arXiv subjects

Gilad Gour

Publications and source records attributed to Gilad Gour.

At least 91 records · Page 5Linked to original sources

arXiv2014

Mutually unbiased measurements in finite dimensions

We generalize the concept of mutually unbiased bases (MUB) to measurements which are not necessarily described by rank one projectors. As such, these measurements can be a useful tool to study the long standing problem of the existence of MUB. We derive their general form, and show that in a finite, $d$-dimensional Hilbert space, one can construct a complete set of $d+1$ mutually unbiased measurements. Beside of their intrinsic link to MUB, we show, that these measurements' statistics provide complete information about the state of the system. Moreover, they capture the physical essence of unbiasedness, and in particular, they satisfy non-trivial entropic uncertainty relation similar to $d+1$ MUB.

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arXiv2013

Universal Uncertainty Relations

Uncertainty relations are a distinctive characteristic of quantum theory that impose intrinsic limitations on the precision with which physical properties can be simultaneously determined. The modern work on uncertainty relations employs \emph{entropic measures} to quantify the lack of knowledge associated with measuring non-commuting observables. However, there is no fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation. Starting from a very reasonable assumption of invariance under mere relabelling of the measurement outcomes, we show that Schur-concave functions are the most general uncertainty quantifiers. We then discover a fine-grained uncertainty relation that is given in terms of the majorization order between two probability vectors, \textcolor{black}{significantly extending a majorization-based uncertainty relation first introduced in [M. H. Partovi, Phys. Rev. A \textbf{84}, 052117 (2011)].} Such a vector-type uncertainty relation generates an infinite family of distinct scalar uncertainty relations via the application of arbitrary uncertainty quantifiers. Our relation is therefore universal and captures the essence of uncertainty in quantum theory.

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arXiv2013

Classification of multipartite entanglement in all dimensions

We provide a systematic classification of multiparticle entanglement in terms of equivalence classes of states under stochastic local operations and classical communication (SLOCC). We show that such an SLOCC equivalency class of states is characterized by ratios of homogenous polynomials that are invariant under local action of the special linear group. We then construct the complete set of all such SL-invariant polynomials (SLIPs). Our construction is based on Schur-Weyl duality and applies to any number of qudits in all (finite) dimensions. In addition, we provide an elegant formula for the dimension of the homogenous SLIPs space of a fixed degree as a function of the number of qudits. The expressions for the SLIPs involve in general many terms, but for the case of qubits we also provide much simpler expressions.

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arXiv2013

Simulating Symmetric Time Evolution With Local Operations

In closed systems, dynamical symmetries lead to conservation laws. However, conservation laws are not applicable to open systems that undergo irreversible transformations. More general selection rules are needed to determine whether, given two states, the transition from one state to the other is possible. The usual approach to the problem of finding such rules relies heavily on group theory and involves a detailed study of the structure of the respective symmetry group. We approach the problem in a completely new way by using entanglement to investigate the asymmetry properties of quantum states. To this end, we embed the space state of the system in a tensor product Hilbert space, whereby symmetric transformations between two states are replaced with local operations on their bipartite images.The embedding enables us to use the well-studied theory of entanglement to investigate the consequences of dynamic symmetries. Moreover, under reversible transformations, the entanglement of the bipartite image states becomes a conserved quantity. These entanglement-based conserved quantities are new and different from the conserved quantities based on expectation values of the Hamiltonian symmetry generators. Our method is not group-specific and applies to general symmetries associated with any semi-simple Lie group.

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arXiv2013

Entanglement Sharing Protocol via Quantum Error Correcting Codes

We introduce a new multiparty cryptographic protocol, which we call `entanglement sharing schemes', wherein a dealer retains half of a maximally-entangled bipartite state and encodes the other half into a multipartite state that is distributed among multiple players. In a close analogue to quantum secret sharing, some subsets of players can recover maximal entanglement with the dealer whereas other subsets can recover no entanglement (though they may retain classical correlations with the dealer). We find a lower bound on the share size for such schemes and construct two non-trivial examples based on Shor's $[[9,1,3]]$ and the $[[4,2,2]]$ stabilizer code; we further demonstrate how other examples may be obtained from quantum error correcting codes through classical encryption. Finally, we demonstrate that entanglement sharing schemes can be applied to characterize leaked information in quantum ramp secret sharing.

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arXiv2012

On Symmetric SL-Invariant Polynomials in Four Qubits

We find the generating set of SL-invariant polynomials in four qubits that are also invariant under permutations of the qubits. The set consists of four polynomials of degrees 2,6,8, and 12, for which we find an elegant expression in the space of critical states. In addition, we show that the Hyperdeterminant in four qubits is the only SL-invariant polynomial (up to powers of itself) that is non-vanishing precisely on the set of generic states.

math-ph↗

arXiv2012

Multipartite Entanglement Evolution Under Separable Operations

We study how multi-partite entanglement evolves under the paradigm of separable operations, which include the local operations and classical communication (LOCC) as a special case. We prove that the average "decay" of entanglement induced by a separable operation is measure independent (among SL-invariant ones) and state independent: the ratio between the average output entanglement and the initial entanglement is solely a function of the separable operation, regardless of the input state and of the SL-invariant entanglement measure being used. We discuss the "disentangling power" of a quantum channel and show that it exhibits a similar state invariance as the average entanglement decay ratio. Our Rapid Communication significantly extends the bipartite results of Ref. [1-3] as well as the multi-partite one of Ref. [4], all of the previous work being restricted to one-sided or particular noise models.

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arXiv2012

Limitations to sharing entanglement

We discuss limitations to sharing entanglement known as monogamy of entanglement. Our pedagogical approach commences with simple examples of limited entanglement sharing for pure three-qubit states and progresses to the more general case of mixed-state monogamy relations with multiple qudits.

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arXiv2012

Alignment of reference frames and an operational interpretation for the G-asymmetry

We determine the quantum states and measurements that optimize the accessible information in a reference frame alignment protocol associated with the groups U(1), corresponding to a phase reference, and $\mathbb{Z}_M$, the cyclic group of $M$ elements. Our result provides an operational interpretation for the $G$-asymmetry which is information-theoretic and which was thus far lacking. In particular, we show that in the limit of many copies of the bounded-size quantum reference frame, the accessible information approaches the Holevo bound. This implies that the rate of alignment of reference frames, measured by the (linearized) accessible information per system, is equal to the regularized, linearized $G$-asymmetry. The latter quantity is equal to the number variance in the case where $G=U(1)$. Quite surprisingly, for the case where $G=\mathbb{Z}_{M}$ and $M\geq 4$, it is equal to a quantity that is not additive in general, but instead can be superadditive under tensor product of two distinct bounded-size reference frames. This remarkable phenomenon is purely quantum and has no classical analog.

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arXiv2012

Reducing the quantum communication cost of quantum secret sharing

We demonstrate a new construction for perfect quantum secret sharing (QSS) schemes based on imperfect "ramp" secret sharing combined with classical encryption, in which the individual parties' shares are split into quantum and classical components, allowing the former to be of lower dimension than the secret itself. We show that such schemes can be performed with smaller quantum components and lower overall quantum communication than required for existing methods. We further demonstrate that one may combine both imperfect quantum and imperfect classical secret sharing to produce an overall perfect QSS scheme, and that examples of such scheme (which we construct) can have the smallest quantum and classical share components possible for their access structures, something provably not achievable using perfect underlying schemes. Our construction has significant potential for being adapted to other QSS schemes based on stabiliser codes.

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arXiv2011

The minimum entropy output of a quantum channel is locally additive

We show that the minimum von-Neumann entropy output of a quantum channel is locally additive. Hasting's counterexample for the additivity conjecture, makes this result quite surprising. In particular, it indicates that the non-additivity of the minimum entropy output is a global effect of quantum channels.

quant-ph↗

arXiv2011

Constructing monotones for quantum phase references in totally dephasing channels

Restrictions on quantum operations give rise to resource theories. Total lack of a shared reference frame for transformations associated with a group G between two parties is equivalent to having, in effect, an invariant channel between the parties and a corresponding superselection rule. The resource associated with the absence of the reference frame is known as "frameness" or "asymmetry." We show that any entanglement monotone for pure bipartite states can be adapted as a pure-state frameness monotone for phase-invariant channels [equivalently U(1) superselection rules] and extended to the case of mixed states via the convex-roof extension. As an application, we construct a family of concurrence monotones for U(1) frameness for general finite-dimensional Hilbert spaces. Furthermore, we study "frameness of formation" for mixed states analogous to entanglement of formation. In the case of a qubit, we show that it can be expressed as an analytical function of the concurrence analogously to the Wootters formula for entanglement of formation. Our results highlight deep links between entanglement and frameness resource theories.

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arXiv2011

Necessary and sufficient conditions for local manipulation of multipartite pure quantum states

Suppose several parties jointly possess a pure multipartite state, |ψ>. Using local operations on their respective systems and classical communication (i.e. LOCC) it may be possible for the parties to transform deterministically |ψ> into another joint state |ϕ>. In the bipartite case, Nielsen majorization theorem gives the necessary and sufficient conditions for this process of entanglement transformation to be possible. In the multipartite case, such a deterministic local transformation is possible only if both the states in the same stochastic LOCC (SLOCC) class. Here we generalize Nielsen majorization theorem to the multipartite case, and find necessary and sufficient conditions for the existence of a local separable transformation between two multipartite states in the same SLOCC class. When such a deterministic conversion is not possible, we find an expression for the maximum probability to convert one state to another by local separable operations. In addition, we find necessary and sufficient conditions for the existence of a separable transformation that converts a multipartite pure state into one of a set of possible final states all in the same SLOCC class. Our results are expressed in terms of (1) the stabilizer group of the state representing the SLOCC orbit, and (2) the associate density matrices (ADMs) of the two multipartite states. The ADMs play a similar role to that of the reduced density matrices, when considering local transformations that involves pure bipartite states. We show in particular that the requirement that one ADM majorize another is a necessary condition but in general far from being also sufficient as it happens in the bipartite case.

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arXiv2011

Local extrema of entropy functions under tensor products

We show that under a certain condition of local commutativity the minimum von-Neumann entropy output of a quantum channel is locally additive. We also show that local minima of the 2-norm entropy functions are closed under tensor products if one of the subspaces has dimension 2.

math-ph↗

arXiv2010

Evolution and Symmetry of Multipartite Entanglement

We discover a simple factorization law describing how multipartite entanglement of a composite quantum system evolves when one of the subsystems undergoes an arbitrary physical process. This multipartite entanglement decay is determined uniquely by a single factor we call the entanglement resilient factor (ERF). Since the ERF is a function of the quantum channel alone, we find that multipartite entanglement evolves in exactly the same way as bipartite (two qudits) entanglement. For the two qubits case, our factorization law reduces to the main result of Nature Physics 4, 99 (2008). In addition, for a permutation $P$, we provide an operational definition of $P$-asymmetry of entanglement, and find the conditions when a permuted version of a state can be achieved by local means.

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arXiv2010

Closed formula for the relative entropy of entanglement in all dimensions

The relative entropy of entanglement is defined in terms of the relative entropy between an entangled state and its closest separable state (CSS). Given a multipartite-state on the boundary of the set of separable states, we find a closed formula for all the entangled state for which this state is a CSS. Quite amazing, our formula holds for multipartite states in all dimensions. In addition we show that if an entangled state is full rank, then its CSS is unique. For the bipartite case of two qubits our formula reduce to the one given in Phys. Rev. A 78, 032310 (2008).

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arXiv2010

All Maximally Entangled Four Qubits States

We find an operational interpretation for the 4-tangle as a type of residual entanglement, somewhat similar to the interpretation of the 3-tangle. Using this remarkable interpretation, we are able to find the class of maximally entangled four-qubits states which is characterized by four real parameters. The states in the class are maximally entangled in the sense that their average bipartite entanglement with respect to all possible bi-partite cuts is maximal. We show that while all the states in the class maximize the average tangle, there are only few states in the class that maximize the average Tsillas or Renyi $α$-entropy of entanglement. Quite remarkably, we find that up to local unitaries, there exists two unique states, one maximizing the average $α$-Tsallis entropy of entanglement for all $α\geq 2$, while the other maximizing it for all $0<α\leq 2$ (including the von-Neumann case of $α=1$). Furthermore, among the maximally entangled four qubits states, there are only 3 maximally entangled states that have the property that for 2, out of the 3 bipartite cuts consisting of 2-qubits verses 2-qubits, the entanglement is 2 ebits and for the remaining bipartite cut the entanglement between the two groups of two qubits is 1ebit. The unique 3 maximally entangled states are the 3 cluster states that are related by a swap operator. We also show that the cluster states are the only states (up to local unitaries) that maximize the average $α$-Renyi entropy of entanglement for all $α\geq 2$.

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arXiv2009

Time-reversal frameness and superselection

We show that appropriate superpositions of motional states are a reference frame resource that enables breaking of time -reversal superselection so that two parties lacking knowledge about the other's direction of time can still communicate. We identify the time-reversal reference frame resource states and determine the corresponding frameness monotone, which connects time-reversal frameness to entanglement. In contradistinction to other studies of reference frame quantum resources, this is the first analysis that involves an antiunitary rather than unitary representation.

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