Covariant Ergodic Quantum Markov Semigroups via Systems of Imprimitivity
aa r X i v : . [ qu a n t - ph ] F e b Covariant Ergodic Quantum Markov Semigroups viaSystems of Imprimitivity
Radhakrishnan BaluArmy Research Laboratory Adelphi, MD, 21005-5069, [email protected] of Mathematics & Norbert Wiener Center for HarmonicAnalysis and Applications, University of Maryland, College Park, [email protected]: date / Accepted: February 22, 2021
Abstract
We construct relativistic quantum Markov semigroups from covariant completelypositive maps. We proceed by generalizing a step in Stinespring’s dilation to a gen-eral system of imprimitivity and basing it on Poincar`e group. The resulting noisechannels are relativistically consistent and the method is applicable to any funda-mental particle, though we demonstrate it for the case of light-like particles. TheKrauss decomposition of the relativistically consistent completely positive identitypreserving maps (our set up is in Heisenberg picture) enables us to construct thecovariant quantum Markov semigroups that are uniformly continuous. We inducerepresentations from the little groups to ensure the quantum Markov semigroupsthat are ergodic due to transitive systems imprimitivity.
Quantum channels that are described by completely positive unital maps are important in-gradients in quantum information processing (QIP). Quantum Markov semigroups (QMS)that are composed of completely positive maps describe open quantum systems that formthe basis for investigating quantum optics circuits [2], [13], [4] and quantum filtering [10]among other examples where the interactions with a bath are taken into account. The1nteracting bath can be explicitly described by an infinite set of quantum harmonic oscil-lators leading to quantum stochastic differential equations [3] or using quantum Markovsemigroups by tracing out the environmental degrees of freedom leading to Lindbladianquantum master equations. In both situations the Markovianity of the dynamics is as-sumed and the noise channels in terms of Krauss operators are obtained. In the QMSdescription discrete noise channels with quantum Poisson statistics will be absent. Thenoise channels in relativistic quantum quantum information (RQI) [8] can be investigatedwithin the framework of covariant QMS which is the focus of this work.QMS that are covariant with respect to a group action has been studied by Fagnolaet al. [5] with several results on invariant subspaces obtained specific to compact groups.In quantum information processing invariant subspaces are important as they providepathways for coherent evolution of quantum systems to ensure fidelity of quantum com-putation or communication. In this work our group of interest is the locally compactPoincar`e relevant in relativistic quantum physics and the QMS is constructed from co-variant completely positive unital maps. In our earlier work we treated covariant freequantum fields via Lorentz Group Representation of Weyl Operators[7] constructed fockspaces for massless Weyl fermions and covariant white noises. The relativistically con-sistent quantum white noises [6] are a rigorous description of noises of an open quantumsystem. In this work we focus on the system degrees of freedom along with the channelsat the level of QMS that can indeed be dilated to the description of quantum stochasticdifferential equations or trajectories as is known in the physics community.Systems of imprimitivity [11], [14], [18] is a compact characterization of dynamicalsystems, when the symmetry of the kinematics of a quantum system is described bya group, from which infinitesimal forms in terms of differential equations (
Shr ¨ odinger ,Heisenberg, and Dirac etc), localizability, and the canonical commutation relations can bederived [15] and [16]. For example, the kinematics of a finite dimensional quantum systemcan be characterized up to an unitary isomorphism by a transitive system of imprimitivitywith the underlying symmetry described by the cyclic group Z n where n is the dimensionof the system [9]. In the context of quantum information processing the Pauli Z operatorcan be considered as the position observable and the conjugate momentum-like opertaoris the Pauli X operator. Our important tool is Mackey machinery that characterizes suchsystems in terms of induced representations, specifically from little groups, applied toPoincar`e group.To build covariant dilations of completely positive maps we break up steps of theproof of Stinespring theorem into construction of a map from the kernel induced by thecompletely positive map, a derived *-unital homomorphism and an isometry composite,and an unitary isomorphism to an extended Hilbert space. The last step was accomplishedby considering a system of imprimitivity, based on the Poincar´e group, endowed withan homogeneous projection valued measure (PVM) and by deriving a PVM of highermultiplicity. The Krauss operators are constructed from the unitary operators, that areirreducible representations of the group, and the isometry. By chaining the completely2ositive maps as a semigroup we obtain a covariant QMS. When the SI is transitive weget an ergodic QMS. Most of the steps are standard and our contribution in selectingthe transitive SI, by inducing representations from little groups, to guarantee an ergodicQMS. Some of the different systems of imprimitivity that live on the orbits of the stabilizersubgroups are described below. It is good to keep in mind the picture that SI is anirreducible unitary representation of Poincar` e group P + induced from the representationof a subgroup such as SO , which is a subgroup of homogeneous Lorentz, as ( U m ( g ) ψ )( k ) = e i { k,g } ψ ( R − m k ) where g belongs to the R portion of the Poincar` e group, m is a member ofthe rotation group, and the duality between the configuration space R and the momentumspace P is expressed using the character the irreducible representation of the group R as: { k, g } = k g − k g − k g − k g , p ∈ P . (1)ˆ p : x → e i { k,g } . (2) { Lx, Lp } = { x, p } . (3)ˆ p ( L − x ) = ˆ Lp ( x ) . (4)In the above L is a matrix representation of Lorentz group acting on R as well as P and it is easy to see that p → Lp is the adjoint of L action on P . The R space is calledthe configuration space and the dual P is the momentum space of a relativistic quantumparticle.The stabilizer subgroup of the Poincar` e group P + (Space-like particle) can be de-scribed as follows: [19]: The Lorentz frame in which the walker is at rest has momentumproportional to (0,1,0,0) and the little group is again SO(3, 1) and this time the rotationswill change the helicity. The states of a freely evolving relativistic quantum particles are described by unitaryirreducible representations of Poincar` e , which is a semidirect product of two groups, groupthat has a geometric interpretation in terms of fiber bundles. Definition 1
Let A and H be two groups and for each h ∈ H let t h : a → h [ a ] bean automorphism (defined below) of the group A. Further, we assume that h → t h is a omomorphism of H into the group of automorphisms of A so that h [ a ] = hah − , ∀ a ∈ A. (5) h = e H , the identity element of H . (6) t h h = t h t h . (7) Now, G = H ⋊ A is a group with the multiplication rule of ( h , a )( h , a ) = ( h h , a t h [ a ]) .The identity element is ( e H , e A ) and the inverse is given by ( h, a ) − = ( h − , h − [ a − ] .When H is the homogeneous Lorentz group and A is R we get the Poincar ` e group P = H ⋊ A via this construction. The covering group of inhomogeneous Lorentz isalso a semidirect product as P ∗ = H ∗ ⋊ R and as every irreducible projective representa-tion of P is uniquely induced from a representation of P ∗ we will work with the coveringgroup, whose orbits in momentum space are smooth, in the following. We will need the following lemma for our discussions on constructing induced represen-tations using characters of an abelian group as in the case of equation (1).
Lemma 2 (Lemma 6.12 [14]) Let h ∈ H . Then, ∀ x ∈ ˆ A where ˆ A is the set of charactersof the group A (which in our case is R ), there exists one and only y ∈ ˆ A such that y ( a ) = x ( h − [ a ]) , ∀ a ∈ A . If we write y = h[x], then h, x → h [ x ] is continuous from H × ˆ A into ˆ A and ˆ A becomes a H-space. Here, y can be thought as the adjoint for actionof H on ˆ A and the map ˆ p in equation (1) is such an example that is of interest to ourconstructions. In essence, we have Fourier analysis when restricted to the abelian groupA of the semidirect product. We need to described few ingredients to construct the Hilbert space of space-like fermionsnamely, the fiber bundle, the fiber vector space, an inner product for the fibers, and an in-variant measure. The 3+1 spacetime Lorentz group ˆ O (3 , R , where the systems of imprimitivity established will live, described by the symmetryˆ O (3 , ⋊ R . The orbits have an invariant measure α + m whose existence is guaranteedas the groups and the stabilizer groups concerned are unimodular and in fact it is theLorentz invariant measure dpp for the case of forward mass hyperboloid. The orbits aredefined as: Y + , / im = { p : p − p − p − p = − m , p > } , single sheet hyperboloid . (8)Each of these orbits are invariant with respect to ˆ O (3 ,
1) and let us consider the stabilizersubgroup of the first orbit at p=(0,1,0,0). Now, assuming that the spin of the particleis 1/2 and mass m > SL (2 , C ).ˆ B + , / im = { ( p, v ) p ∈ ˆ Y +1 / im , v ∈ C , X r =0 p r γ r v = mv } . (9)ˆ π : ( p, v ) → p. Projection from the total space ˆ B + , im to the base ˆ Y + , / im . (10)It is easy to see that if ( p, v ) ∈ B + , / im then so is also ( δ ( h ) p, S ( h ∗− ) v ). Thus, we havethe following Poincar` e group symmetric action on the bundle that encodes spinors intothe fibers: h, ( p, v ) → ( p, v ) h = ( δ ( h ) p, S ( h ∗− ) v ) . (11)We define the states of the light like particles [7] on the Hilbert space ˆ H + , / , squareintegrable functions on Borel sections of the bundle ˆ B + , = { ( p, v ) : ( p, v ) ∈ B +0 , Γ v = ∓ v } .The states of the particles are defined on the Hilbert space ˆ H + , / m , square integrablefunctions on Borel sections of the bundle ˆ B + , m with respect to the invariant measure β + , ,whose norm induced by the inner product is given below: k φ k = Z X + m p − h φp, φp i dβ + , ( p ) . (12)The invariant measure and the induced representation of the Poincar´e group from that ofthe Weyl fermion are given below: dβ + , ( p ) = dp dp dp p + p + p ) . (13)( U h,x φ )( p ) = exp i { x, p } φ ( δ ( h ) − p ) h . (14) Theorem 3 [21] Let H , H be Hilbert spaces and T : B ( H ) → B ( H ) be a linearoperator satisfying the following conditions:(1) T(1) = 1, T ( X ∗ ) = T ( X ) ∗ ; (ii) k T ( X ) k < k X k ; (iii) if X n → X weakly in H then T ( X n ) → T ( X ) weakly in H ; (iv) ∀ n = 1 , , . . . the correspondence (( X ij )) → (( T ( X ij ))) , ≤ i, j ≤ n from B ( H ⊗ C n )) into B ( H ⊗ C n )) preserves positivity.Then, there exists a Hilbert space k, an isometry V from H into H ⊗ k such that(a) T ( X ) = V ∗ X ⊗ V, ∀ X ∈ B ( H ) ; (b) { ( X ⊗ V u | X ∈ B ( H ) , u ∈ H } is total in H ⊗ k ; (c) This association is valid up to an unitary isomorphism. Positive definite kernel:
Let H be any set, possibly countably infinite. A positivedefinite kernel on H is a complex number valued map K : H × H → C satisfying X i,j ¯ α i α j K ( x i , x j ) ≥ , α i ∈ C , x i ∈ H . (15) Proposition 4
Let T : B ( H ) → B ( H ) be a completely positive map. Let us define akernel K ( X , Y , u ; X , Y , u ) = h u , Y ∗ T ( X ∗ X ) Y u i , (16) ∀ ( X i , Y i , u i ) ∈ B ( H ) × B ( H ) × H , i = 1 , . (17) Then, K is a positive definite kernel on B ( H ) × B ( H ) × H Next, we extract a map λ that is part of the Gelfand pair via the following proposition. Proposition 5
Let T : B ( H ) → B ( H ) be a completely positive map. Then, thereexists a Hilbert space H and a map λ such that the following holds up to an unitary:(i) { λ ( X, u ) | X ∈ B ( H , u ∈ H } is total in H ;(ii) λ is linear in each of the variables X, u;(iii) h λ ( X , u ) , λ ( X , u ) i = h u , T ( X ∗ , X ) u i ;(iv) In particular the map V : u → λ (1 , u ) is an isometry from H into H . This leads to the existence of a homomorphism to the bath Hilbert space that is vital insynthesizing the noise operators.
Proposition 6
Let
T, λ, V be as above. Then, there exists a *-unital homomorphism π : B ( H ) → B ( H ) such that λ ( X, u ) = π ( X ) V u, ∀ X ∈ B ( H ) , u ∈ H . The next step, that decomposes the bath space operators via Hahn-Hellinger theorem, isour focus and so let us state and look at the details of the proof.
Proposition 7
Dilation step: Let π a *-unital homomorphism π : B ( H ) → B ( H ) .There exists,a Hilbert space k and an unitary isomorphism Γ : H → H ⊗ k such that π ( X ) = Γ − X ⊗ , ∀ X ∈ B ( H ) where 1 is the identity operator in k. The algebragenerated by U, V is B ( H ) and linear combinations of U s V t , s, t ∈ R are weakly densein B ( H ) it follows that π ( X ) = Γ − X ⊗ . Let us consider only the case when dim H = n < ∞ and refer the readers tothe above reference for the infinite case. Then up to unitary equivalence there exists aunique irreducible pair U, V of unitary operators in H such that U n = V n = 1 and6 V = V U exp πin , that is they obey Weyl commutator relation. By analogue of theStone-von Neumann theorem for the group Z n there exists an Hilbert space k such thatΓ − U ⊗
1Γ = π ( U ) , Γ − V ⊗
1Γ = π ( V ) . We get the noise channels as
V u = L u ⊕ L u ⊕· · · ⊕ L m u, m ≤ n −
1. It is easy to verify that the dimension of k is bounded by n n that form the ancilla qubits in quantum information processing.Let us reformulate the Weyl commutator relation in terms of SI [9] with { U g | g ∈ Z n } as the unitary representation of the group in the Hilbert space H , and E = { E ( S ) , S is Borel subset of M } .M = { , , . . . , n − } .U g E ( S ) U − g = E ( g − S ) . It is easy to see that the above unitary representation is induced from that of the subgroup H = { } of Z n .To get the usual picture of a system interacting with a bath let us consider the space h ⊗ H . We can define an unitary isomorphism W : h ⊗ H → h ⊕ h ⊕ . . . as W u ⊗ v = ⊕h f i , v i u where { f i } forms the orthonormal basis of h . Then, the operator Lcan be viewed as acting on the system Hilbert space as Lu = L j L j u, u ∈ h .We need a way to unitarily transport an SI relation from an arbitrary Hilbert space toone that is a disjoint union of countably many (infinite or finite) of them that in turn ismediated by transporting a homogeneous projection valued measure [14]. We then applyit to Poincar`e group to build the covariant completely positive maps and QMS. Definition 8
Let H = L ( X, K , α ) and P = P ( K , α ) be a projection valued measure(PVM) E → P E where P E f = χ E f, f ∈ H . The PVM acting on a separable Hilbertspace is said to be homogeneous if it is equivalent to a P ( K , α ) for suitable K and α .The dimension of K is the multiplicity of P. Given any measure class C on X and anyinteger ≤ n ≤ ∞ there is a unique PVM P with C as the measure class of P and n itsmultiplicity. Theorem 9
Let (U, P) be a system of imprimitivity acting in H such that P is homo-geneous of multiplicity ≤ n ≤ ∞ . Let α be a σ -finite measure in the measure class C P of P. Then (U, P) is equivalent to a system ( U ′ , P n,α ) acting in H n,α . Let us state and prove the main result on covariant QMS.
Proposition 10
Let T : B ( ˆ H + , / ) → B ( ˆ H + , / ) be a completely positive map. Then,T can be dilated into a covariant map V : B ( ˆ H + , / ) → B ( ˆ H + , / ) ⊗ B ( k ) . A relativis-tically consistent QMS can be constructed by composing V t parameterized by time. (Sketch of proof ): In the dilation step of the Stinespring theorem we need to transportan SI relation that can be guaranteed if the PVM involved is a homogeneous one. This7s the case for finite dimensional Hilbert spaces, as there is only one SI up to an unitarytransformation, and for the infinite case we can ensure it by starting with a transitivesystem of imprimitivity. This can be achieved by inducing a representation from a closedsubgroup which is light-like little group in our case as follows:The states of the particles are defined on the Hilbert space [7] ˆ H + , , square integrablefunctions on Borel sections of the bundle ˆ B + , with respect to the invariant measure β + , ,whose norm induced by the inner product is given below: k φ k = Z X + m p − h φp, φp i dβ + , ( p ) . (18)The invariant measure and the induced representation of the Poincar´e group from that ofthe Weyl fermion are given below: dβ + , ( p ) = dp dp dp p + p + p ) . (19)( U h,x φ )( p ) = exp i { x, p } φ ( δ ( h ) − p ) h . (20)The representation ( U h,x φ ) that gives rise to a transitive SI can be transported across the*-unital homomorphism π in the dilation step. The transitivity of the system ensures thatit is also ergodic. It is important to note that in the case of infinite dimensional systemsmore than one dilation is possible for a given completely positive map and we have realizedthe dilation based on light-like particles. The construction of a covrainat QMS can proceedin the usual manner as a one parameter semigroup of the above completely positive mapT. Suppose T has the Krauss operator representation as T ( X ) = X j L ∗ j XL j , X j L ∗ j L j = I. (21)then the generator of the QMS is given byΘ( X ) = − λ X j L ∗ j L j X + XL ∗ j L j − L ∗ j XL j . (22) We derived the covariant QMS using Mackey’s machinery on systems of imprimitivity. Wedilated a completely positive operator by first building a transitive system of imprimitivityand transported across an unitary at the dilation step of the Stinespring construction. Ourconstruction based on light-like particle is one of several possible ways to dilate a givencompletely positive map. We continued with the construction of a covriant QMS with theusual approach of gluing together completely positive maps as a semigroup. Our work onQMS covariant with respect to the actions by locally compact groups that are relevant inRQI opens up opportunities for extending invariant subspace results of compact groups.8 eferences and Notes [1] Holevo, A.S.: Quantum systems, channels, information: A mathematical introduc-tion. Berlin, DeGruyter (2012).[2] Heinz-Peter Breuer and Francesco Petruccione: The Theory of Open Quantum Sys-tems, Oxford University Press (March 29, 2007).[3] K. R. Parthasarathy: An Introduction to Quantum Stochastic Calculus, Birkhauser,Basel (1992)[4] Rolando Rebolledo. Complete positivity and the Markov structure of open quantumsystems. In Open quantum systems. II, volume 1881 of Lecture Notes in Math., pages149, 182. Springer, Berlin, 2006.[5] Fagnola, Franco; Sasso, Emanuela; and Umanit`a, Veronica (2020) ”Invariant Projec-tions for Covariant Quantum Markov Semigroups,” Journal of Stochastic Analysis:Vol. 1 : No. 4 , Article 3.[6] Hida, T.; Streit, L. On quantum theory in terms of white noise. Nagoya Math. J. 68(1977), 21–34.[7] Radhakrishnan Balu: Covariant Quantum White Noise from Light-like QuantumFields, Special issue on Quantum Probability and Related Topics, JSOA (Accepted(2020)).[8] D. R. Terno, Introduction to Relativistic Quantum Information, in: Quantum Infor-mation Processing: From Theory to Experiment, edited by D. G. Angelakis et al.(Institute of Physics Press, 2006), p. 61.[9] J. Tolar, “A classification of finite quantum kinematics,” Jour-nal of Physics: Conference Series 538 no. 1, (2014) 012020.http://stacks.iop.org/1742-6596/538/i=1/a=012020.[10] J. Gough, V. P. Belavkin, and O. G. Smolyanov, Hamilton-Jacobi-Bellman equationsfor quantum filtering and control, J. Opt. B: Quantum Semiclass. Opt., 7 (2005), pp.237 - 244.[11] Infinite dimensional group representations, Bulletin of the American MathematicalSociety, 69, 628 (1963).[12] Charles Schwartz, ”Toward a quantum theory of tachyon fields,” Int. J. Mod. Phys.A 31, 1650041 (2016). 913] Tezak, N., Niederberger, A., Pavlichin, D. S., Sarma, G., and Mabuchi, H.: Specifica-tion of photonic circuits using quantum hardware description language. Philosophicaltransactions A, 370, 5270 (2012).[14] V.S. Varadarajan: Geometry of quantum theory, Springer (1985)[15] Newton T D and Wigner E P 1949 Localized states for elementary systems Rev.Mod. Phys. 21 400[16] Radhakrishnan Balu: Kinematics and Dynamics of Quantum Walk in terms of Sys-tems of Imprimitivity, J. Phys. A: Mathematical and Theoretical (2019).[17] Von Neumann: On rings of operators. Reduction theory, Annals of Mathematics, 50,401 (1949).[18] Wigner: Unitary representations of the inhomogeneous Lorentz group, Annals ofmathematics, 40, 149 (1939).[19] Y.S. Kim and E.S. Noz: Phase Space Picture of Quantum Mechanics: Group The-oretical Approach, Lecture Notes in Physics Series, World Scientific Pub Co Inc(1991).[20] Wigner: Unitary representations of the inhomogeneous Lorentz group: Ann. Math.,40, 845 (1939).[21] W. F. Stinespring, Positive Functions on C*-algebras, Proceedings of the AmericanMathematical Society, 6, 211–216 (1955).10[1] Holevo, A.S.: Quantum systems, channels, information: A mathematical introduc-tion. Berlin, DeGruyter (2012).[2] Heinz-Peter Breuer and Francesco Petruccione: The Theory of Open Quantum Sys-tems, Oxford University Press (March 29, 2007).[3] K. R. Parthasarathy: An Introduction to Quantum Stochastic Calculus, Birkhauser,Basel (1992)[4] Rolando Rebolledo. Complete positivity and the Markov structure of open quantumsystems. In Open quantum systems. II, volume 1881 of Lecture Notes in Math., pages149, 182. Springer, Berlin, 2006.[5] Fagnola, Franco; Sasso, Emanuela; and Umanit`a, Veronica (2020) ”Invariant Projec-tions for Covariant Quantum Markov Semigroups,” Journal of Stochastic Analysis:Vol. 1 : No. 4 , Article 3.[6] Hida, T.; Streit, L. On quantum theory in terms of white noise. Nagoya Math. J. 68(1977), 21–34.[7] Radhakrishnan Balu: Covariant Quantum White Noise from Light-like QuantumFields, Special issue on Quantum Probability and Related Topics, JSOA (Accepted(2020)).[8] D. R. Terno, Introduction to Relativistic Quantum Information, in: Quantum Infor-mation Processing: From Theory to Experiment, edited by D. G. Angelakis et al.(Institute of Physics Press, 2006), p. 61.[9] J. Tolar, “A classification of finite quantum kinematics,” Jour-nal of Physics: Conference Series 538 no. 1, (2014) 012020.http://stacks.iop.org/1742-6596/538/i=1/a=012020.[10] J. Gough, V. P. Belavkin, and O. G. Smolyanov, Hamilton-Jacobi-Bellman equationsfor quantum filtering and control, J. Opt. B: Quantum Semiclass. Opt., 7 (2005), pp.237 - 244.[11] Infinite dimensional group representations, Bulletin of the American MathematicalSociety, 69, 628 (1963).[12] Charles Schwartz, ”Toward a quantum theory of tachyon fields,” Int. J. Mod. Phys.A 31, 1650041 (2016). 913] Tezak, N., Niederberger, A., Pavlichin, D. S., Sarma, G., and Mabuchi, H.: Specifica-tion of photonic circuits using quantum hardware description language. Philosophicaltransactions A, 370, 5270 (2012).[14] V.S. Varadarajan: Geometry of quantum theory, Springer (1985)[15] Newton T D and Wigner E P 1949 Localized states for elementary systems Rev.Mod. Phys. 21 400[16] Radhakrishnan Balu: Kinematics and Dynamics of Quantum Walk in terms of Sys-tems of Imprimitivity, J. Phys. A: Mathematical and Theoretical (2019).[17] Von Neumann: On rings of operators. Reduction theory, Annals of Mathematics, 50,401 (1949).[18] Wigner: Unitary representations of the inhomogeneous Lorentz group, Annals ofmathematics, 40, 149 (1939).[19] Y.S. Kim and E.S. Noz: Phase Space Picture of Quantum Mechanics: Group The-oretical Approach, Lecture Notes in Physics Series, World Scientific Pub Co Inc(1991).[20] Wigner: Unitary representations of the inhomogeneous Lorentz group: Ann. Math.,40, 845 (1939).[21] W. F. Stinespring, Positive Functions on C*-algebras, Proceedings of the AmericanMathematical Society, 6, 211–216 (1955).10