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Quantum channels and their capacities: An introduction Ana Kontrec December 2, 2013 Elements of Classical Information Theory. What is information? Claude Shannon (1948.) related information with uncertainty: the more rare the event, the more information we gain if we know that it happened Denition Consider an event described by a random variable X with probability distribution p (x ). Then we dene the measure of uncertainty of getting the outcome x as the surprisal (or self-information) I (x ) := − log p (x ). Denition The expected amount of surprisal of X is the H (X ) := − X Shannon entropy H (X ), p (x ) log p (x ). Elements of Classical Information Theory. What is information? Claude Shannon (1948.) related information with uncertainty: the more rare the event, the more information we gain if we know that it happened Denition Consider an event described by a random variable X with probability distribution p (x ). Then we dene the measure of uncertainty of getting the outcome x as the surprisal (or self-information) I (x ) := − log p (x ). Denition The expected amount of surprisal of X is the H (X ) := − X Shannon entropy H (X ), p (x ) log p (x ). Elements of Classical Information Theory. What is information? Claude Shannon (1948.) related information with uncertainty: the more rare the event, the more information we gain if we know that it happened Denition Consider an event described by a random variable X with probability distribution p (x ). Then we dene the measure of uncertainty of getting the outcome x as the surprisal (or self-information) I (x ) := − log p (x ). Denition The expected amount of surprisal of X is the H (X ) := − X Shannon entropy H (X ), p (x ) log p (x ). Elements of Classical Information Theory. Shannon's model for a classical channel both the input X and the output Y (with probability distributions {πi } are modeled as random variables and {qj }, respectively); we assume output is correlated with input Denition The classical channel is a linear map N π = (π1 , π2 , ...) and q = (q1 , q1 , ...). between probability distributions It acts independently on each input letter, according to the xed stochastic matrix pij = P (Y = j | X = i )): N (π) = q , N (π)j = {pij },(where X i pij πi . Elements of Classical Information Theory. Shannon's model for a classical channel both the input X and the output Y (with probability distributions {πi } are modeled as random variables and {qj }, respectively); we assume output is correlated with input Denition The classical channel is a linear map N π = (π1 , π2 , ...) and q = (q1 , q1 , ...). between probability distributions It acts independently on each input letter, according to the xed stochastic matrix pij = P (Y = j | X = i )): N (π) = q , N (π)j = {pij },(where X i pij πi . Elements of Classical Information Theory. Shannon channel capacity (memoryless) channel capacity: maximal number of classical bits which can be transmitted per channel use, in a reliable manner (i.e. with arbitrarily small error at the receiver) Theorem The channel capacity of a discrete memoryless channel C where I (X : Y ) (N ) = max I {π(x )} N is given by (X : Y ) , is the mutual information of the random variables X and Y , and the maximum is taken over all possible input distributions {π(x )} . Mutual information I (X : Y ) := P p i ,j πi pij log q =⇒ about X when we know Y ij j amount of information that we gain Elements of Classical Information Theory. Shannon channel capacity (memoryless) channel capacity: maximal number of classical bits which can be transmitted per channel use, in a reliable manner (i.e. with arbitrarily small error at the receiver) Theorem The channel capacity of a discrete memoryless channel C where I (X : Y ) (N ) = max I {π(x )} N is given by (X : Y ) , is the mutual information of the random variables X and Y , and the maximum is taken over all possible input distributions {π(x )} . Mutual information I (X : Y ) := P p i ,j πi pij log q =⇒ about X when we know Y ij j amount of information that we gain Elements of Classical Information Theory. Shannon channel capacity (memoryless) channel capacity: maximal number of classical bits which can be transmitted per channel use, in a reliable manner (i.e. with arbitrarily small error at the receiver) Theorem The channel capacity of a discrete memoryless channel C where I (X : Y ) (N ) = max I {π(x )} N is given by (X : Y ) , is the mutual information of the random variables X and Y , and the maximum is taken over all possible input distributions {π(x )} . Mutual information I (X : Y ) := P p i ,j πi pij log q =⇒ about X when we know Y ij j amount of information that we gain The quantum channel Denition Quantum channel is a linear completely positive trace-preserving map (CPTP) between quantum states. Why is a quantum channel given by a linear CPTP map (i) Linearity: Φ : Mn → Mm ? Φ(aρ1 + aρ2 ) = aΦ(ρ1 ) + bΦ(ρ2 ) =⇒ allows for a probabilistic interpretation of any mixed state (ii) Trace-preserving (TP): Tr (iii) Positive: if =⇒ ρ ≥ 0, then (Φ(ρ)) = Tr (ρ) Φ (ρ) ≥ 0. these criteria ensure that the image under Φ of a quantum state is indeed a quantum state. Is this enough to produce a physically realizable transformation? The quantum channel Denition Quantum channel is a linear completely positive trace-preserving map (CPTP) between quantum states. Why is a quantum channel given by a linear CPTP map (i) Linearity: Φ : Mn → Mm ? Φ(aρ1 + aρ2 ) = aΦ(ρ1 ) + bΦ(ρ2 ) =⇒ allows for a probabilistic interpretation of any mixed state (ii) Trace-preserving (TP): Tr (iii) Positive: if =⇒ ρ ≥ 0, then (Φ(ρ)) = Tr (ρ) Φ (ρ) ≥ 0. these criteria ensure that the image under Φ of a quantum state is indeed a quantum state. Is this enough to produce a physically realizable transformation? The quantum channel Denition Quantum channel is a linear completely positive trace-preserving map (CPTP) between quantum states. Why is a quantum channel given by a linear CPTP map (i) Linearity: Φ : Mn → Mm ? Φ(aρ1 + aρ2 ) = aΦ(ρ1 ) + bΦ(ρ2 ) =⇒ allows for a probabilistic interpretation of any mixed state (ii) Trace-preserving (TP): Tr (iii) Positive: if =⇒ ρ ≥ 0, then (Φ(ρ)) = Tr (ρ) Φ (ρ) ≥ 0. these criteria ensure that the image under Φ of a quantum state is indeed a quantum state. Is this enough to produce a physically realizable transformation? The quantum channel Denition Quantum channel is a linear completely positive trace-preserving map (CPTP) between quantum states. Why is a quantum channel given by a linear CPTP map (i) Linearity: Φ : Mn → Mm ? Φ(aρ1 + aρ2 ) = aΦ(ρ1 ) + bΦ(ρ2 ) =⇒ allows for a probabilistic interpretation of any mixed state (ii) Trace-preserving (TP): Tr (iii) Positive: if =⇒ ρ ≥ 0, then (Φ(ρ)) = Tr (ρ) Φ (ρ) ≥ 0. these criteria ensure that the image under Φ of a quantum state is indeed a quantum state. Is this enough to produce a physically realizable transformation? The quantum channel Denition Quantum channel is a linear completely positive trace-preserving map (CPTP) between quantum states. Why is a quantum channel given by a linear CPTP map (i) Linearity: Φ : Mn → Mm ? Φ(aρ1 + aρ2 ) = aΦ(ρ1 ) + bΦ(ρ2 ) =⇒ allows for a probabilistic interpretation of any mixed state (ii) Trace-preserving (TP): Tr (iii) Positive: if =⇒ ρ ≥ 0, then (Φ(ρ)) = Tr (ρ) Φ (ρ) ≥ 0. these criteria ensure that the image under Φ of a quantum state is indeed a quantum state. Is this enough to produce a physically realizable transformation? The quantum channel mathematical formalism of quantum channels describes the dynamics of open quantum systems =⇒ systems which have interactions with an environment system The quantum channel If we extend the primary space of states HA ⊗ Henv HA to the tensor product (by adding an ancilla), then the quantum channel Φ must take density matrices of this joint system again to density matrices. (iv) Completely positive (CP): a linear map called completely positive if Φ : Mn → Mm Φ ⊗ Ik : Mn ⊗ Mk → Mm ⊗ Ik is positivity preserving for every k ≥ 1. is The quantum channel If we extend the primary space of states HA ⊗ Henv HA to the tensor product (by adding an ancilla), then the quantum channel Φ must take density matrices of this joint system again to density matrices. (iv) Completely positive (CP): a linear map called completely positive if Φ : Mn → Mm Φ ⊗ Ik : Mn ⊗ Mk → Mm ⊗ Ik is positivity preserving for every k ≥ 1. is The quantum channel If we extend the primary space of states HA ⊗ Henv HA to the tensor product (by adding an ancilla), then the quantum channel Φ must take density matrices of this joint system again to density matrices. (iv) Completely positive (CP): a linear map called completely positive if Φ : Mn → Mm Φ ⊗ Ik : Mn ⊗ Mk → Mm ⊗ Ik is positivity preserving for every k ≥ 1. is Examples of single qubit channels Depolarising channel Denition A depolarising channel is a channel which: leaves the qubit unaected with probability q and replaces the state of the qubit with a completely mixed state I 2 with probability 1 − q, I Φ (ρ) := (1 − q ) ρ + q . 2 Examples of single qubit channels Amplitude-damping channel Amplitude damping channel describes energy dissipation of a quantum system (for example, the dynamics of a 2-level atom which is spontaneuosly emmiting a photon). Unital channels We say that a channel is unital if Φ (I ) = I . Amplitude damping channel is not unital! Examples of single qubit channels Amplitude-damping channel Amplitude damping channel describes energy dissipation of a quantum system (for example, the dynamics of a 2-level atom which is spontaneuosly emmiting a photon). Unital channels We say that a channel is unital if Φ (I ) = I . Amplitude damping channel is not unital! Classical capacities of quantum channels Shannon capacity of quantum channel the most classical of quantum channel capacities =⇒ no entangled inputs, no joint measurements Protocol 1 Each input letter (from a nite alphabet) is encoded in a quantum state ρi Φ (ρi ) 2 The channel 3 At the output a measurement (POVM) is performed Φ maps it to an output state Classical capacities of quantum channels Shannon capacity of quantum channel the most classical of quantum channel capacities =⇒ no entangled inputs, no joint measurements Protocol 1 Each input letter (from a nite alphabet) is encoded in a quantum state ρi Φ (ρi ) 2 The channel 3 At the output a measurement (POVM) is performed Φ maps it to an output state Classical capacities of quantum channels Shannon capacity of quantum channel This is actually a particular realization of a classical channel: Alice chooses a set of states chooses a POVM {Ej } the stochastic matrix {ρi } to encode the input letters, and Bob to measure the output {pij } of the channel is given by pij = Tr (Ej Φ (ρi )). Denition The Shannon capacity of a quantum channel is dened as CShan (Φ) := max I {ρ ,E } i (X : Y ) , j where X and Y are classical random variables corresponding to the source (which emmits input letters) and the outcome of Bob's POVM, respectively. Classical capacities of quantum channels Holevo capacity of quantum channels What if Alice encodes her messages using product states only, but Bob is allowed to decode them using entangled measurements? (:= product-state capacity) Theorem (Holevo-Schumacher-Westmoreland (HSW) theorem) Dene the Holevo capacity of the channel Φ as " χ (Φ) := where S χ (Φ) sup {pi ,ρi } (ρ) = −Tr ρ log ρ !! S Φ X i pi ρi # − X i pi S (Φ (ρi )) , is the Von Neumann entropy. Then the quantity is the product-state capacity of the channel Φ. Holevo capacity is the maximal amount of classical information that can be transmitted through a noisy quantum channel using product-state inputs. Classical capacities of quantum channels Holevo capacity of quantum channels What if Alice encodes her messages using product states only, but Bob is allowed to decode them using entangled measurements? (:= product-state capacity) Theorem (Holevo-Schumacher-Westmoreland (HSW) theorem) Dene the Holevo capacity of the channel Φ as " χ (Φ) := where S χ (Φ) sup {pi ,ρi } (ρ) = −Tr ρ log ρ !! S Φ X i pi ρi # − X i pi S (Φ (ρi )) , is the Von Neumann entropy. Then the quantity is the product-state capacity of the channel Φ. Holevo capacity is the maximal amount of classical information that can be transmitted through a noisy quantum channel using product-state inputs. Classical capacities of quantum channels Capacities of product channels If we allow both entangled inputs and entangled measurements, it can be shown that there exist channels satisfying the superadditivity property CShan (Φ ⊗ Φ) > 2CShan (Φ) . This leads to the question of nding the asymptotic (or ultimate) capacity Cult (Φ) = C Φ⊗n lim n→∞ n Shan 1 Fact From the HSW theorem it follows that Cult (Φ) = lim 1 n→∞ n χ Φ⊗n . Classical capacities of quantum channels Capacities of product channels If we allow both entangled inputs and entangled measurements, it can be shown that there exist channels satisfying the superadditivity property CShan (Φ ⊗ Φ) > 2CShan (Φ) . This leads to the question of nding the asymptotic (or ultimate) capacity Cult (Φ) = C Φ⊗n lim n→∞ n Shan 1 Fact From the HSW theorem it follows that Cult (Φ) = lim 1 n→∞ n χ Φ⊗n . Classical capacities of quantum channels Capacities of product channels If we allow both entangled inputs and entangled measurements, it can be shown that there exist channels satisfying the superadditivity property CShan (Φ ⊗ Φ) > 2CShan (Φ) . This leads to the question of nding the asymptotic (or ultimate) capacity Cult (Φ) = C Φ⊗n lim n→∞ n Shan 1 Fact From the HSW theorem it follows that Cult (Φ) = lim 1 n→∞ n χ Φ⊗n . Additivity conjectures Conjecture (additivity of Holevo capacity For any two quantum channels Φ and Ω, ) it holds that χ (Φ ⊗ Ω) = χ (Φ) + χ (Ω) . This would imply that χ (Φ⊗n ) = nχ (Φ), Cult and hence (Φ) = χ (Φ) . Physical meaning of additivity conjecture The channel capacity is achieved by encoding on product states only, i.e. using entangled inputs does not increase the capacity. Additivity conjectures Conjecture (additivity of Holevo capacity For any two quantum channels Φ and Ω, ) it holds that χ (Φ ⊗ Ω) = χ (Φ) + χ (Ω) . This would imply that χ (Φ⊗n ) = nχ (Φ), Cult and hence (Φ) = χ (Φ) . Physical meaning of additivity conjecture The channel capacity is achieved by encoding on product states only, i.e. using entangled inputs does not increase the capacity. Additivity conjectures Conjecture (additivity of Holevo capacity For any two quantum channels Φ and Ω, ) it holds that χ (Φ ⊗ Ω) = χ (Φ) + χ (Ω) . This would imply that χ (Φ⊗n ) = nχ (Φ), Cult and hence (Φ) = χ (Φ) . Physical meaning of additivity conjecture The channel capacity is achieved by encoding on product states only, i.e. using entangled inputs does not increase the capacity. Additivity conjectures Can we do without entangled inputs? The following theorem was proved in 2003 by Shor: Theorem The following conjectures are equivalent: (1) additivity of the Holevo capacity of a quantum channel (2) additivity of the minimal entropy output Smin (Φ) of a quantum channel, where Smin (Φ) = inf S (Φ (ρ)) , ρ (3) additivity of the entanglement of formation EF ρ, (ρ) of a bipartite state where EF (ρ) = min E X i pi S (TrB (| ρi >< ρi |)) . In 2008, Hastings proved the non-additivity of the minimal output entropy, hence disproving all of the above conjectures. Additivity conjectures Can we do without entangled inputs? The following theorem was proved in 2003 by Shor: Theorem The following conjectures are equivalent: (1) additivity of the Holevo capacity of a quantum channel (2) additivity of the minimal entropy output Smin (Φ) of a quantum channel, where Smin (Φ) = inf S (Φ (ρ)) , ρ (3) additivity of the entanglement of formation EF ρ, (ρ) of a bipartite state where EF (ρ) = min E X i pi S (TrB (| ρi >< ρi |)) . In 2008, Hastings proved the non-additivity of the minimal output entropy, hence disproving all of the above conjectures. Additivity for unital qubit channels Theorem (King, 2001) Let Φ be a unital qubit channel. Then for any (!) channel χ (Φ ⊗ Ω) = χ (Φ) + χ (Ω) . Ω, it holds that What next? entanglement-assisted classical capacity (what if the sender and receiver share some prior entanglement?) I protocols of quantum teleportation and superdense coding quantum capacities of quantum channels What next? entanglement-assisted classical capacity (what if the sender and receiver share some prior entanglement?) I protocols of quantum teleportation and superdense coding quantum capacities of quantum channels Summary quantum information has features which are distinctly dierent from classical information, mainly due to the phenomenon of entanglement this opens the questions not only of quantifying entanglement, but of nding new mathematical approaches to the quantum case (geometry?) a quote by Wojciech Zurek: Indeed, quantum computing inevitably poses questions that probe to the very core of the distinction between quantum and classical. (...) Questions originally asked for the most impractical of reasons - questions about the EPR paradox, the quantum-to-classical transition, the role of information, and the interpretation of the quantum state vector - have become relevant for practical applications such as quantum criptography and quantum computation. Summary quantum information has features which are distinctly dierent from classical information, mainly due to the phenomenon of entanglement this opens the questions not only of quantifying entanglement, but of nding new mathematical approaches to the quantum case (geometry?) a quote by Wojciech Zurek: Indeed, quantum computing inevitably poses questions that probe to the very core of the distinction between quantum and classical. (...) Questions originally asked for the most impractical of reasons - questions about the EPR paradox, the quantum-to-classical transition, the role of information, and the interpretation of the quantum state vector - have become relevant for practical applications such as quantum criptography and quantum computation. Summary quantum information has features which are distinctly dierent from classical information, mainly due to the phenomenon of entanglement this opens the questions not only of quantifying entanglement, but of nding new mathematical approaches to the quantum case (geometry?) a quote by Wojciech Zurek: Indeed, quantum computing inevitably poses questions that probe to the very core of the distinction between quantum and classical. (...) Questions originally asked for the most impractical of reasons - questions about the EPR paradox, the quantum-to-classical transition, the role of information, and the interpretation of the quantum state vector - have become relevant for practical applications such as quantum criptography and quantum computation.