The work is intended to represent some interesting and apparently peculiar features of entangled system in both pure as well as mixed states level. In the pure state level, we are largely concerned about the existence and characteristics of bipartite incomparable pair of states. We have found some schemes for local transformation of one pure bipartite state to another, incomparable in nature, by exploiting free unpreserved entanglement. Also, we observed that different incomparable pair of states in $3\times 3$ may co-operate with each other to cross the barrier of incomparability. Our next goal was to probe critically the roots of the existence of incomparable states. We have succeeded in connecting incomparability with some famous no-go theorems of quantum mechanics. We have established it with no-flipping principle, general anti-unitary operators and lastly with inner product preserving operations. Thus we are able to pose incomparability as a good detector of some non-physical operations. In the next part, we provide a new class of multipartite bound entanglement in mixed state level. The class of states are activable bound entangled. There are exactly four orthogonal activable bound entangled states in any even number of qubit system. They have some interesting features like Bell correlation, permutation symmetry, local Pauli connection, non-compatibility, etc. Apart from those properties, the class of bound entangled states have the remarkable feature of local indistinguishability in quantum systems.
The work is intended to represent some interesting and apparently peculiar features of entangled system in both pure as well as mixed states level. In the pure state level, we are largely concerned about the existence and characteristics of bipartite incomparable pair of states. We have found some schemes for local transformation of one pure bipartite state to another, incomparable in nature, by exploiting free unpreserved entanglement. Also, we observed that different incomparable pair of states in 3×3 may co-operate with each other to cross the barrier of incomparability. Our next goal was to probe critically the roots of the existence of incomparable states. We have succeeded in connecting incomparability with some famous no-go theorems of quantum mechanics. We have established it with no-flipping principle, general anti-unitary operators and lastly with inner product preserving operations. Thus we are able to pose incomparability as a good detector of some non-physical operations. In the next part, we provide a new class of multipartite bound entanglement in mixed state level. The class of states are activable bound entangled. There are exactly four orthogonal activable bound entangled states in any even number of qubit system. They have some interesting features like Bell correlation, permutation symmetry, local Pauli connection, non-compatibility, etc. Apart from those properties, the class of bound entangled states have the remarkable feature of local indistinguishability in quantum systems.
While we possess a basic understanding of bipartite entangled systems, multipartite entanglement
remains mostly unexplored. Specifically, it is quite unclear how several concepts
developed for the bipartite case can be generalized in a meaningful way to study multipartite
scenarios. At the heart of this research on multiparty quantum states lies the problem
of testing the entanglement present in a given state. It has been proved, that in the most
general scenario, the question of determining the entanglement or separability of a state is
NP-hard.
From this perspective, it is desirable to have a construction of states for which
some prior knowledge about entanglement properties already exist. One particular approach
to this problem relies on the construction of Completely Entangled Subspaces
(CESes), i.e., subspaces void of product states. It follows that states in such subspaces
are necessarily entangled and form a convex set. This means we possess prior knowledge
of entanglement of all states in that subspace. In our study of multiparty systems, we have
constructed a specific subclass of the CESes such that all the states residing in the newly
constructed subspace is not only entangled but also Genuinely Entangled. A given state
is Genuinely Entangled if it is not just entangled for any pair among the multiparty but is
entangled with all the parties present and at the same time.
The notion of a CES is intimately connected with the idea of Unextendible Product
Bases (UPBs). UPBs are sets of twisted orthogonal product states/vectors spanning a
proper subset of a given Hilbert space. They have the property that no other product vector
exists in the ortho-complement of their span. While UPBs have been relatively well
studied for bipartite systems, no known multipartite orthogonal UPBs give rise to Genuinely
Entangled Subspace (GES). In our study, we provide schemes for the construction
of GES for a tripartite scenario. The construction proposed is linked to a Symmetric Block
structure such that it can be extended to any arbitrary dimension.One of the primary motivations for construction of UPBs is to create entangled
states which remain Positive under Partial Transposition (PPT). These classes of Entangled
States are known to be un-distillable and hence called Bound Entangled, i.e., states
in which the resource of Entanglement is not free to be used. These Bound Entangled states are essential to our understanding the basics of Entanglement Theory such as distillability,
quantification of entanglement, and the relationship between nonlocality and
entanglement (apart from their applications in other quantum information tasks). The
construction itself guarantees the PPTness of the state residing in the complementary subspace
i.e. formed by the equal mixture of the basis spanning the complementary space.
In our study, we propose a symmetric and an asymmetric construction of the Biseperable
and Genuinely Entangled states. Via the symmetric construction, we have derived two
subspaces, an eight-dimensional CES and a five-dimensional GES within the CES mentioned
above. The state formed by the equal mixtures of the CES is a Genuinely Bound
Entangled PPT state i.e., it entangled as well as PPT in each cut. To the best of author’s
knowledge, the proposed state is the only know Genuinely Entangled PPT state derived
from a UPB construction.
Additionally, the state lying in the ortho-complement of UPB (i.e., states formed
by an equal mixture of the basis of CES) is known to be a PPT Bound Entangled state.
However, unlike in case of UPBs, the state in the ortho-complement of Unextendible
Biseparable Product basis (i.e. states formed by an equal mixture of the basis of GES)
is found to be NPT states across all bipartitions. This is striking and prompts us to believe
that the state in the ortho-complement of CES is the only examples of bound entangled
state and distinct from other states in the space. We further expect that the state we obtain
must exhibit some distinctive features. We verify that this state is one copy undistillable.
However, the n-copy undistillablity needs to be probed further, and no conclusive
statement regarding the same can be made with our current analysis. The existence of
NPT-bound entangled state would prove that the distillable entanglement is nonadditive
and not convex.
The constructed subspace of Genuinely Entangled States in-turn reveals some
peculiar features. As mentioned above, unlike the Bennett et al., 1999 case, the state in the
ortho-complement is an NPT state across all bipartition. Furthermore, it was numerically
verified by taking one hundred thousand random probability distributions that the entire
five-dimensional genuine subspace obtained happens to be NPT. This leads us to conjecture
that the subspace obtained is a Genuinly Entangled NPT subspace. To the best of
the author’s knowledge, no known genuine NPT subspace exists for the multipartite scenario.
The exists only one other NPT subspace for Biseparable States and was suggested
by Johnston, 2013. Here, the construction is not related to UPBs. We further argue that
if our conjecture on the NPTness of the entire subspace is correct, any four-dimensional
subspace of the constructed tripartite NPT GES is also distillable. This would then be the
only known distillable subspace in a multipartite scenario. We believe that our study brings us closer to a systematic analysis of multipartite
entanglement and provides us with tools to probe further into the current mess of an
entangled (read communicating) world of multiple actors.
Abstract Consider a bipartite quantum system, where Alice and Bob jointly possess a pure state ∣ψ⟩ | ψ ⟩ . Using local quantum operations on their respective subsystems, and unlimited classical communication, Alice and Bob may be able to transform ∣ψ⟩ | ψ ⟩ into another state ∣ϕ⟩ | ϕ ⟩ . Famously, Nielsen’s theorem [28] provides a necessary and sufficient algebraic criterion for such a transformation to be possible (namely, the entanglement spectrum of ∣ϕ⟩ | ϕ ⟩ should majorise the entanglement spectrum of ∣ψ⟩ | ψ ⟩ ). In the paper where Nielsen proved this theorem, he conjectured that in the limit of large dimensionality, for almost all pairs of states ∣ψ⟩,∣ϕ⟩ | ψ ⟩ , | ϕ ⟩ (according to the natural unitary invariant measure) such a transformation is not possible. That is to say, typical pairs of quantum states ∣ψ⟩,∣ϕ⟩ | ψ ⟩ , | ϕ ⟩ are entangled in fundamentally different ways, that cannot be converted to each other via local operations and classical communication. Via Nielsen’s theorem, this conjecture can be equivalently stated as a conjecture about majorisation of spectra of random matrices from the so-called trace-normalised complex Wishart–Laguerre ensemble . Concretely, let X and Y be independent n×m n × m random matrices whose entries are i.i.d. standard complex Gaussians; then Nielsen’s conjecture says that the probability that the spectrum of XX†/tr(XX†) X X † / tr ( X X † ) majorises the spectrum of YY†/tr(YY†) Y Y † / tr ( Y Y † ) tends to zero as both n and m grow large. We prove this conjecture, and we also confirm some related predictions of Cunden et al. [12].
In this work we show that the most general class of anti-unitary operators are nonphysical in nature through the existence of incomparable pure bipartite entangled states. It is also shown that a large class of inner-product-preserving operations defined only on the three qubits having spin-directions along x,y and z are impossible. If we perform such an operation locally on a particular pure bipartite state then it will exactly transform to another pure bipartite state that is incomparable with the original one. As subcases of the above results we find the nonphysical nature of universal exact flipping operation and existence of universal Hadamard gate. Beyond the information conservation in terms of entanglement, this work shows how an impossible local operation evolve with the joint system in a nonphysical way.
In the past decades, quantum entanglement has been recognized to be the basic resource in quantum information theory. A fundamental need is then the understanding its qualification and its quantification: Is the quantum state entangled, and if it is, then how much entanglement is carried by that? These questions introduce the topics of separability criteria and entanglement measures, both of which are based on the issue of classification of multipartite entanglement. In this dissertation, after reviewing these three fundamental topics for finite dimensional Hilbert spaces, I present my contribution to knowledge. My main result is the elaboration of the partial separability classification of mixed states of quantum systems composed of arbitrary number of subsystems of Hilbert spaces of arbitrary dimensions. This problem is simple for pure states, however, for mixed states it has not been considered in full detail yet. I give not only the classification but also necessary and sufficient criteria for the classes, which make it possible to determine to which class a mixed state belongs. Moreover, these criteria are given by the vanishing of quantities measuring entanglement. Apart from these, I present some side results related to the entanglement of mixed states. These results are obtained in the learning phase of my studies and give some illustrations and examples.
Entanglement is a central phenomenon in modern physics, ubiquitous in quantum systems. From a quantum information theory perspective, entanglement is a resource when spatially separated labs are only able to act locally. The set of quantum operations in this setting is known as local operations and classical communication (LOCC). Studying LOCC transformations allows one to quantify entanglement. With this approach, bipartite entanglement has been well understood. However, multipartite entanglement presents a greater challenge. In particular, for large enough systems, states are generically incomparable via LOCC. Motivated by these results, we study multipartite entanglement via two approaches: firstly, by considering physically motivated modifications to LOCC; secondly, by focusing on physically relevant states. This thesis is divided into two parts. The first part takes the first of the two approaches and studies physically motivated modifications of LOCC. In particular, we study multi-state LOCC and approximate LOCC. In multi-state LOCC, we consider transformations of multiple states. This setting is physically motivated; multi-state transformations are natural if one can store states and act on them simultaneously. We show this setting is considerably more powerful than single-state LOCC, with even local unitaries (LUs) able to change the stochastic LOCC (SLOCC) class of the individual states. We also demonstrate novel multipartite features, including multipartite catalysis, and characterise LU transformations of multiple bipartite states. In approximate LOCC, we study transformations which nearly reach the desired state, rather than exactly reaching it. This setting is also physically motivated as, in any real lab, one never transforms states exactly to a desired state, but instead transforms states to an ensemble of states that are near the desired state. Here, we show certain features of LOCC simplify in approximate transformations. We then define a hierarchy of physically relevant approximate transformations and show that, unlike in the bipartite setting, this hierarchy does not collapse for multipartite approximate transformations. The second part of this thesis takes the second approach and considers a physically relevant class of states: matrix product states (MPS). Two MPS are said to belong to the same phase with respect to a symmetry if they can be converted into one another via short-depth, symmetric circuits. With this definition, new exotic phases of matter emerge, including symmetry-protected topological (SPT) order. We study how symmetric measurements and feedforward alter the phase classification of MPS. We demonstrate that for all on-site finite abelian symmetries, the phase classification trivialises. Moreover, for certain non-abelian symmetry groups, we provide protocols that implement asymptotically deterministic transformations between any two phases. We also study the connection between SPT phases and SLOCC classes of MPS. In particular, we characterise the normal MPS SLOCC classes, with bond and physical dimension equal to two, that are capable of corresponding to SPT phases.