“Quantum Bounds, Estimation, and Metrology” limits and possibilities offered by the theory in the process of extracting info from Quantum Systems - Quantum Communication (this lecture) - Quantum Metrology Vittorio Giovannetti NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR ICTP- TRIESTE 2017
Quantum Communication = A theory of OPEN QUANTUM SYSTEM propagation of “quantum signals” (information carriers) through an environmental (noisy) medium (e.g. light pulses in an optical fiber); INPUT/OUPUT output state of the carrier FORMALISM ρ � = Φ ( ρ ) Φ ρ BOB ALICE input state of the carrier v e r ) r ) ( t h e r e c e i ( t h e s e n d e
• Quantum Channels • Classical Theory • Quantum capacities Outlook • Quantum Channels (QCs) .............................. • Classical Theory of Communication ............. • Capacities of a quantum channel ................... ‣ sending classical messages on a QC ..... ‣ sending quantum info on a QC .............. ‣ entanglement as a resource for QC ..... • Example: Bosonic Gaussian channels .............
• Quantum Channels • Classical Theory • Quantum capacities QUANTUM CHANNEL = EVOLUTION of a QUANTUM SYSTEM input state The most general discrete-time evolution of a quantum system is described by assigning a map (channel) which connects the input Φ state of the system to its output ρ ∈ S ( H S ) output state counterpart ρ � ∈ S ( H S ) ρ � = Φ ( ρ ) Φ ρ Three different but equivalent definitions: 1. Physical , Extrinsic (Stinespring) 2. Intrinsic (Kraus) 3. Axiomatic
• Quantum Channels • Classical Theory • Quantum capacities environment E 1. Physical , Extrinsic U ( ρ � | 0 ⇤⇥ 0 | E ) U † ρ � | 0 ⇤⇥ 0 | E (Stinespring) system 2. unitary coupling 3. partial I. extend Unitary trace over the space the space to include dilation o f t h e environe environm ment ent Φ ( ρ ) = Tr E [ U ( ρ � | 0 ⇤ E ⇥ 0 | ) U † ] Φ ρ
• Quantum Channels • Classical Theory REPRESENTATIONS • Quantum capacities environment E 1. Physical , Extrinsic U ( ρ � | 0 ⇤⇥ 0 | E ) U † ρ � | 0 ⇤⇥ 0 | E (Stinespring) system 2. unitary coupling 3. partial I. extend Unitary trace over the space the space to include dilation o f t h e environe environm ment ent Φ ( ρ ) = Tr E [ U ( ρ � | 0 ⇤ E ⇥ 0 | ) U † ] Φ ρ 2. Intrinsic (Kraus) M k ρ M † � Φ ( ρ ) = Φ ρ k k M † � k M k = I k
• Quantum Channels • Classical Theory REPRESENTATIONS • Quantum capacities A Quantum Channel must be LINEAR in the larger space of the operator 3. Axiomatic algebra, COMPLETELY POSITIVE, and TRACE-PRESERVING (CPT). (i) LINEARITY Φ : B ( H S ) → B ( H S ) H S algebra of all linear operators acting on ∀ Θ 1 , Θ 2 ∈ B ( H S ) Φ ( a Θ 1 + b Θ 2 ) = a Φ ( Θ 1 ) + b Φ ( Θ 2 ) ∀ a, b complex G ( i i ) T R A C E P R E S E R V I N Tr[ Φ ( Θ )] = Tr[ Θ ] ∀ Θ ∈ B ( H S ) identity C O M P L E T E P O S I T I V I T Y ( i i i ) map on A [ Φ ⊗ I ]( ρ SA ) � 0 ∀ ρ SA ∈ B ( H S ⊗ H A ) ⇒ = generic auxiliary system P O S I T I V I T Y ∀ ρ ∈ B ( H S ) Φ ( ρ ) � 0 ⇒ =
• Quantum Channels • Classical Theory REPRESENTATIONS • Quantum capacities C O M P L E T E P O S I T I V I T Y ( i i i ) environment environment Φ ( ρ S ) ρ S E E system system S S [ Φ ⊗ I ]( ρ SA ) ρ SA A A There exist MAPS which are POSITIVE but NOT COMPLETELY POSITIVE (e.g. partial transpose): they do not represent PHYSICAL TRANSFORMATION of the system.
Q-CHANNELS ARE NON EXPANSIVE CPT maps tend to decrease the distance between states (i.e. to increase the fidelity between them). A special case is that of UNITARY TRANSFORMATION which preserve the distance among all inputs. ρ 2 ρ 1 Φ ( ρ 2 ) Φ ( ρ 1 )
Q-CHANNELS ARE NON EXPANSIVE CPT maps tend to decrease the distance between states (i.e. to increase the fidelity between them). A special case is that of UNITARY TRANSFORMATION which preserve the distance among all inputs. ρ 2 x x ρ 1 hence Φ ( ρ 2 ) Φ ( ρ 1 ) - C P T a r e N O T ( p h y s i c a l l y ) INVERTIBLE…. - CPT always admit at least a fixed point
• Quantum Channels • Classical Theory • Quantum capacities EXAMPLES: (i) Unitary evolution: it is the only (fully) Φ ( ρ ) = V ρ V † invertible (noiseless) transformation. V V † = V † V = I Describes the evolution of closed systems. Most trivial example is of course the identity channel . I ( ρ ) = ρ (ii) Fully depolarizing channel: ALL inputs are Φ ( ρ ) = I/d mapped into the totally mixed state. ρ
E N T A N G L E M E N T BREAKING CHANNELS: when acting on half of a Φ p ⇤ ρ ( ⇤ ) A ⊗ ρ ( ⇤ ) composite system they � ( Φ ⊗ I )( ρ AB ) = ρ AB B a l w a y s p r o d u c e s ⇤ separable state SEPARABLE outputs. (The real enemy of QI). Theorem (Shor-Ruskai-Horodecki): CPT map is EB iff and only if, there exist a POVM { E j , j = 1 , 2 , · · · } and a class of states such that, { ρ j , j = 1 , 2 , · · · } � Φ ( ρ ) = Tr[ ρ M j ] ρ j Proof: via CJ isomorphism ⇒ = j POVM measure state preparation QC CQ Φ = “CRYPTO-CLASSICAL” channels classical info quantum info
Φ 2 , Φ 1 COMPOSITION RULES Given CPT maps on S, one can construct the following CPT maps (i)CONVEX SUM [ p Φ 1 + (1 − p ) Φ 2 ]( ρ ) = p Φ 1 ( ρ ) + (1 − p ) Φ 2 ( ρ ) given p ∈ [0 , 1] NB: this implies that the set of CPT maps is CONVEX (ii) CONCATENATION [ Φ 2 � Φ 1 ]( ρ ) = Φ 2 ( Φ 1 ( ρ )) Φ 1 Φ 2 Φ 2 � Φ 1 = NB: this implies that the CPT maps form a (non Abelian) SEMIGROUP (the identity map being the identity element). (iii) TENSOR PRODUCT given two copies of S, S1 and Φ 1 Φ 1 S2, we can define a tensor product channel by acting with ⊗ the first operation on S1 and Φ 2 with the second on S2. Φ 2
• Quantum Channels • Classical Theory • Quantum capacities La Palma Tenerife BOB v e r ) ( t h e r e c e i ALICE Schmitt-Manderbach et al., PRL 98, 010504 (2007) r ) ( t h e s e n d e How reliably CLASSICAL messages can be transferred on a quantum channel?
• Quantum Channels • Classical Theory • Quantum capacities In classical information theory a (discrete) communication channel is fully specified by assigning the conditional probabilities that measure the probability, that given a certain input symbol, the receiver will receive a given output. p ( y | x ) BOB ALICE Y X NOISY v e r ) ( t h e r e c e i r ) ( t h e s e n d e CHANNEL y ∈ A x ∈ A L A S S I C A L C C L A S S I C A L U T P U T O INFO SOURCE p ( Y = y | X = x ) ≡ p ( y | x ) CONDITIONAL PROBABILITY THAT BOB RECEIVES y WHILST ALICE IS SENDING x; BAYES’s RULE � p ( y ) = p ( y | x ) p ( x ) p ( x, y ) = p ( y | x ) p ( x ) x ∈ A
• Quantum Channels • Classical Theory • Quantum capacities x, y ∈ A = { 0 , 1 } Example: binary symmetric channel 0 0 p p ( Y = x | X = x ) = p when Alice sends a symbol X Bob 1 − p receives Y=X with probability p , and Y=X +1 with probability 1-p. 1 1 p ( Y = ¯ x | X = x ) = 1 − p p How to improve the reliability of the communication? INDEPENDENT CHANNEL USES ALICE USE REDUNDANCY first carrier (e.g. repeat the message r ) ( t h e s e n d e 0 p ( y | x ) sufficiently many time, if second carrier the error probability is 0 p ( y | x ) sufficiently small, Bob third carrier 0 p ( y | x ) could then deduce the ... e . g . u s e 3 correct message via i n d e p e n d e n t majority voting). Memoryless channel: the noise model channels uses to encode a single 0 acts the same way on all the elements of the sequences of inputs
• Quantum Channels • Classical Theory • Quantum capacities #channel uses = log 2 M #Bits RATE = R = N
• Quantum Channels • Classical Theory • Quantum capacities #channel uses = log 2 M #Bits RATE = R = N The capacity of the channel is the maximum of the ACHIEVABLE rates. ⇥ log 2 M ⇤ � C = achievable R = lim max � → 0 lim sup such that P err ( C ) < � � ∃ C M,N � N N →∞ CHANNEL USES BOB p ( y | x ) ALICE x 1 y 1 v e r ) ( t h e r e c e i r ) ( t h e s e n d e x 2 y 2 m � ∈ M m ∈ M ≡ ⇥ ⇥ C D x y ≡ · · · · · · x N y N ENCODING TRANSFERRING DECODING N � p ( ⌅ y | ⌅ x ) = p ( y 1 | x 1 ) p ( y 2 | x 2 ) · · · p ( y N | x N ) = p ( y ⇥ | x ⇥ ) ⇥ =1
• Quantum Channels • Classical Theory • Quantum capacities ⇥ log 2 M ⇤ � C = achievable R = lim max � → 0 lim sup such that P err ( C ) < � � ∃ C M,N � N N →∞ Shannon NOISY CHANNEL CODING THEOREM C = max p ( x ) H ( X : Y ) H ( X : Y ) = H ( X ) + H ( Y ) − H ( X, Y ) MUTUAL INFORMATION of X, Y � H ( X ) = − p ( x ) log 2 p ( x ) x
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