Observables can be tailored to change the entanglement of any pure state

Abstract

We show that for a finite-dimensional Hilbert space, there exist observables that induce a tensor product structure such that the entanglement properties of any pure state can be tailored. In particular, we provide an explicit, finite method for constructing observables in an unstructured d-dimensional system so that an arbitrary known pure state has any Schmidt decomposition with respect to an induced bipartite tensor product structure. In effect, this article demonstrates that in a finite-dimensional Hilbert space, entanglement properties can always be shifted from the state to the observables and all pure states are equivalent as entanglement resources in the ideal case of complete control of observables.PACS numbers: 03.65.Aa, 03.65.UdThe entanglement of a quantum state is only defined with respect to a tensor product structure within the Hilbert space that represents the quantum system. In turn, a tensor product structure of the Hilbert space is induced by the algebra of observables. Zanardi and colleagues [1,2] have provided criteria for the algebra of observables of a finite-dimensional system to induce a tensor product structure. The algebra of observables must be partitioned into subalgebras that satisfy two mathematical requirements, the subalgebras must be independent and complete (see Corollary 3 for a precise formulation of Zanardi's Theorem), and one physical requirement, the subalgebras must be locally accessible. Such observableinduced partitions of the Hilbert space have been referred to as virtual subsystems and can be thought of as a generalization from entanglement between subsystems to entanglement between degrees of freedom (see also [3,4]). This mathematical framework has found applications to studies of multi-level encoding [5], decoherence [6], operator quantum error correction [7], entanglement in fermionic systems [8], single-particle entanglement [9,10], and entanglement in scattering systems [11].In this Letter, we extend this mathematical framework and prove what we call the Tailored Observables Theorem (Theorem 6): observables can be constructed such that any pure state in a finite-dimensional Hilbert space H = C d has any amount of entanglement possible for any given factorization of the dimension d of H. This means all pure states are equivalent as entanglement resources in the ideal case of complete control of observables. To establish the framework, we provide a brief, relatively self-contained introduction to Zanardi's Theorem and obtain some necessary preliminary results about observable algebras in finite dimensions. We then prove Theorem 6, which applies to bipartite tensor product structures, and present an illustrative example. We will also provide a corollary of the theorem (Corollary 7) applied to multipartite tensor product structures. Before delving into the technical details, we present a more intuitive discussion of this result. this Hilbert space could represent states of a quantum system composed from N subsystems each represented by Hilbert spaces H i = C ki . Fo...

Quantumness of correlations in indistinguishable particles

“…[14] for a detailed discussion), and based on this identification we shall set up an "activation protocol" for indistinguishable particles. The importance to study the correlations, particularly the entanglement, in terms of subalgebras of observables has been emphasized in [14,22,[34][35][36][37], proving to be a useful approach for such analysis. The algebra of single-particle observables is generated by,…”

Quantumness and the role of locality on quantum correlations

“…As pointed out in the Introduction, the relative character of quantum correlations with respect to the chosen partition of a system into subsystems has been carefully studied in the case of pure states [4][5][6][7]. In our presentation, we focus on the case of mixed states under a locality restriction: we allow only local unitary operations (over the enlarged Hilbert space) to explore the observables' subspaces of each local subsystem.…”

“…Also, in Zanardi et al [5], the authors studied the role of the relevant observables in the tensor product structure of the Hilbert space. Harshmann and Ranade [7] gave a formal proof of the fact that, in the pure states case, one can tailor the observables so as to go from a situation with no entanglement to a maximal entanglement one, for any fixed state. However, for mixed states the situation is radically different -as it can be seen, for example, taking the maximally mixed state, which is the same for any observables we choose-and the tailoring of observables cannot place all the states on an equal footing.…”

“…Harshman and Ranade gave in Ref. [7] the formal proof that, for any pure state in C N , where N ∈ N is not a prime, the observables can be 'tailored' to induce a subsystem decomposition for which the state has a desired level of entanglement, from a product state to a maximally entangled one. The situation is rather different for mixed states.…”

“…Later, Barnum et al [6] gave a subsystem-independent notion of entanglement. Harshmann and Ranade [7] proved that all pure states of a finite-dimensional (and unstructured) Hilbert space are equivalent as entanglement resources in the ideal case that one has complete access and control of observables. Given that CC implies separability and given that the question about separability becomes relative to the preferred observables (the ones that determine the local subsystems), the question about the correlations on CC states becomes relative too.…”