Classical and Quantum Channel Simulations

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1 Classical and Quantum Channel Simulations Mario Berta (based on joint work with Fernando Brandão, Matthias Christandl, Renato Renner, Joseph Renes, Stephanie Wehner, Mark Wilde)

2 Outline Classical Shannon Theory

3 Outline Classical Shannon Theory Classical Channel Simulations: Classical Reverse Shannon Theorem

4 Outline Classical Shannon Theory Classical Channel Simulations: Classical Reverse Shannon Theorem Quantum Shannon Theory

5 Outline Classical Shannon Theory Classical Channel Simulations: Classical Reverse Shannon Theorem Quantum Shannon Theory Quantum Channel Simulations: Quantum Reverse Shannon Theorem Information Gain of Quantum Measurements Entanglement Cost of Quantum Channels

6 Outline Classical Shannon Theory Classical Channel Simulations: Classical Reverse Shannon Theorem Quantum Shannon Theory Quantum Channel Simulations: Quantum Reverse Shannon Theorem Information Gain of Quantum Measurements Entanglement Cost of Quantum Channels Proof Idea: Post-Selection Technique for (Quantum) Channels Randomness Extractors with (Quantum) Side Information

7 Outline Classical Shannon Theory Classical Channel Simulations: Classical Reverse Shannon Theorem Quantum Shannon Theory Quantum Channel Simulations: Quantum Reverse Shannon Theorem Information Gain of Quantum Measurements Entanglement Cost of Quantum Channels Proof Idea: Post-Selection Technique for (Quantum) Channels Randomness Extractors with (Quantum) Side Information Extensions and Applications

8 Outline Classical Shannon Theory Classical Channel Simulations: Classical Reverse Shannon Theorem Quantum Shannon Theory Quantum Channel Simulations: Quantum Reverse Shannon Theorem Information Gain of Quantum Measurements Entanglement Cost of Quantum Channels Proof Idea: Post-Selection Technique for (Quantum) Channels Randomness Extractors with (Quantum) Side Information Extensions and Applications

9 Classical Shannon Theory Independent and identical distribution (IID) + interested in asymptotic rates.

10 Classical Shannon Theory Independent and identical distribution (IID) + interested in asymptotic rates. At what rate can a channel Λ simulate the identity channel? Transmitter Alice Receiver Bob Λ Λ: channel

11 Classical Shannon Theory Independent and identical distribution (IID) + interested in asymptotic rates. At what rate can a channel Λ simulate the identity channel? Transmitter Alice Encoder Λ Receiver Bob Decoder Λ: channel id: identity channel id

12 Classical Shannon Theory Independent and identical distribution (IID) + interested in asymptotic rates. At what rate can a channel Λ simulate the identity channel? Transmitter Alice Encoder Λ Receiver Bob Decoder Λ: channel id: identity channel id Shannon s noisy channel coding theorem, channel capacity [1]: C( ) = max I(X : (X)) I(X : Y )=H(X)+H(Y) H(XY ) H(X) = X X x p x log p x [1] Shannon, Bst J 27:379,623, 1948

13 Classical Shannon Theory Independent and identical distribution (IID) + interested in asymptotic rates. At what rate can a channel Λ simulate the identity channel? Transmitter Alice Encoder Λ Receiver Bob Decoder Λ: channel id: identity channel id Shannon s noisy channel coding theorem, channel capacity [1]: C( ) = max I(X : (X)) I(X : Y )=H(X)+H(Y) H(XY ) H(X) = X X x p x log p x Neither back communication nor shared randomness help. [1] Shannon, Bst J 27:379,623, 1948

14 Outline Classical Shannon Theory Classical Channel Simulations: Classical Reverse Shannon Theorem Quantum Shannon Theory Quantum Channel Simulations: Quantum Reverse Shannon Theorem Information Gain of Quantum Measurements Entanglement Cost of Quantum Channels Proof Idea: Post-Selection Technique for (Quantum) Channels Randomness Extractors with (Quantum) Side Information Extensions and Applications

15 Classical Channel Simulations At what rate can the identity channel simulate a channel Λ (using shared randomness)? Alice Bob Encoder id Decoder shared randomness id: identity channel Λ: channel Λ

16 Classical Channel Simulations At what rate can the identity channel simulate a channel Λ (using shared randomness)? Alice Bob Encoder id Decoder shared randomness id: identity channel Λ: channel Λ Classical reverse Shannon theorem, channel simulation is possible if and only if [2,3]: c max I(X : (X)) c + r max H( (X)) X X C CRST ( ) = max X I(X : (X)) = C( ) [2] Bennett et al., IEEE TIF 48(10):2637, 2002 [3] Bennett et al., arxiv: v2

17 Outline Classical Shannon Theory Classical Channel Simulations: Classical Reverse Shannon Theorem Quantum Shannon Theory Quantum Channel Simulations: Quantum Reverse Shannon Theorem Information Gain of Quantum Measurements Entanglement Cost of Quantum Channels Proof Idea: Post-Selection Technique for (Quantum) Channels Randomness Extractors with (Quantum) Side Information Extensions and Applications

18 Quantum Shannon Theory Independent and identical distribution (IID) + interested in asymptotic rates.

19 Quantum Shannon Theory Independent and identical distribution (IID) + interested in asymptotic rates. At what rate can a channel simulate the identity channel (using additional resources)? Alice Bob Encoder E Decoder additional resources* E: quantum channel id: quantum identity channel id *e.g. entanglement, classic communication (forward, backward, two-way)

20 Quantum Shannon Theory Independent and identical distribution (IID) + interested in asymptotic rates. At what rate can a channel simulate the identity channel (using additional resources)? Alice Bob Encoder E Decoder additional resources* E: quantum channel id: quantum identity channel id *e.g. entanglement, classic communication (forward, backward, two-way) Quantum Channel Capacities [...]: Q, Q E,Q!,Q,Q $

21 Quantum Shannon Theory Independent and identical distribution (IID) + interested in asymptotic rates. At what rate can a channel simulate the identity channel (using additional resources)? Alice Bob Encoder E Decoder additional resources* E: quantum channel id: quantum identity channel id *e.g. entanglement, classic communication (forward, backward, two-way) Quantum Channel Capacities [...]: Q, Q E,Q!,Q,Q $ Entanglement-assisted quantum capacity [2]: I(A : B) = H(A) + H(B) H(AB) H(A) = tr[ A log A ] [2] Bennett et al., IEEE TIF 48(10):2637, 2002 Q E (E) = 1 2 max I(B : R) (E id)( )

22 Outline Classical Shannon Theory Classical Channel Simulations: Classical Reverse Shannon Theorem Quantum Shannon Theory Quantum Channel Simulations: Quantum Reverse Shannon Theorem Information Gain of Quantum Measurements Entanglement Cost of Quantum Channels Proof Idea: Post-Selection Technique for (Quantum) Channels Randomness Extractors with (Quantum) Side Information Extensions and Applications

23 Quantum Reverse Shannon Theorem Using entanglement, at what rate can the quantum identity channel simulate a quantum channel? Alice Bob Encoder id Decoder entanglement id: quantum identity channel E: quantum channel E

24 Quantum Reverse Shannon Theorem Using entanglement, at what rate can the quantum identity channel simulate a quantum channel? Alice Bob Encoder id Decoder entanglement id: quantum identity channel E: quantum channel E Quantum reverse Shannon theorem, channel simulation is possible for [3,4]: q = 1 2 max I(B : R) (E id)( ) e = 1 (embezzling states) [3] Bennett et al., arxiv: v2 [4] Berta et al., CMP 306(3):579, 2011

25 Quantum Reverse Shannon Theorem Using entanglement, at what rate can the quantum identity channel simulate a quantum channel? Alice Bob Encoder id Decoder entanglement id: quantum identity channel E: quantum channel E Quantum reverse Shannon theorem, channel simulation is possible for [3,4]: q = 1 2 max I(B : R) (E id)( ) e = 1 (embezzling states) Q QRST (E) = 1 2 max I(B : R) (E id)( ) = Q E (E) Communication optimal, for more communication other tradeoffs are possible [3]. [3] Bennett et al., arxiv: v2 [4] Berta et al., CMP 306(3):579, 2011

26 Outline Classical Shannon Theory Classical Channel Simulations: Classical Reverse Shannon Theorem Quantum Shannon Theory Quantum Channel Simulations: Quantum Reverse Shannon Theorem Information Gain of Quantum Measurements Entanglement Cost of Quantum Channels Proof Idea: Post-Selection Technique for (Quantum) Channels Randomness Extractors with (Quantum) Side Information Extensions and Applications

27 Information Gain of Quantum Measurements At what rate can the classical identity channel simulate a quantum measurement (using shared randomness)? Alice Bob Encoder id Decoder shared randomness id: classical identity channel M: quantum measurement M

28 Information Gain of Quantum Measurements At what rate can the classical identity channel simulate a quantum measurement (using shared randomness)? Alice Bob Encoder id Decoder shared randomness id: classical identity channel M: quantum measurement M Measurement simulation is possible if and only if (universal measurement compression) [5]: c max I(X B : R) (M id)( ) c + r max H(X B ) M( ) [5] Berta et al., arxiv: v1

29 Information Gain of Quantum Measurements At what rate can the classical identity channel simulate a quantum measurement (using shared randomness)? Alice Bob Encoder id Decoder shared randomness id: classical identity channel M: quantum measurement M Measurement simulation is possible if and only if (universal measurement compression) [5]: c max I(X B : R) (M id)( ) c + r max H(X B ) M( ) C IG (M) = max I(X B : R) (M id)( ) (= C E (M)) Following Winter [6]: C IG (M) is the information gained by the measurement! [5] Berta et al., arxiv: v1 [6] Winter, CMP 244(1):157, 2004

30 Outline Classical Shannon Theory Classical Channel Simulations: Classical Reverse Shannon Theorem Quantum Shannon Theory Quantum Channel Simulations: Quantum Reverse Shannon Theorem Information Gain of Quantum Measurements Entanglement Cost of Quantum Channels Proof Idea: Post-Selection Technique for (Quantum) Channels Randomness Extractors with (Quantum) Side Information Extensions and Applications

31 Entanglement Cost of Quantum Channels Using classical communication, at what rate can the quantum identity channel simulate a quantum channel? Alice Bob Encoder id Decoder classical communication id: quantum identity channel E: quantum channel E

32 Entanglement Cost of Quantum Channels Using classical communication, at what rate can the quantum identity channel simulate a quantum channel? Alice Bob Encoder id Decoder classical communication id: quantum identity channel E: quantum channel E Quantum communication equivalent to ebits, channel simulation possible for [7]: 1 q = e = lim n!1 n max E X n F ((E n I)( n)) E F ( AB )= inf p i H(A) i AB = X {p i, i } i i c = 1 (unknown) p i i AB [7] Berta et al., IEEE ISIT Proc. p. 900, 2012

33 Entanglement Cost of Quantum Channels Using classical communication, at what rate can the quantum identity channel simulate a quantum channel? Alice Bob Encoder id Decoder classical communication id: quantum identity channel E: quantum channel E Quantum communication equivalent to ebits, channel simulation possible for [7]: 1 q = e = lim n!1 n max E X n F ((E n I)( n)) E F ( AB )= inf p i H(A) i AB = X {p i, i } i i c = 1 (unknown) 1 E C (E) = lim n!1 n max E n F ((E n I)( n)) ( Q $ (E)) p i i AB Entanglement cost optimal, for less communication other tradeoffs are possible [3]. [7] Berta et al., IEEE ISIT Proc. p. 900, 2012 [3] Bennett et al., arxiv: v2

34 Outline Classical Shannon Theory Classical Channel Simulations: Classical Reverse Shannon Theorem Quantum Shannon Theory Quantum Channel Simulations: Quantum Reverse Shannon Theorem Information Gain of Quantum Measurements Entanglement Cost of Quantum Channels Proof Idea: Post-Selection Technique for (Quantum) Channels Randomness Extractors with (Quantum) Side Information Extensions and Applications

35 Proof Idea: Post-Selection Technique for Channels E n A!B F n," A!B : to simulate : channel simulation with cost x (1) (F n," A!B ) Alice id Bob E

36 Proof Idea: Post-Selection Technique for Channels E n A!B F n," A!B : to simulate : channel simulation with cost x (1) (F n," A!B ) Alice id Bob Channel Simulation has to work for all (entangled) inputs! E

37 Proof Idea: Post-Selection Technique for Channels E n A!B F n," A!B : to simulate : channel simulation with cost x (1) (F n," A!B ) Alice id Bob Channel Simulation has to work for all (entangled) inputs! The diamond norm of a quantum operation is defined as kek =sup k2n sup k(e id k )( )k 1 k k 1 =tr( p ) k k 1 apple1 E

38 Proof Idea: Post-Selection Technique for Channels : to simulate : channel simulation with cost E n A!B Channel Simulation has to work for all (entangled) inputs! The diamond norm of a quantum operation is defined as kek =sup k2n We need: F n," A!B sup k(e id k )( )k 1 k k 1 =tr( p ) k k 1 apple1 lim lim n ke!0 n!1 A!B F n, A!B k =0 lim lim!0 n!1 x (1) (F n," A!B ) Alice 1 n x(1) (F n," A!B )=X(E) id E Bob

39 Proof Idea: Post-Selection Technique for Channels : to simulate : channel simulation with cost E n A!B Channel Simulation has to work for all (entangled) inputs! The diamond norm of a quantum operation is defined as kek =sup k2n We need: F n," A!B sup k(e id k )( )k 1 k k 1 =tr( p ) k k 1 apple1 lim lim n ke!0 n!1 A!B F n, A!B k =0 lim lim!0 n!1 Post-selection technique for quantum channels [8]: x (1) (F n," A!B ) Alice 1 n x(1) (F n," A!B )=X(E) id E Bob ke n F n," k apple poly(n) k((e n F n," ) id)( n )k 1 n is the purification of a special de Finetti state (a state which consists of n identical and independent copies of a state on a single subsystem). No IID structure! [8] Christandl et al., PRL 102(2):020504, 2009

40 Proof Idea: Post-Selection Technique for Channels : to simulate : channel simulation with cost E n A!B Channel Simulation has to work for all (entangled) inputs! The diamond norm of a quantum operation is defined as kek =sup k2n We need: F n," A!B sup k(e id k )( )k 1 k k 1 =tr( p ) k k 1 apple1 lim lim n ke!0 n!1 A!B F n, A!B k =0 lim lim!0 n!1 Post-selection technique for quantum channels [5]: x (1) (F n," A!B ) Alice 1 n x(1) (F n," A!B )=X(E) id E Bob ke n F n," k apple poly(n) k((e n F n," ) id)( n )k 1 n is the purification of a special de Finetti state (a state which consists of n identical and independent copies of a state on a single subsystem). No IID structure! Basic idea: create n = E n ( n ) locally at Alice s side, send it over to Bob s side. This defines the channel simulation! [8] Christandl et al., PRL 102(2):020504, 2009 F n,"

41 Outline Classical Shannon Theory Classical Channel Simulations: Classical Reverse Shannon Theorem Quantum Shannon Theory Quantum Channel Simulations: Quantum Reverse Shannon Theorem Information Gain of Quantum Measurements Entanglement Cost of Quantum Channels Proof Idea: Post-Selection Technique for (Quantum) Channels Randomness Extractors with (Quantum) Side Information Extensions and Applications

42 Proof Idea: Randomness Extractors with Side Information State transfer from Alice to Bob: n = E n ( n ) Use random coding schemes and analyze how well they perform in the one-shot regime --> randomness extractors, also called decoupling (cf. Renner s and Dupuis talk)!

43 Proof Idea: Randomness Extractors with Side Information State transfer from Alice to Bob: n = E n ( n ) Use random coding schemes and analyze how well they perform in the one-shot regime --> randomness extractors, also called decoupling (cf. Renner s and Dupuis talk)! One-shot information theory, smooth entropy formalism [9,10]. [9] Renner, PhD Thesis, ETHZ, 2005 [10] Tomamichel, PhD Thesis, ETHZ, 2011

44 Proof Idea: Randomness Extractors with Side Information State transfer from Alice to Bob: n = E n ( n ) Use random coding schemes and analyze how well they perform in the one-shot regime --> randomness extractors, also called decoupling (cf. Renner s and Dupuis talk)! One-shot information theory, smooth entropy formalism [9,10]. Technically: Classical reverse Shannon theorem: classical randomness extractor with classical side information (e.g. random permutation). Information gain of quantum measurements: classical randomness extractor with quantum side information (e.g. random permutation). Entanglement cost of quantum channels: quantum randomness extractor (e.g. random unitary). Quantum reverse Shannon theorem: quantum randomness extractor with quantum side information (e.g. random unitary). [9] Renner, PhD Thesis, ETHZ, 2005 [10] Tomamichel, PhD Thesis, ETHZ, 2011

45 Outline Classical Shannon Theory Classical Channel Simulations: Classical Reverse Shannon Theorem Quantum Shannon Theory Quantum Channel Simulations: Quantum Reverse Shannon Theorem Information Gain of Quantum Measurements Entanglement Cost of Quantum Channels Proof Idea: Post-Selection Technique for (Quantum) Channels Randomness Extractors with (Quantum) Side Information Extensions and Applications

46 Extensions and Applications Results: C CRST ( ) = max X C IG (M) = max I(X : (X)) I(X B : R) (M id)( ) classical reverse Shannon information gain of measurements Q QRST (E) = 1 2 max E C (E) = lim n!1 I(B : R) (E id)( ) 1 n max E n F ((E n I)( n)) quantum reverse Shannon entanglement cost

47 Extensions and Applications Results: C CRST ( ) = max X C IG (M) = max I(X : (X)) I(X B : R) (M id)( ) classical reverse Shannon information gain of measurements Q QRST (E) = 1 2 max E C (E) = lim n!1 Other tradeoffs are possible [3]. I(B : R) (E id)( ) 1 n max E n F ((E n I)( n)) quantum reverse Shannon entanglement cost Feedback vs. non-feedback simulations (classical and quantum) [3,5]. [3] Bennett et al., arxiv: v2 [5] Berta et al., arxiv: v1

48 Extensions and Applications Results: C CRST ( ) = max X C IG (M) = max I(X : (X)) I(X B : R) (M id)( ) classical reverse Shannon information gain of measurements Q QRST (E) = 1 2 max E C (E) = lim n!1 Other tradeoffs are possible [3]. I(B : R) (E id)( ) 1 n max E n F ((E n I)( n)) quantum reverse Shannon entanglement cost Feedback vs. non-feedback simulations (classical and quantum) [3,5]. Purely information theoretic interest: classification of channels. Determine upper bounds on strong converse capacities (applications in quantum cryptography), cf. misc. talks at this workshop. Quantum rate distortion theory (lossy data compression), cf. Wilde s talk Friday 14:30. [3] Bennett et al., arxiv: v2 [5] Berta et al., arxiv: v1

49 Extensions and Applications Results: C CRST ( ) = max X C IG (M) = max I(X : (X)) I(X B : R) (M id)( ) classical reverse Shannon information gain of measurements Q QRST (E) = 1 2 max E C (E) = lim n!1 Other tradeoffs are possible [3]. I(B : R) (E id)( ) 1 n max E n F ((E n I)( n)) quantum reverse Shannon entanglement cost Feedback vs. non-feedback simulations (classical and quantum) [3,5]. Purely information theoretic interest: classification of channels. Determine upper bounds on strong converse capacities (applications in quantum cryptography), cf. misc. talks at this workshop. Quantum rate distortion theory (lossy data compression), cf. Wilde s talk Friday 14:30. That s it... [3] Bennett et al., arxiv: v2 [5] Berta et al., arxiv: v1

50 Example for Classification of Channels Capacity of a classical channel Λ to simulate another classical channel Λ in the presence of free shared randomness is given by: C R (, 0 )= C( ) C( 0 ) Alice Λ Λ Bob Capacity of a quantum channel to simulate another quantum channel in the presence of free entanglement is given by: C E (E, F) = C E(E) C E (F) Alice E Bob F

Example of Embezzling States Introduced by Van Dam and Hayden [11] Definition: A pure, bipartite state of the form kx µ(k)i AB = 1 p G(k) where G(k) =, is called embezzling state of index k.

51 Example of Embezzling States Introduced by Van Dam and Hayden [11] Definition: A pure, bipartite state of the form kx µ(k)i AB = 1 p G(k) where G(k) =, is called embezzling state of index k. j=1 1 j Proposition: Let >0 and let 'i AB be a pure bipartite state of Schmidt rank m. Then the transformation kx j=1 1 p j jji AB µ(k)i AB 7! µ(k)i AB 'i AB can be accomplished with fidelity better than (1 ) for k>m 1/ with local isometries at A and B. Definition: The fidelity between two density matrices ρ and σ is defined as and it is a notion of distance on the set of density matrices. [11] Pra, Rc 67:060302(R), 2003 F (, q p p )=(tr( )) 2

Example: Quantum State Merging/State Splitting R state merging R i BA 0 R i BA 0 R R: reference system : entanglement A B A B Alice Bob Alice Bob How much of a given resource is needed to do this?

52 Example: Quantum State Merging/State Splitting R state merging R i BA 0 R i BA 0 R R: reference system : entanglement A B A B Alice Bob Alice Bob How much of a given resource is needed to do this? Our case: BA 0! BA 0 R = ih BA 0 R purification, free entanglement, classical communication to quantify. Horodecki et al. [12], n i BA 0 R with classical communication cost c n : c = lim n!1 1 n c n = H( B )+H( R ) H( BR )=I(B : R) One-shot version, with classical communication cost for an error [4]: c = I max(b : R) state splitting i BA 0 R c [12] Nature 436: , 2005 [4] Berta et al., CMP 306(3):579, 2011

53 Details: The Post-Selection Technique Christandl et al. [8]: Let E A n and F A n be quantum operations that act permutation-covariant on a n-partite system H A n = H n A. Then ke A n F A nk apple poly(n)k((e A n F A n) id R n R 0 )( A n R n R 0 )k 1 where A n R n R 0 is a purification of the (de Finetti type) state An R n = Z! n AR d(! AR)! AR H R = H A H R n = H n with a pure state on H A H R,, R and d(.) the measure on the normalized pure states on H A H R induced by the Haar Z measure on the unitary group acting on H A H R, normalized to. d(.) =1 [8] Christandl et al., PRL 102(2):020504, 2009

Quantum to Classical Randomness Extractors

Quantum to Classical Randomness Extractors Mario Berta, Omar Fawzi, Stephanie Wehner - Full version preprint available at arxiv: 1111.2026v3 08/23/2012 - CRYPTO University of California, Santa Barbara