗ | 0 �� 0 | G )] . Can depict conventional quantum channel discrimination as E’ E’ |0 〉 F B Prover |0 〉 R R V B F A B Verifier Wi G E |0 〉 R 1 π where π = ( | 0 �� 0 | + | 1 �� 1 | ) / 2 is the maximally mixed qubit state, representing random choice of channel selection Mark M. Wilde Coherent quantum channel discrimination 7 / 17
Coherent quantum channel discrimination Main idea behind coherent quantum channel discrimination is to “coherify” each step of a channel discrimination protocol E’ E’ |0 〉 F F Prover |0 〉 R R V B B A B A B F Verifier † Wi Wi E G G E |0 〉 R 1 |+ 〉 Replaces initial maximally mixed state with coherent version | + �� + | , includes an uncomputing step, and final measurement is a Bell measurement Mark M. Wilde Coherent quantum channel discrimination 8 / 17
Details of coherent channel discrimination Step 1 → | + � R 1 | ψ � RA 1 � | i � R 1 W i Step 2 → √ AG → BE | ψ � RA | 0 � G 2 i ∈{ 0 , 1 } 1 � | i � R 1 V RBE ′ F W i Step 3 → √ AG → BE | ψ � RA | 000 � GE ′ F . 2 i ∈{ 0 , 1 } 1 � | i � R 1 W i † VW i | ψ � RA | 000 � GE ′ F , √ Step 4 → 2 i ∈{ 0 , 1 } Step 5 → Measure { Φ R 1 F ⊗ | 0 �� 0 | G , I R 1 FG − Φ R 1 F ⊗ | 0 �� 0 | G } Success if first outcome occurs. Probability of success: 2 � � � � 1 � | i � R 1 W i † VW i | ψ � RA | 000 � GE ′ F � � � Φ | R 1 F � 0 | G � � 2 � � i ∈{ 0 , 1 } � � 2 Mark M. Wilde Coherent quantum channel discrimination 9 / 17
Optimal success probability of coh. channel discrimination Optimizing over all input states | ψ �� ψ | RA and unitaries V of the prover, the optimal success probability is 2 � � � � 1 � p coh ( N 0 , N 1 ) = � ( N i A → B ) † ( P i � sup RB → RBE ′ ) s � � 2 { P i } i ∈{ 0 , 1 } : � � i ∈{ 0 , 1 } � � ∞ i P i † P i = I RB � This can be compared with an alternate expression for success probability of conventional channel discrimination: � � � � s ( N 0 , N 1 ) = 1 � p inc � ( N i A → B ) † (Λ i � sup RB ) � � 2 { Λ i RB } i ∈{ 0 , 1 } � � i ∈{ 0 , 1 } � � ∞ Mark M. Wilde Coherent quantum channel discrimination 10 / 17
Bounding the optimal success probability Optimal success probability is bounded as 1 / 2 ≤ p coh ( N 0 , N 1 ) ≤ 1 s 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 � p coh ( N 0 , N 1 ) ≤ p inc s ( N 0 , N 1 ) ≤ p coh ( N 0 , N 1 ) s s Mark M. Wilde Coherent quantum channel discrimination 11 / 17
Quantum superchannels Most general physical transformation of a quantum channel is a superchannel , which accepts as input a quantum channel and outputs a quantum channel The superchannel Θ ( A → B ) → ( C → D ) takes as input a quantum channel N A → B and outputs a quantum channel K C → D , which we denote by Θ ( A → B ) → ( C → D ) ( N A → B ) = K C → D . Mark M. Wilde Coherent quantum channel discrimination 12 / 17
Physical realizations of quantum superchannels Superchannel has a physical realization in terms of pre- and post-processing quantum channels: Θ ( A → B ) → ( C → D ) ( N A → B ) = D BM → D ◦ N A → B ◦ E C → AM , where E C → AM and D BM → D are pre- and post-processing channels A B N C D E D M Mark M. Wilde Coherent quantum channel discrimination 13 / 17
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: E’ E’ |0 〉 F F Prover |0 〉 R R V D D M 1 D D G’ M 1 |0 〉 G’ C C M W E M W D W D † W E † L M 2 M 2 L |0 〉 A B B A B F Verifier † Wi Wi G E E G |0 〉 R 1 |+ 〉 Mark M. Wilde Coherent quantum channel discrimination 14 / 17
Optimal success probability as SDP Optimal success probability can be computed by a semi-definite program (SDP): � � Tr[ Y R 1 FBE σ R 1 FBE ] : Tr BF [ σ R 1 FBE ] = Z ρ sup , R 1 E σ R 1 FBE , ρ A where σ R 1 FBE and ρ A are density operators and Y R 1 FBE := 1 � k N j † | ii �� jj | R 1 F ⊗ N i ℓ ⊗ | k �� ℓ | E , 2 i , j ∈{ 0 , 1 } , k ,ℓ R 1 E := 1 Z ρ Tr[ N j † � ℓ N i k ρ A ] | i �� j | R 1 ⊗ | k �� ℓ | E , 2 i , j ∈{ 0 , 1 } , k ,ℓ with { N i k } k a set of Kraus operators for the channel N i A → B for i ∈ { 0 , 1 } . Mark M. Wilde Coherent quantum channel discrimination 15 / 17
Comparing conventional to coherent channel discrimination SDP allows for comparing conventional (incoherent) to coherent quantum channel discrimination for simple examples. 0.55 Incoherent Channel Discrimination 0.545 Coherent Channel Discrimination 0.54 0.535 Success probability 0.53 0.525 0.52 0.515 0.51 0.505 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Noise parameter N 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. Mark M. Wilde Coherent quantum channel discrimination 16 / 17
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