When Is a Measurement Reversible?

Transcription

1 When Is a Measurement Reversible? Information Gain and Disturbance in Quantum Measurements (revisited) Physical Review A 93, (2016) Francesco Buscemi, Siddhartha Das, Mark M. Wilde contributed talk at AQIS 16 Academia Sinica, Taipei, Taiwan 1 September 2016 these slides are available for download at tinyurl.com/bdw-aqis16 Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

2 Quantum Measurement Processes an operation transforming a quantum input (the density operator representing the state of the system before the measurement, i.e., the pre-measurement state ) into a classical output (the outcome of the measurement) and a quantum output (the state of the system conditional on the observation of some outcome, i.e., the post-measurement state ) Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

3 Quantum Measurement Processes an operation transforming a quantum input (the density operator representing the state of the system before the measurement, i.e., the pre-measurement state ) into a classical output (the outcome of the measurement) and a quantum output (the state of the system conditional on the observation of some outcome, i.e., the post-measurement state ) mathematically, a measurement process is represented by a completely positive instrument (Ozawa, 1984), namely, a family M = {E m : m M } of completely positive maps E m, indexed by the outcome set M = {m}, and such that m trace-preserving (completeness relation) Em is Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

4 Quantum Measurement Processes an operation transforming a quantum input (the density operator representing the state of the system before the measurement, i.e., the pre-measurement state ) into a classical output (the outcome of the measurement) and a quantum output (the state of the system conditional on the observation of some outcome, i.e., the post-measurement state ) mathematically, a measurement process is represented by a completely positive instrument (Ozawa, 1984), namely, a family M = {E m : m M } of completely positive maps E m, indexed by the outcome set M = {m}, and such that m trace-preserving (completeness relation) upon input ρ, the outcome m is obtained with probability p(m) = Tr[E m(ρ)] and the corresponding post-measurement state is σ m = E m(ρ)/p(m) Em is Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

5 Measurement Reversibility information extraction requires measurement, but measurements occur by interaction with a probe, and interactions cause disturbance = information disturbance tradeoff in quantum measurements Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

6 Measurement Reversibility information extraction requires measurement, but measurements occur by interaction with a probe, and interactions cause disturbance = information disturbance tradeoff in quantum measurements the intuition is clear, less so the mathematical formulation (state dependent/independent, figures of merit, average/single-outcome, with/without feedback, etc); in particular, are measurements always irreversible? Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

7 Measurement Reversibility information extraction requires measurement, but measurements occur by interaction with a probe, and interactions cause disturbance = information disturbance tradeoff in quantum measurements the intuition is clear, less so the mathematical formulation (state dependent/independent, figures of merit, average/single-outcome, with/without feedback, etc); in particular, are measurements always irreversible? correction scheme: where {C m : m M } is a family of error-correcting CPTP maps (independent of ρ) Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

8 Measurement Reversibility information extraction requires measurement, but measurements occur by interaction with a probe, and interactions cause disturbance = information disturbance tradeoff in quantum measurements the intuition is clear, less so the mathematical formulation (state dependent/independent, figures of merit, average/single-outcome, with/without feedback, etc); in particular, are measurements always irreversible? correction scheme: where {C m : m M } is a family of error-correcting CPTP maps (independent of ρ) average disturbance: distance between ρ and ρ = m p(m) ρm = m (Cm Em)(ρ) Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

9 Measurement Reversibility information extraction requires measurement, but measurements occur by interaction with a probe, and interactions cause disturbance = information disturbance tradeoff in quantum measurements the intuition is clear, less so the mathematical formulation (state dependent/independent, figures of merit, average/single-outcome, with/without feedback, etc); in particular, are measurements always irreversible? correction scheme: where {C m : m M } is a family of error-correcting CPTP maps (independent of ρ) average disturbance: distance between ρ and ρ = m p(m) ρm = m (Cm Em)(ρ) Fact. Among all measurements with the same outcome statistics, efficient measurements (i.e., E m( ) = E m E m) are the least disturbing one: we will hence focus on these only Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

10 Information Gain in Efficient Quantum Measurements let H(M) be the Shannon entropy of the outcome distribution: is this amount all information? Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

11 Information Gain in Efficient Quantum Measurements let H(M) be the Shannon entropy of the outcome distribution: is this amount all information? the Groenewold-Lindblad-Ozawa information gain is defined as the average entropy decrease due to measurement: I G(ρ, M) H(ρ) m p(m)h(σm) Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

12 Information Gain in Efficient Quantum Measurements let H(M) be the Shannon entropy of the outcome distribution: is this amount all information? the Groenewold-Lindblad-Ozawa information gain is defined as the average entropy decrease due to measurement: I G(ρ, M) H(ρ) m p(m)h(σm) for efficient measurements (i.e., σ m E mρe m), I G(ρ, M) is always non-negative and equals the optimal rate of measurement compression: minimum rate at which classical information needs to be sent in order to communicate the measurement outcome to a receiver (Winter, 2004) Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

13 Information Gain in Efficient Quantum Measurements let H(M) be the Shannon entropy of the outcome distribution: is this amount all information? the Groenewold-Lindblad-Ozawa information gain is defined as the average entropy decrease due to measurement: I G(ρ, M) H(ρ) m p(m)h(σm) for efficient measurements (i.e., σ m E mρe m), I G(ρ, M) is always non-negative and equals the optimal rate of measurement compression: minimum rate at which classical information needs to be sent in order to communicate the measurement outcome to a receiver (Winter, 2004) I G(ρ, M) quantifies intrinsic/useful data; the difference H(M) I G(ρ, M) quantifies extrinsic/useless data Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

14 Information Gain in Efficient Quantum Measurements let H(M) be the Shannon entropy of the outcome distribution: is this amount all information? the Groenewold-Lindblad-Ozawa information gain is defined as the average entropy decrease due to measurement: I G(ρ, M) H(ρ) m p(m)h(σm) for efficient measurements (i.e., σ m E mρe m), I G(ρ, M) is always non-negative and equals the optimal rate of measurement compression: minimum rate at which classical information needs to be sent in order to communicate the measurement outcome to a receiver (Winter, 2004) I G(ρ, M) quantifies intrinsic/useful data; the difference H(M) I G(ρ, M) quantifies extrinsic/useless data tempting analogy with work and heat in thermodynamics Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

15 The Technical Lemma Theorem Let Φ be a subunital (i.e., Φ(I) I) CPTP map. Then there exists a CPTP map Ψ such that, for any ρ, H[Φ(ρ)] H(ρ) D[ρ (Ψ Φ)(ρ)]. Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

16 The Technical Lemma Theorem Let Φ be a subunital (i.e., Φ(I) I) CPTP map. Then there exists a CPTP map Ψ such that, for any ρ, H[Φ(ρ)] H(ρ) D[ρ (Ψ Φ)(ρ)]. Proof. 1 Φ is CPTP subunital map Φ is CP trace-non-increasing unital map Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

17 The Technical Lemma Theorem Let Φ be a subunital (i.e., Φ(I) I) CPTP map. Then there exists a CPTP map Ψ such that, for any ρ, H[Φ(ρ)] H(ρ) D[ρ (Ψ Φ)(ρ)]. Proof. 1 Φ is CPTP subunital map Φ is CP trace-non-increasing unital map 2 H[Φ(ρ)] H(ρ) = Tr[ρ log ρ] Tr[Φ(ρ) log Φ(ρ)] = Tr[ρ log ρ] Tr [ ρ Φ (log Φ(ρ)) ] Tr[ρ log ρ] Tr [ ρ log{(φ Φ)(ρ)} ] = D[ρ (Φ Φ)(ρ)], where inequality follows from operator concavity of the logarithm and the operator Jensen inequality for positive unital maps Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

18 The Technical Lemma Theorem Let Φ be a subunital (i.e., Φ(I) I) CPTP map. Then there exists a CPTP map Ψ such that, for any ρ, H[Φ(ρ)] H(ρ) D[ρ (Ψ Φ)(ρ)]. Proof. 1 Φ is CPTP subunital map Φ is CP trace-non-increasing unital map 2 H[Φ(ρ)] H(ρ) = Tr[ρ log ρ] Tr[Φ(ρ) log Φ(ρ)] = Tr[ρ log ρ] Tr [ ρ Φ (log Φ(ρ)) ] Tr[ρ log ρ] Tr [ ρ log{(φ Φ)(ρ)} ] = D[ρ (Φ Φ)(ρ)], where inequality follows from operator concavity of the logarithm and the operator Jensen inequality for positive unital maps 3 Ψ( ) Φ ( ) + Tr [ (id Φ )( ) ] ω, where ω is arbitrary but fixed density operator Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

19 The Technical Lemma Theorem Let Φ be a subunital (i.e., Φ(I) I) CPTP map. Then there exists a CPTP map Ψ such that, for any ρ, H[Φ(ρ)] H(ρ) D[ρ (Ψ Φ)(ρ)]. Proof. 1 Φ is CPTP subunital map Φ is CP trace-non-increasing unital map 2 H[Φ(ρ)] H(ρ) = Tr[ρ log ρ] Tr[Φ(ρ) log Φ(ρ)] = Tr[ρ log ρ] Tr [ ρ Φ (log Φ(ρ)) ] Tr[ρ log ρ] Tr [ ρ log{(φ Φ)(ρ)} ] = D[ρ (Φ Φ)(ρ)], where inequality follows from operator concavity of the logarithm and the operator Jensen inequality for positive unital maps 3 Ψ( ) Φ ( ) + Tr [ (id Φ )( ) ] ω, where ω is arbitrary but fixed density operator 4 since Ψ( ) Φ ( ), it follows that D[ρ (Φ Φ)(ρ)] D[ρ (Ψ Φ)(ρ)] Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

20 The Technical Lemma Theorem Let Φ be a subunital (i.e., Φ(I) I) CPTP map. Then there exists a CPTP map Ψ such that, for any ρ, H[Φ(ρ)] H(ρ) D[ρ (Ψ Φ)(ρ)]. Proof. 1 Φ is CPTP subunital map Φ is CP trace-non-increasing unital map 2 H[Φ(ρ)] H(ρ) = Tr[ρ log ρ] Tr[Φ(ρ) log Φ(ρ)] = Tr[ρ log ρ] Tr [ ρ Φ (log Φ(ρ)) ] Tr[ρ log ρ] Tr [ ρ log{(φ Φ)(ρ)} ] = D[ρ (Φ Φ)(ρ)], where inequality follows from operator concavity of the logarithm and the operator Jensen inequality for positive unital maps 3 Ψ( ) Φ ( ) + Tr [ (id Φ )( ) ] ω, where ω is arbitrary but fixed density operator 4 since Ψ( ) Φ ( ), it follows that D[ρ (Φ Φ)(ρ)] D[ρ (Ψ Φ)(ρ)] Stronger than any other (currently known) recoverability results: 1 the lemma holds also for positive (not necessarily completely positive) Φ 2 the explicit form of correction Ψ is known 3 D(ρ σ) D M (ρ σ) log F (ρ, σ), where F (ρ, σ) ρ σ 2 1 Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

21 Application to Efficient Quantum Measurements given an efficient measurement process M = {E m}, construct the CPTP map M( ) = m Em E m m m Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

22 Application to Efficient Quantum Measurements given an efficient measurement process M = {E m}, construct the CPTP map M( ) = m Em E m m m then M(I) = m E 1E E me m 0 E 2E 2 0 m m = 0 0 E 3E Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

23 Application to Efficient Quantum Measurements given an efficient measurement process M = {E m}, construct the CPTP map M( ) = m Em E m m m then M(I) = m E 1E E me m 0 E 2E 2 0 m m = 0 0 E 3E m E me m = I = E me m I for all m = E me m I for all m = M(I) I I Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

24 Application to Efficient Quantum Measurements given an efficient measurement process M = {E m}, construct the CPTP map M( ) = m Em E m m m then M(I) = m E 1E E me m 0 E 2E 2 0 m m = 0 0 E 3E m E me m = I = E me m I for all m = E me m I for all m = M(I) I I in other words, the channel M is subunital and the Technical Lemma can be applied Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

25 Strengthened Second Law for Efficient Quantum Measurements recall the definition: M(ρ) = m EmρE m m m Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

26 Strengthened Second Law for Efficient Quantum Measurements recall the definition: M(ρ) = m EmρE m m m the Technical Lemma implies that there exists a correction scheme {C m} such that H[M(ρ)] H(ρ) D[ρ m (Cm Em)(ρ)] for all ρ Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

27 Strengthened Second Law for Efficient Quantum Measurements recall the definition: M(ρ) = m EmρE m m m the Technical Lemma implies that there exists a correction scheme {C m} such that H[M(ρ)] H(ρ) D[ρ m (Cm Em)(ρ)] for all ρ H[M(ρ)] H(ρ) = H(M) [H(ρ) m p(m)h(σm)] = H(M) IG(ρ, M) Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

28 Strengthened Second Law for Efficient Quantum Measurements recall the definition: M(ρ) = m EmρE m m m the Technical Lemma implies that there exists a correction scheme {C m} such that H[M(ρ)] H(ρ) D[ρ m (Cm Em)(ρ)] for all ρ H[M(ρ)] H(ρ) = H(M) [H(ρ) m p(m)h(σm)] = H(M) IG(ρ, M) Theorem (Strengthened Second Law for Efficient Quantum Measurements) Let M = {E m} be an efficient quantum measurement process. Then there exists a correction scheme {C m} such that ( ) H(M) I G(ρ, M) D ρ (C m E m)(ρ), for all ρ. m Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

29 Strengthened Second Law for Efficient Quantum Measurements recall the definition: M(ρ) = m EmρE m m m the Technical Lemma implies that there exists a correction scheme {C m} such that H[M(ρ)] H(ρ) D[ρ m (Cm Em)(ρ)] for all ρ H[M(ρ)] H(ρ) = H(M) [H(ρ) m p(m)h(σm)] = H(M) IG(ρ, M) Theorem (Strengthened Second Law for Efficient Quantum Measurements) Let M = {E m} be an efficient quantum measurement process. Then there exists a correction scheme {C m} such that ( ) H(M) I G(ρ, M) D ρ (C m E m)(ρ), for all ρ. m hence, the extrinsic noise measures the irreversibility of efficient measurements (as does entropy for adiabatic processes) Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

30 Strengthened Second Law for Efficient Quantum Measurements recall the definition: M(ρ) = m EmρE m m m the Technical Lemma implies that there exists a correction scheme {C m} such that H[M(ρ)] H(ρ) D[ρ m (Cm Em)(ρ)] for all ρ H[M(ρ)] H(ρ) = H(M) [H(ρ) m p(m)h(σm)] = H(M) IG(ρ, M) Theorem (Strengthened Second Law for Efficient Quantum Measurements) Let M = {E m} be an efficient quantum measurement process. Then there exists a correction scheme {C m} such that ( ) H(M) I G(ρ, M) D ρ (C m E m)(ρ), for all ρ. m hence, the extrinsic noise measures the irreversibility of efficient measurements (as does entropy for adiabatic processes) indeed, in [K. Jacobs, PRA 80, (2009)] the inequality H(M) I G(ρ, M) 0 is interpreted as the second law for quantum measurements; the above Theorem considerably strengthens this Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8

31 Thank You slides available for download at tinyurl.com/bdw-aqis16 Buscemi, Das, Wilde When Is a Measurement Reversible? 1 September / 8