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arXiv:quant-ph/0212062 · Physical Review A

Conal representation of quantum states and non-trace-preserving quantum operations

Pablo J. Arrighi, Christophe Patricot

Published Oct 10, 2003• 11 citations
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Abstract

We represent generalized density matrices of a d-complex dimensional quantum system as a subcone of a real pointed cone of revolution in ${R}^{{d}^{2}},$ or indeed a Minkowskian cone in ${E}^{{1,d}^{2}\ensuremath{-}1}.$ Generalized pure states correspond to certain future-directed lightlike vectors of ${E}^{{1,d}^{2}\ensuremath{-}1}.$ This extension of the generalized Bloch sphere enables us to cater for non-trace-preserving quantum operations, and in particular to view the per-outcome effects of generalized measurements. We show that these consist of the product of an orthogonal transform about the axis of the cone of revolution and a positive real linear transform. We give detailed formulas for the one-qubit case and express the post-measurement states in terms of the initial-state vectors and measurement vectors. We apply these results in order to find the information gain versus disturbance trade-off in the case of two equiprobable pure states. Thus we recover Fuchs and Peres's formula in an elegant manner.

DOI
10.1103/physreva.68.042310

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We represent generalized density matrices of a d-complex dimensional quantum system as a subcone of a real pointed cone of revolution in Rd2,{R}^{{d}^{2}},Rd2, or indeed a Minkowskian cone in E1,d2\ensuremath−1.{E}^{{1,d}^{2}\ensuremath{-}1}.E1,d2\ensuremath−1. Generalized pure states correspond to certain future-directed lightlike vectors of E1,d2\ensuremath−1.{E}^{{1,d}^{2}\ensuremath{-}1}.E1,d2\ensuremath−1. This extension of the generalized Bloch sphere enables us to cater for non-trace-preserving quantum operations, and in particular to view the per-outcome effects of generalized measurements. We show that these consist of the product of an orthogonal transform about the axis of the cone of revolution and a positive real linear transform. We give detailed formulas for the one-qubit case and express the post-measurement states in terms of the initial-state vectors and measurement vectors. We apply these results in order to find the information gain versus disturbance trade-off in the case of two equiprobable pure states. Thus we recover Fuchs and Peres's formula in an elegant manner.
Representations of quantum operations – We start by introducing a geometrical representation (real vector space) of quantum states and quantum operations. To do so we exploit an isomorphism from positive matrices to a subcone of the Minkowski future light-cone. Pure states map onto certain light-like vectors, whilst the axis of revolution encodes the overall probability of occurrence for the state. This extension of the Generalized Bloch Sphere enables us to cater for non-trace-preserving quantum operations, and in particular to view the per-outcome effects of generalized measurements. We show that these consist of the product of an orthogonal transform about the axis of the cone of revolution and a positive real symmetric linear transform. In the case of a qubit the representation becomes all the more interesting since it elegantly associates, to each measurement element of a generalized measurement, a Lorentz transformation in Minkowski space. We formalize explicitly this correspondence between ‘observation of a quantum system’ and ‘special relativistic change of inertial frame’. To end this part we review the state-operator correspondence, which was successfully exploited by Choi to derive the operator-sum representation of quantum operations. We go further and show that all of the important theorems concerning quantum operations can in fact be derived as simple corollaries of those concerning quantum states. Using this methodology we derive novel composition laws upon quantum states and quantum operations, Schmidt-type decompositions for bipartite pure states and some powerful formulae relating to the correspondence. Quantum cryptography – The key principle of quantum cryptography could be summarized as follows. Honest parties communicate using quantum states. To the eavesdropper these states are random and non-orthogonal. In order to gather information she must measure them, but this may cause irreversible damage. Honest parties seek to detect her mischief by checking whether certain quantum states are left intact. Thus tradeoff between the eavesdropper’s information gain, and the disturbance she necessarily induces, can be viewed as the power engine behind quantum cryptographic protocols. We begin by quantifying this tradeoff in the case of a measure distinguishing two non-orthogonal equiprobable pure states. A formula for this tradeoff was first obtained by Fuchs and Peres, but we provide a shorter, geometrical derivation (within the framework of the above mentioned conal representation). Next we proceed to analyze the Information gain versus disturbance tradeoff in a scenario where Alice and Bob interleave, at random, pairwise superpositions of two message words within their otherwise classical communications. This work constitutes one of the few results currently available regarding d-level systems quantum cryptography, and seems to provide a good general primitive for building such protocols. The proof crucially relies on the state-operator correspondence formulae derived in the first part, together with some methods by Banaszek. Finally we make use of this analysis to prove the security of a ‘blind quantum computation’ protocol, whereby Alice gets Bob to perform some quantum algorithm for her, but prevents him from learning her input to this quantum algorithm.
We exploit a well-known isomorphism between complex Hermitian 2 × 2 matrices and 4 , which yields a convenient real vector representation of qubit states. Because these do not need to be normalized we find that they map onto a Minkowskian future cone in 1,3 , whose vertical cross-sections are nothing but Bloch spheres. Pure states are represented by light-like vectors, unitary operations correspond to special orthogonal transforms about the axis of the cone, positive operations correspond to pure Lorentz boosts. We formalize the equivalence between the generalized measurement formalism on qubit states and the Lorentz transformations of special relativity, or more precisely elements of the restricted Lorentz group together with future-directed null boosts. The note ends with a discussion of the equivalence and some of its possible consequences.
While the Bloch sphere provides a canonical geometric representation for qubits, its generalization to pure qudit states (d>2d>2d>2) is not unique. This representation can be extended following two different approaches: increasing the number of points, as in the Majorana representation, or using a single point but in a space of larger dimension, as in the generalized Bloch representation. In this work, we compare these representations of pure qudit states by finding explicit analytical relations between them. In particular, a general formula of the Bloch vector in terms of the Majorana points of a given state is provided. We also analyze some suitable extensions of these representations in the case of pure permutation-symmetric two- and three-qubit states, deriving the corresponding relation between them. These analytical results allow us to study, from a geometric point of view, quantum correlations of special classes of two-qubit states.
We propose a generalization of the Bloch sphere representation for arbitrary spin states. It provides a compact and elegant representation of spin density matrices in terms of tensors that share the most important properties of Bloch vectors. Our representation, based on covariant matrices introduced by Weinberg in the context of quantum field theory, allows for a simple parametrization of coherent spin states, and a straightforward transformation of density matrices under local unitary and partial tracing operations. It enables us to provide a criterion for anticoherence, relevant in a broader context such as quantum polarization of light.
We establish a geometric generalization of the Bloch sphere for two-level quantum systems with non-Hermitian Hamiltonians using the Spacetime Algebra (STA) formulation. By lifting the state density operator from the even subalgebra to the full STA, we show that the state space expands from the unit 2-sphere to a future light cone. The non-unitary time evolution generated by a general non-Hermitian Hamiltonian corresponds to proper orthochronous Lorentz transformations on the null vectors. We classify the Hamiltonian dynamics into four distinct geometric classes: spatial rotations (corresponding to PT\mathcal{PT}PT-symmetric systems), pure boosts (anti-PT\mathcal{PT}PT-symmetric systems), null rotations (exceptional points), and general mixtures. We also use this formulation to study several results from PT-symmetric quantum mechanics, including the topological features of the exceptional points.
Quantum correlations often defy an explanation in terms of fundamental notions of classical physics, such as causality, locality, and realism. While the mathematical theory underpinning quantum correlations between spacelike separated systems has been well-established since the 1930s, the mathematical theory for correlations between non-spacelike separated systems is much less developed. In this work, we develop the theory of what we refer to as "local-density operators", which we view as joint states for possibly non-spacelike separated quantum systems. Local-density operators are unit trace operators whose marginals are genuine density operators, which we show not only subsumes the notion of density operator, but also several extensions of the notion of density operator into the spatiotemporal domain, such as pseudo-density operators and quantum states over time. More importantly, we prove a result which establishes a one-to-one correspondence between local-density operators and what we refer to as "Dirac measures", which are complex-valued measures on the space of separable projectors associated with two quantum systems. In the case that one of the systems is the trivial quantum system with a one-dimensional Hilbert space, our result recovers the fundamental result known as Gleason's Theorem, which implies that the Born rule from quantum theory is the only way in which one may assign probabilities to the outcomes of measurements performed on quantum systems in a non-contextual manner. As such, our results establish a direct generalization of Gleason's Theorem to measurements performed on possibly non-spacelike separated systems, thus extending the mathematical theory of quantum correlations across space to quantum correlations across space and time.