Private quantum subsystems and error correction

Transcription

1 Private quantum subsystems and error correction Sarah Plosker Department of Mathematics and Computer Science Brandon University September 26, 2014

2 Outline 1 Classical Versus Quantum Setting Classical Setting Quantum Setting 2 Thought Experiments (1930 s) 3 QIT: States and Channels 4 Quantum Cryptography Private Quantum Subsystem Example 5 Connection with Quantum Error Correction Complementarity of Q. Cryptography and QEC From QEC to QCrypto? 6 Conclusion

3 The Setup Alice (Sender) channel Φ Bob (Receiver) Input Message noise Output Message

4 States Definition A bit is a variable that can take as value either 0 or 1. Its realized value is called the state of the system (on/off, up/down, +/, true/false,... ).

5 States Definition A bit is a variable that can take as value either 0 or 1. Its realized value is called the state of the system (on/off, up/down, +/, true/false,... ). Classically: (KABOOM!)

6 States Definition A qubit is a system that is the quantum analogue of a bit. Its state is a two-dimensional complex vector φ = α 0 + β 1, where α and β are probability amplitudes, ( ) meaning ( ) 1 0 α 2 + β 2 = 1, α, β C, and 0 =, 1 =, using Dirac 0 1 (bra-ket) notation.

7 States Definition A qubit is a system that is the quantum analogue of a bit. Its state is a two-dimensional complex vector φ = α 0 + β 1, where α and β are probability amplitudes, ( ) meaning ( ) 1 0 α 2 + β 2 = 1, α, β C, and 0 =, 1 =, using Dirac 0 1 (bra-ket) notation. Thus a quantum system is in a state that, with probability α 2, will be measured as 0, and with probability β 2, will be measured as 1 ; though it is truly a superposition of both of the states.

8 Copenhagen Interpretation Interpretation of quantum mechanics by Bohr and Heisenberg, supported by von Neumann, Pauli. Opposed by Einstein, Schrödinger, Russell. 1 The Uncertainty Principle (also called the Indeterminacy Principle) of Heisenberg: The state of every particle is described by a wavefunction. The act of measurement causes the probabilities of the wavefunction to collapse to the value defined by the measurement. 2 Principle of Complementarity of Bohr: Wave-particle duality: light, electrons,..., in short, all energy (and thus all matter) exhibits both wave-like and particle-like properties.

9 Schrödinger s Cat Thought experiment (1935): A cat is put in a sealed box, along with a small amount of radioactive substance and a flask of poison. If any radioactive decay is detected, the flash of poison is broken and the cat dies. If no radioactive decay is detected, then nothing happens. After a while, the cat is simultaneously dead and alive. Yet, when we look in the box, we see the cat either dead or alive, not both. cat = 1 2 dead alive

10 Schrödinger s Cat (cont.)

11 Schrödinger s Cat (cont.)

12 EPR Paradox Entangled states violate the classical principle of locality the idea that an object is directly influenced only by its immediate surroundings.

13 EPR Paradox Entangled states violate the classical principle of locality the idea that an object is directly influenced only by its immediate surroundings. The easiest way to see evidence of entanglement is to measure one component of an entangled state. This measurement fixes the value of the other component of the state, implying non-local communication between the two parts.

14 EPR Paradox Entangled states violate the classical principle of locality the idea that an object is directly influenced only by its immediate surroundings. The easiest way to see evidence of entanglement is to measure one component of an entangled state. This measurement fixes the value of the other component of the state, implying non-local communication between the two parts. Entanglement can be used as a resource in order to implement quantum teleportation.

15 Example of Entanglement Electron spin, when measured, can be either of two states: spin up or spin down. In the absence of measurement, electron spin is in a superposition of the two states / (even without entanglement).

16 Example of Entanglement Electron spin, when measured, can be either of two states: spin up or spin down. In the absence of measurement, electron spin is in a superposition of the two states / (even without entanglement). Consider an entangled electron pair; suppose the state of one of the electrons is measured as spin up. This measurement fixes the state of the other electron it will be spin down.

17 Example of Entanglement Electron spin, when measured, can be either of two states: spin up or spin down. In the absence of measurement, electron spin is in a superposition of the two states / (even without entanglement). Consider an entangled electron pair; suppose the state of one of the electrons is measured as spin up. This measurement fixes the state of the other electron it will be spin down. In this sense, even when spatially separated, entangled electron pairs behave as a single quantum object.

18 States pure state: ψ unit vector in C n rank one projection associated to a pure state: ρ ψ = ψ ψ M n mixed state (density matrix): ρ = i p i ψ ψ 0, Trρ = 1.

19 Separable vs Entangled Pure States In quantum mechanics, one often looks at tensor product quantum systems, for which different users (Alice, Bob, Charlie,... ) have access to the individual tensored Hilbert spaces.

20 Separable vs Entangled Pure States In quantum mechanics, one often looks at tensor product quantum systems, for which different users (Alice, Bob, Charlie,... ) have access to the individual tensored Hilbert spaces. A pure state ψ C m C n is called separable if it can be written as an elementary tensor: ψ = ϕ φ where ϕ C m, φ C n. Otherwise, ψ is said to be entangled.

21 Separable vs Entangled Mixed States In the case of mixed states, we say that ρ M m M n is separable if it can be written as a convex combination of separable pure states: ρ = i p i ρ ϕi ρ φi, where ρ ϕi = ϕ i ϕ i M m and ρ φi = φ i φ i M n. Otherwise, ρ is said to be entangled.

22 Completely Positive Maps Definition We say that a linear map Φ : M m M n is completely positive (CP) if the induced mappings Φ d : M d M m M d M n, defined by Φ d = id d Φ, are positive for all d.

23 Completely Positive Maps Definition We say that a linear map Φ : M m M n is completely positive (CP) if the induced mappings Φ d : M d M m M d M n, defined by Φ d = id d Φ, are positive for all d. A map is called trace preserving if Tr(Φ(X )) = Tr(X ) for all X.

24 Quantum Channels Definition A (quantum) channel Φ is a linear, CP, trace preserving (CPTP) map.

25 Quantum Channels Definition A (quantum) channel Φ is a linear, CP, trace preserving (CPTP) map. We can always write a quantum channel in the form Φ(ρ) = i V iρvi, where i V i V i = I. We call V i the Choi/Kraus operators of Φ.

26 Random Unitary Channels A quantum channel Φ : M m M m is called a unitary channel if we can write Φ(ρ) = UρU for all ρ M m. Such a channel has a single Kraus operator U M m. Trace-preservation forces U to be unitary.

27 Random Unitary Channels A quantum channel Φ : M m M m is called a unitary channel if we can write Φ(ρ) = UρU for all ρ M m. Such a channel has a single Kraus operator U M m. Trace-preservation forces U to be unitary. As a natural generalization of this example, a channel Φ is called a random unitary channel if it admits a decomposition Φ(ρ) = i p i U i ρu i for all ρ, where p i form a probability distribution and U i are unitary operators.

28 Partial Trace Definition The map Tr k : M m M k M m defined by Tr k = id m Tr is called the partial trace (with respect to M k ) and we say that we trace out the system M k. We could similarly define the partial trace with respect to M m. The partial trace is in fact a linear, trace preserving, completely positive map, and as such it is a quantum channel.

29 Stinespring s theorem Schrödinger picture Theorem Suppose that Φ : M m M n is a completely positive trace preserving linear map. Then for some k mn, there exists a partial isometry U M nk mk, and a pure state ψ C k such that Φ(ρ) = Tr k (U(ρ ρ ψ )U ) for all ρ M m. (1)

30 This is the Schrödinger picture for the (discrete) time evolution of quantum states. If we define V φ := U( φ ψ ) for all pure states φ C m, for some fixed pure state ψ C k, then we can write equation (1) more succinctly as Φ(ρ) = Tr k (V ρv ). The general form for U is V 1 V 2 V k U = ( V ), where V =..

31 Purification of Mixed States Fix a density operator ρ 0 M m, and consider the quantum channel Φ : C M m defined by Φ(c 1) = c ρ 0 for all c C.

32 Purification of Mixed States Fix a density operator ρ 0 M m, and consider the quantum channel Φ : C M m defined by Φ(c 1) = c ρ 0 for all c C. Then, with M k = C M m = M m and U M m 2 m, the Stinespring Theorem gives ρ 0 = Φ(1) = Tr k (U(1 ρ ψ )U ) = Tr k (ρ ψ ), where ψ M m 2 = M m M m is a purification of ρ 0, and the freedom in U (and thus in ψ ) yields all possible purifications ρ ψ for ρ 0.

33 Purification of Mixed States Fix a density operator ρ 0 M m, and consider the quantum channel Φ : C M m defined by Φ(c 1) = c ρ 0 for all c C. Then, with M k = C M m = M m and U M m 2 m, the Stinespring Theorem gives ρ 0 = Φ(1) = Tr k (U(1 ρ ψ )U ) = Tr k (ρ ψ ), where ψ M m 2 = M m M m is a purification of ρ 0, and the freedom in U (and thus in ψ ) yields all possible purifications ρ ψ for ρ 0. This idea of dilation constructions of channels and states is known as going to the church of the larger Hilbert space, a phrase credited to John Smolin.

34 Conjugate/Complementary Channels Definition Given a quantum channel Φ : M m M n, consider the Stinespring representation given by V M nk m and M k for which Φ(ρ) = Tr k (V ρv ). Then the corresponding conjugate (or complementary) channel is the quantum channel Φ : M m M k given by Φ (ρ) = Tr n (V ρv ).

35 Computing Kraus Operators for Conjugates We can compute the Kraus operators of the complementary channel by stacking the j-th row of each of the Kraus operators V i of Φ one below the next, to obtain the j-th Kraus operator R j of Φ.

36 Motivation Question: How can we protect information stored in a quantum state from an external observer?

37 Motivation Question: How can we protect information stored in a quantum state from an external observer? Quantum cryptography is used to hide information when sending quantum information through a quantum channel so that the original message cannot be recovered by a third party.

38 Example Alice wishes to send a message ψ from a particular set S to Bob without an eavesdropper, Eve, being able to learn the message.

39 Example Alice wishes to send a message ψ from a particular set S to Bob without an eavesdropper, Eve, being able to learn the message. Alice can perform any unitary operation from a set {U i } with probability p i ({U i } and {p i } are known publicly)

40 Example Alice wishes to send a message ψ from a particular set S to Bob without an eavesdropper, Eve, being able to learn the message. Alice can perform any unitary operation from a set {U i } with probability p i ({U i } and {p i } are known publicly) Alice and Bob share a key, i 0 (privately), and Alice applies U i0 corresponding to i 0 to her message ρ ψ ; that is, ρ ψ U i0 ρ ψ U i 0.

41 Example Alice wishes to send a message ψ from a particular set S to Bob without an eavesdropper, Eve, being able to learn the message. Alice can perform any unitary operation from a set {U i } with probability p i ({U i } and {p i } are known publicly) Alice and Bob share a key, i 0 (privately), and Alice applies U i0 corresponding to i 0 to her message ρ ψ ; that is, ρ ψ U i0 ρ ψ U i 0. Bob receives the output message, and, knowing i 0, can undo Alice s operation to recover the original message.

42 Example Alice wishes to send a message ψ from a particular set S to Bob without an eavesdropper, Eve, being able to learn the message. Alice can perform any unitary operation from a set {U i } with probability p i ({U i } and {p i } are known publicly) Alice and Bob share a key, i 0 (privately), and Alice applies U i0 corresponding to i 0 to her message ρ ψ ; that is, ρ ψ U i0 ρ ψ U i 0. Bob receives the output message, and, knowing i 0, can undo Alice s operation to recover the original message. This is secure against Eve, who does not know i 0, and always sees the same output ρ 0 = i p iu i (ρ)ui regardless of Alice s input message.

43 Example Alice wishes to send a message ψ from a particular set S to Bob without an eavesdropper, Eve, being able to learn the message. Alice can perform any unitary operation from a set {U i } with probability p i ({U i } and {p i } are known publicly) Alice and Bob share a key, i 0 (privately), and Alice applies U i0 corresponding to i 0 to her message ρ ψ ; that is, ρ ψ U i0 ρ ψ U i 0. Bob receives the output message, and, knowing i 0, can undo Alice s operation to recover the original message. This is secure against Eve, who does not know i 0, and always sees the same output ρ 0 = i p iu i (ρ)ui regardless of Alice s input message.

44 Private Quantum Channels Definition Definition Let S C m be a subspace of pure states, let Φ : M m M k M n be a quantum channel, let ρ a M k be an auxiliary density matrix, and let ρ 0 M n. Then [S, Φ, ρ a, ρ 0 ] is a private quantum channel if we have Φ(ρ ψ ρ a ) = ρ 0 for all ψ S. Motivating class of examples: random unitary channels, where Φ(ρ) = i p iu i ρu i.

45 Private Quantum Channels Note: In previous works, the ancilla state ρ a is used theoretically, but does not appear in physical examples (Ambainis et al, Boykin-Roychowdhury, 2000) ρ 0 is often taken to be the maximally mixed state 1 d I (Boykin-Roychowdhury).

46 Completely Depolarizing Channel The simplest example of a private quantum channel is the completely depolarizing channel E acting on M 2. Define the Pauli matrices as X = ( ) Y = ( ) 0 i i 0 Z = ( )

47 Completely Depolarizing Channel The simplest example of a private quantum channel is the completely depolarizing channel E acting on M 2. Define the Pauli matrices as X = ( ) Y = ( ) 0 i i 0 Z = Then E(ρ) = 1 4 (I ρ I +X ρx + Y ρy + ZρZ) = 1 2 I. ( ) The completely depolarizing channel sends all density matrices to the maximally mixed state.

48 Private Subsystem We call A and B subsystems of C n if we can write C n = (A B) (A B). The subspaces of C n can be viewed as subsystems B for which A is one-dimensional. A subscript on a state will indicate to which subsystem the state belongs, e.g. σ A means the operator belongs to M na.

49 Private Subsystem We call A and B subsystems of C n if we can write C n = (A B) (A B). The subspaces of C n can be viewed as subsystems B for which A is one-dimensional. A subscript on a state will indicate to which subsystem the state belongs, e.g. σ A means the operator belongs to M na. Subsystem encodings first arose in quantum information in 2004 through the work of Bartlett-Rudolph-Spekkens (quantum cryptography) and Kribs-Laflamme-Poulin ( operator quantum error correction).

50 Private Subsystem Definition Let Φ : M n M n be a channel and let B be a subsystem of C n. Then B is a completely private subsystem for Φ if there is a ρ 0 M n and σ A M na such that Definition Φ(σ A σ B ) = ρ 0 for all σ B. Let Φ : M n M n be a channel and let B be a subsystem of C n. Then B is a operator private subsystem for Φ if there is a ρ 0 M n such that Φ(σ A σ B ) = ρ 0 for all σ A, for all σ B

51 Phase Damping Channel Consider the following phase damping channels acting on M 4 : Λ i (ρ) = 1 2 (ρ + Z iρz i ), for all ρ M 4, where Z 1 = Z I 2 and Z 2 = I 2 Z.

52 Phase Damping Channel Consider the following phase damping channels acting on M 4 : Λ i (ρ) = 1 2 (ρ + Z iρz i ), for all ρ M 4, where Z 1 = Z I 2 and Z 2 = I 2 Z. For i = 1, 2, Λ i is not private: the diagonal entries of the input ρ are preserved Eve would be able to learn some information about the original message based on what diagonal terms she observes.

53 Privacy for phase damping channel? Let Λ be the composition of the application of the maps Λ 1 and Λ 2 : Λ(ρ) = Λ 2 Λ 1.

54 Privacy for phase damping channel? Let Λ be the composition of the application of the maps Λ 1 and Λ 2 : Λ(ρ) = Λ 2 Λ 1. The Kraus operators for Λ are { 1 2 I I, 1 2 I Z, 1 2 Z I, 1 2 ZZ}.

55 Privacy for phase damping channel? Let Λ be the composition of the application of the maps Λ 1 and Λ 2 : Λ(ρ) = Λ 2 Λ 1. The Kraus operators for Λ are { 1 2 I I, 1 2 I Z, 1 2 Z I, 1 2 ZZ}. Similar to before, we find Λ preserves the diagonal of the input ρ (in this case the output is a diagonal matrix); no private subspace exists.

56 Privacy for phase damping channel? However, consider a density matrix ρ L = 1 (I I +αxx + βy I +γzx ). 4 We note that Λ(ρ L ) = 1 4 I 2 I 2, for any ρ L of this form. Thus Λ is private for some set of states, which we can show forms a subsystem.

57 Common Physical Gates The T -gate, K = 1 ( ) 1 1, 2 i i CNOT 1,2 = , CNOT 2,1 = ,

58 Private subsystem for phase damping Let U = CNOT 1,2 CNOT 2,1 (K I 2 ). The composition U( )U acts as XX I Y YI I Z ZX I X. Thus, we obtain ρ L ρ = 1 4 (I 2 I 2 +γ I X + α I Y + β I Z). we see the set of operators Uρ L U generate the algebra I 2 M 2. In other words, the set M 4 (S) spanned by all ρ L is isomorphic to I 2 M 2.

59 Quantum Error Correction How can we protect our quantum information against sources of quantum noise?

60 Quantum Error Correction How can we protect our quantum information against sources of quantum noise? Quantum error correction is used to recover information from errors introduced by noise that occurs when sending quantum information through a quantum channel.

61 Quantum Error Correction How can we protect our quantum information against sources of quantum noise? Quantum error correction is used to recover information from errors introduced by noise that occurs when sending quantum information through a quantum channel. Definition Let Φ : M n M n be a channel and let B be a subsystem of M n. Then B is a correctable subsystem for Φ if there exists a quantum channel R, called a recovery operation, such that for all σ A, for all σ B, there exists a τ A we have R Φ(σ A σ B ) = τ A σ B.

62 Quantum Error Correction How can we protect our quantum information against sources of quantum noise? Quantum error correction is used to recover information from errors introduced by noise that occurs when sending quantum information through a quantum channel. Definition Let Φ : M n M n be a channel and let B be a subsystem of M n. Then B is a correctable subsystem for Φ if there exists a quantum channel R, called a recovery operation, such that for all σ A, for all σ B, there exists a τ A we have R Φ(σ A σ B ) = τ A σ B. Note: For correctable subspace S, we have R Φ(ρ) = ρ for all ρ M(S).

63 Complementarity of Private Subspaces Both quantum error correction and quantum cryptography protect against the unwanted leakage, or loss, of quantum information and there is an established connection between such notions: Theorem (Kretschmann-Kribs-Spekkens) Given a conjugate pair of quantum channels Φ, Φ, a subsystem S is error-correctable for one if and only if it is an operator private subsystem for the other. The extreme example of this phenomena is given by a unitary channel paired with the completely depolarizing channel where the entire Hilbert space is the correctable (private) subspace.

64 Example (cont.) For our phase damping channel Λ, we can easily compute the Kraus operators of Λ : We find Λ (U( 1 2 I A σ B )U ) = 1 4 I 2 I 2. Instead of being correctable on the private subsystem I 2 M 2, the complementary channel Λ (with the proper unitary transformation) is completely depolarizing! All information is lost, so there is no possibility of the channel being quantum error correctable.

65 Quantum Circuit Diagram Let Φ : M m M n. Let (V Φ, M k ) be the Stinespring representation for Φ. M n Φ(ρ) = ρ 0 Tr k ρ Mm V Φ Tr n M k M k U R R Φ (ρ) = ρ

66 The big picture ρ Z 0 0 K Z Φ(ρ) + + Mixing ancilla + + Φ (ρ) + + The dotted box ( ) represents the mixed state encoding The dashed box ( ) encodes the state ρ into the mixed state encoding ρ The solid box ( ) performed the private quantum channel

67 From QEC to QCrypto? General Program: Using the algebraic bridge given by the notion of conjugate channels, we wish to investigate what results, techniques, special types of codes, applications, etc,. from the well-studied field of quantum error correction have analogues or versions in the world of private quantum codes and quantum cryptography. At the least, we can use work from QEC as motivation for studies in this different setting.

68 Knill-Laflamme Theorem for QEC Theorem Let S be a subspace of C n and let P be the projector onto M n (S). Suppose Φ is a quantum channel with Kraus operators {E i }. Then there exists an error-correction operation R correcting Φ on S if and only if PEi E jp = α ij P for some Hermitian matrix α = [α ij ] ij of complex numbers.

69 Knill-Laflamme Type Conditions for Private Subspaces Using the bridge between privacy and quantum error correction, we can formulate Knill-Laflamme type conditions for the private setting. Theorem (Kribs P.) Let S be a subspace of C n and let P be the projector onto M n (S). Then S is private for a quantum channel Φ with output state ρ 0 ; i.e., Φ(ρ) = ρ 0 for all ρ M(S) if and only if for all X λ X C : PΦ (X )P = λ X P. In this case, λ X = Tr(ρ 0 X ), where Φ (X ) = i V i XV i.

70 Knill-Laflamme Type Conditions for Private Subsystems Theorem (Jochym-O Connor Kribs Laflamme P.) A subsystem B is private for a channel Φ(ρ) = i V iρvi with fixed σ A M(H A ) and output state ρ 0 if and only if there are complex scalars λ ijkl forming an isometry matrix λ = (λ ijkl ) such that p k V j ψ A,k = i,l λ ijkl ql φ l ψ B,i, where ψ A,k (p k ) and φ l (q l ) are eigenstates (eigenvalues) of σ A and ρ 0 respectively, ψ B,i is an orthonormal basis for B, and where ψ A,k is viewed as a channel from B into S.

71 Necessary Condition for Negative Private Subspace Result Theorem (Jochym-O Connor Kribs Laflamme P.) Let Φ be a quantum channel whose Kraus operators are a set of commuting unitary operators { p i U i }, where p i is a scaling factor such that i p i = 1. Then there exists no private subspace S for Φ.

72 Conclusion We ve seen several variations of private quantum channel/code/subspace/operator subsystem/completely (private) subsystem... Knill-Laflamme analogue for private setting via complementary channels. The theorem linking QEC with quantum cryptography doesn t work for completely private subsystems. Relaxed definition of conjugate channel?

73 Big Picture Over the last 10 years, there s been lots of progress in QEC motivated by operator algebraic approach At this point, much less mathematical development on PQC side; still at an early stage. Still lots to uncover in particular, an overarching mathematical formalism of private subsystems, similar to the Stabilizer formalism in QEC.