Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0706.2617 · Open Systems & Information Dynamics

On the Relation between States and Maps in Infinite Dimensions

Janusz Grabowski, Marek Kuś, Giuseppe Marmo

Published Dec 1, 2007• 18 citations
arXivHTMLPDF

Abstract

Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.

DOI
10.1007/s11080-007-9061-3

Related papers

Beyond trace class -- Tensor products of Hilbert spaces and operator ideals in quantum physics

Frank Oertel

Starting from the meaning of the conjugate of a complex Hilbert space, including a related application of the theorem of Fréchet-Riesz (by which an analysis of semilinear operators can be reduced to - linear - operator theory) to a revisit of applications of nuclear and absolutely $p$-summing operators in algebraic quantum field theory in the sense of Araki, Haag and Kastler ($p=2$) and more recently in the framework of general probabilistic spaces ($p=1$), we will outline that Banach operator ideals in the sense of Pietsch, or equivalently tensor products of Banach spaces in the sense of Grothendieck are even lurking in the foundations and philosophy of quantum physics and quantum information theory. In particular, we concentrate on their importance in algebraic quantum field theory. In doing so, we establish a canonical isometric isomorphism between the Hilbert spaces $H\otimes_2 (K \otimes_2 L)$ and $(H \otimes_2 K) \otimes_2 L$ (Theorem 3.8) and revisit the role of trace class operators. A few applications are specified, including the appropriateness of the class of Hilbert-Schmidt operators and an implied Banach operator ideal representation of the tensor product of two complex Hilbert spaces $H \otimes_2 K$ (Proposition 3.4) and a purely linear algebraic description of the quantum teleportation process (Example 3.10).

arXiv:2308.04627•Aug 8, 2023•0 citations

Infinite dimensional generalizations of Choi’s Theorem

Shmuel Friedland

In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.

•Jul 1, 2019•7 citations

Choi-Jamiołkowski isomorphism(s) and channel/state duality, naturally!

Paul E. Lammert

Provided a maximally entangled state vector in ${\mathcal{H}}\otimes{\mathcal{H}}$, the Choi-Jamiolkowski isomorphism of quantum information theory (or channel/state duality) produces a mapping of ${\mathsf CP}({\mathcal{H}},{\mathcal{K}})$ (completely positive maps of density operators on ${\mathcal{H}}$ tothose on ${\mathcal{K}}$) onto the density operators on ${\mathcal{K}}\otimes{\mathcal{H}}$, which extends to a unitary map. Here, ${\mathcal{H}}$ and ${\mathcal{K}}$ are {finite-dimensional} Hilbert spaces.The nature of this construction, the form of the associated formulas, and the possibility of versions which do not require an entangled vector seed, are illuminated from the perspective of {naturality}, in the sense of Eilenberg and MacLane.There is a unique such correspondence (``Choi map'') natural in both ${\mathcal{H}}$ and ${\mathcal{K}}$ which produces density operators on either ${\mathsf L}({\mathcal{H}},{\mathcal{K}})$ (Hilbert-Schmidt inner product) or ${\mathcal{K}}\otimes{\mathcal{H}}^*$ instead of ${\mathcal{K}}\otimes{\mathcal{H}}$. In particular, no entangled state input is required. There are no such binatural maps carried on ${\mathcal{K}}\otimes{\mathcal{H}}$, but those natural in only ${\mathcal{K}}$ are canonically associated with unitary maps ${\mathcal{H}}\rightarrow{\mathcal{H}}^*$ and derive from the binatural maps. The construction recovers the standard Choi-Jamiolkowski formulae and shows how they express naturality.

quant-pharXiv:2610.27394•Oct 21, 2026•0 citations

Beyond trace-class and Hilbert-Schmidt -- Interaction between operator ideals and von Neumann algebras in quantum physics

Frank Oertel

Starting from a thorough analysis of the conjugate $\overline{H}$ of a complex Hilbert space $H$, including its significant importance regarding a representation of the tensor product of two complex Hilbert spaces and its impact to the theorem of Fréchet-Riesz over to a revisit of applications of nuclear and absolutely $p$-summing operators in algebraic quantum field theory (AQFT) in the sense of Araki, Haag and Kastler ($p=2$) and more recently in the framework of general probabilistic spaces ($p=1$), we will outline that Banach operator ideals in the sense of Pietsch, or equivalently tensor products of Banach spaces in the sense of Grothendieck are even lurking in the foundations and philosophy of quantum physics and quantum information theory. In particular, we concentrate on their importance in AQFT (Theorem 5.27). In doing so, we revisit the role of trace-class operators in quantum theory and construct the enveloping $\tup{C}^\adj$-algebra, corresponding to an arbitrarily given normed operator ideal (Proposition 5.3 and Theorem 5.5). Applications are presented, including a purely linear algebraic description of the quantum teleportation process, thereby showing a link to quantum information theory, also due to the emergence of the Hadamard-Walsh transform and the controlled NOT gate (Example 4.18). All Hilbert spaces discussed in this paper may be nonseparable (and hence infinite-dimensional).

quant-pharXiv:2605.16025•May 18, 2026•0 citations

The cross states of a composite quantum system: separability and entanglement in any Hilbert space dimension

Paolo Aniello

We introduce a class of states of a composite quantum system, the so-called cross states, that turn out to play a major role in the theory of entanglement for a genuinely infinite-dimensional bipartite system. In the case where at least one of the Hilbert spaces of the bipartition is finite-dimensional, all states are cross states, whereas, in the genuinely infinite-dimensional setting where the dimension of both Hilbert spaces is not finite, the cross states form a trace-norm dense, convex, proper subset of the set of all states. In the latter case, the cross states can be regarded as those physical states that possess a finite amount of entanglement; accordingly, all separable states are of this kind. We prove that, for any Hilbert space dimension, the separable states can be characterized as those cross states that minimize a suitable norm, i.e., the projective norm associated with the projective tensor product of two trace classes; all other cross states are density operators belonging to the projective tensor product space. This is a generalization of the classical cross norm criterion of separability. Finally, we define an extended real-valued entanglement function and study its main properties. Coherently with the interpretation of cross states as finitely entangled states, this function is finite, and coincides with the projective norm, precisely on the cross states of the system.

arXiv:2602.18125•Feb 20, 2026•0 citations

Infinite dimensional generalizations of Choi’s Theorem

Shmuel Friedland

Abstract In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.

arXiv:1806.06938•Special Matrices•Jan 1, 2019•1 citation
Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.
Starting from the meaning of the conjugate of a complex Hilbert space, including a related application of the theorem of Fréchet-Riesz (by which an analysis of semilinear operators can be reduced to - linear - operator theory) to a revisit of applications of nuclear and absolutely ppp-summing operators in algebraic quantum field theory in the sense of Araki, Haag and Kastler (p=2p=2p=2) and more recently in the framework of general probabilistic spaces (p=1p=1p=1), we will outline that Banach operator ideals in the sense of Pietsch, or equivalently tensor products of Banach spaces in the sense of Grothendieck are even lurking in the foundations and philosophy of quantum physics and quantum information theory. In particular, we concentrate on their importance in algebraic quantum field theory. In doing so, we establish a canonical isometric isomorphism between the Hilbert spaces H⊗2(K⊗2L)H\otimes_2 (K \otimes_2 L)H⊗2​(K⊗2​L) and (H⊗2K)⊗2L(H \otimes_2 K) \otimes_2 L(H⊗2​K)⊗2​L (Theorem 3.8) and revisit the role of trace class operators. A few applications are specified, including the appropriateness of the class of Hilbert-Schmidt operators and an implied Banach operator ideal representation of the tensor product of two complex Hilbert spaces H⊗2KH \otimes_2 KH⊗2​K (Proposition 3.4) and a purely linear algebraic description of the quantum teleportation process (Example 3.10).
In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.
Provided a maximally entangled state vector in H⊗H{\mathcal{H}}\otimes{\mathcal{H}}H⊗H, the Choi-Jamiolkowski isomorphism of quantum information theory (or channel/state duality) produces a mapping of CP(H,K){\mathsf CP}({\mathcal{H}},{\mathcal{K}})CP(H,K) (completely positive maps of density operators on H{\mathcal{H}}H tothose on K{\mathcal{K}}K) onto the density operators on K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H, which extends to a unitary map. Here, H{\mathcal{H}}H and K{\mathcal{K}}K are {finite-dimensional} Hilbert spaces.The nature of this construction, the form of the associated formulas, and the possibility of versions which do not require an entangled vector seed, are illuminated from the perspective of {naturality}, in the sense of Eilenberg and MacLane.There is a unique such correspondence (``Choi map'') natural in both H{\mathcal{H}}H and K{\mathcal{K}}K which produces density operators on either L(H,K){\mathsf L}({\mathcal{H}},{\mathcal{K}})L(H,K) (Hilbert-Schmidt inner product) or K⊗H∗{\mathcal{K}}\otimes{\mathcal{H}}^*K⊗H∗ instead of K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H. In particular, no entangled state input is required. There are no such binatural maps carried on K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H, but those natural in only K{\mathcal{K}}K are canonically associated with unitary maps H→H∗{\mathcal{H}}\rightarrow{\mathcal{H}}^*H→H∗ and derive from the binatural maps. The construction recovers the standard Choi-Jamiolkowski formulae and shows how they express naturality.
Starting from a thorough analysis of the conjugate H‾\overline{H}H of a complex Hilbert space HHH, including its significant importance regarding a representation of the tensor product of two complex Hilbert spaces and its impact to the theorem of Fréchet-Riesz over to a revisit of applications of nuclear and absolutely ppp-summing operators in algebraic quantum field theory (AQFT) in the sense of Araki, Haag and Kastler (p=2p=2p=2) and more recently in the framework of general probabilistic spaces (p=1p=1p=1), we will outline that Banach operator ideals in the sense of Pietsch, or equivalently tensor products of Banach spaces in the sense of Grothendieck are even lurking in the foundations and philosophy of quantum physics and quantum information theory. In particular, we concentrate on their importance in AQFT (Theorem 5.27). In doing so, we revisit the role of trace-class operators in quantum theory and construct the enveloping $\tup{C}^\adj$-algebra, corresponding to an arbitrarily given normed operator ideal (Proposition 5.3 and Theorem 5.5). Applications are presented, including a purely linear algebraic description of the quantum teleportation process, thereby showing a link to quantum information theory, also due to the emergence of the Hadamard-Walsh transform and the controlled NOT gate (Example 4.18). All Hilbert spaces discussed in this paper may be nonseparable (and hence infinite-dimensional).
We introduce a class of states of a composite quantum system, the so-called cross states, that turn out to play a major role in the theory of entanglement for a genuinely infinite-dimensional bipartite system. In the case where at least one of the Hilbert spaces of the bipartition is finite-dimensional, all states are cross states, whereas, in the genuinely infinite-dimensional setting where the dimension of both Hilbert spaces is not finite, the cross states form a trace-norm dense, convex, proper subset of the set of all states. In the latter case, the cross states can be regarded as those physical states that possess a finite amount of entanglement; accordingly, all separable states are of this kind. We prove that, for any Hilbert space dimension, the separable states can be characterized as those cross states that minimize a suitable norm, i.e., the projective norm associated with the projective tensor product of two trace classes; all other cross states are density operators belonging to the projective tensor product space. This is a generalization of the classical cross norm criterion of separability. Finally, we define an extended real-valued entanglement function and study its main properties. Coherently with the interpretation of cross states as finitely entangled states, this function is finite, and coincides with the projective norm, precisely on the cross states of the system.
Abstract In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.