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Łukasz Pankowski

Publications and source records attributed to Łukasz Pankowski.

8 recordsLinked to original sources
arXiv2023

Complete extension: the non-signaling analog of quantum purification

Deriving quantum mechanics from information-theoretic postulates is a recent research direction taken, in part, with the view of finding a beyond-quantum theory; once the postulates are clear, we can consider modifications to them. A key postulate is the purification postulate, which we propose to replace by a more generally applicable postulate that we call the complete extension postulate (CEP), i.e., the existence of an extension of a physical system from which one can generate any other extension. This new concept leads to a plethora of open questions and research directions in the study of general theories satisfying the CEP (which may include a theory that hyper-decoheres to quantum theory). For example, we show that the CEP implies the impossibility of bit-commitment. This is exemplified by a case study of the theory of non-signalling behaviors which we show satisfies the CEP. We moreover show that in certain cases the complete extension will not be pure, highlighting the key divergence from the purification postulate.

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arXiv2015

Long-distance quantum communication over noisy networks without long-time quantum memory

The problem of sharing entanglement over large distances is crucial for implementations of quantum cryptography. A possible scheme for long-distance entanglement sharing and quantum communication exploits networks whose nodes share Einstein-Podolsky-Rosen (EPR) pairs. In Perseguers et al. [Phys. Rev. A 78, 062324 (2008)] the authors put forward an important isomorphism between storing quantum information in a dimension $D$ and transmission of quantum information in a $D+1$-dimensional network. We show that it is possible to obtain long-distance entanglement in a noisy two-dimensional (2D) network, even when taking into account that encoding and decoding of a state is exposed to an error. For 3D networks we propose a simple encoding and decoding scheme based solely on syndrome measurements on 2D Kitaev topological quantum memory. Our procedure constitutes an alternative scheme of state injection that can be used for universal quantum computation on 2D Kitaev code. It is shown that the encoding scheme is equivalent to teleporting the state, from a specific node into a whole two-dimensional network, through some virtual EPR pair existing within the rest of network qubits. We present an analytic lower bound on fidelity of the encoding and decoding procedure, using as our main tool a modified metric on space-time lattice, deviating from a taxicab metric at the first and the last time slices.

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arXiv2013

Conjectured strong complementary-correlations tradeoff

We conjecture new uncertainty relations which restrict correlations between results of measurements performed by two separated parties on a shared quantum state. The first uncertainty relation bounds the sum of two mutual informations when one party measures a single observable and the other party measures one of two observables. The uncertainty relation does not follow from Maassen-Uffink uncertainty relation and is much stronger than Hall uncertainty relation derived from the latter. The second uncertainty relation bounds the sum of two mutual informations when each party measures one of two observables. We provide numerical evidence for validity of conjectured uncertainty relations and prove them for large classes of states and observables.

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arXiv2013

Local random quantum circuits are approximate polynomial-designs - numerical results

We numerically investigate the statement that local random quantum circuits acting on n qubits composed of polynomially many nearest neighbour two-qubit gates form an approximate unitary poly(n)-design [F.G.S.L. Brandao et al., arXiv:1208.0692]. Using a group theory formalism, spectral gaps that give a ratio of convergence to a given t-design are evaluated for a different number of qubits n (up to 20) and degrees t (t=2,3,4 and 5), improving previously known results for n=2 in the case of t=2 and 3. Their values lead to a conclusion that the previously used lower bound that bounds spectral gaps values may give very little information about the real situation and in most cases, only tells that a gap is closed. We compare our results to the another lower bounding technique, again showing that its results may not be tight.

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arXiv2010

Low-dimensional quite noisy bound entanglement with cryptographic key

We provide a class of bound entangled states that have positive distillable secure key rate. The smallest state of this kind is 4 \bigotimes 4. Our class is a generalization of the class presented in [1] (IEEE Trans. Inf. Theory 54, 2621 (2008); arXiv:quant-ph/0506203). It is much wider, containing, in particular, states from the boundary of PPT entangled states (all of the states in the class in [1] were of this kind) but also states inside the set of PPT entangled states, even, approaching the separable states. This generalization comes with a price: for the wider class a positive key rate requires, in general, apart from the one-way Devetak-Winter protocol (used in [1]) also the recurrence preprocessing and thus effectively is a two-way protocol. We also analyze the amount of noise that can be admixtured to the states of our class without losing key distillability property which may be crucial for experimental realization. The wider class contains key-distillable states with higher entropy (up to 3.524, as opposed to 2.564 for the class in [1]).

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arXiv2010

A few steps more towards NPT bound entanglement

We consider the problem of existence of bound entangled states with non-positive partial transpose (NPT). As one knows, existence of such states would in particular imply nonadditivity of distillable entanglement. Moreover it would rule out a simple mathematical description of the set of distillable states. Distillability is equivalent to so called n-copy distillability for some n. We consider a particular state, known to be 1-copy nondistillable, which is supposed to be bound entangled. We study the problem of its two-copy distillability, which boils down to show that maximal overlap of some projector Q with Schmidt rank two states does not exceed 1/2. Such property we call the the half-property. We first show that the maximum overlap can be attained on vectors that are not of the simple product form with respect to cut between two copies. We then attack the problem in twofold way: a) prove the half-property for some classes of Schmidt rank two states b) bound the required overlap from above for all Schmidt rank two states. We have succeeded to prove the half-property for wide classes of states, and to bound the overlap from above by c<3/4. Moreover, we translate the problem into the following matrix analysis problem: bound the sum of the squares of the two largest singular values of matrix A \otimes I + I \otimes B with A,B traceless 4x4 matrices, and Tr A^\dagger A + Tr B^\dagger B = 1/4.

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arXiv2007

Can quantum correlations be completely quantum?

Deficit of information zero-way was proposed in HorodeckiHHOSSS2005 as one of possible measures of quantumness of correlations. Numerical calculations suggested that there exist such states for which this quantity is almost equal to mutual information. In this paper we present a family of states for which we have equality between above measure of quantumness of correlations and the measure of total correlations -- mutual information. It means that whole correlations in these states have, in some sense, quantum character and that quantum correlations do not necessarily imply classical correlations. We prove this intriguing feature for a subclass of 2x2 separable states. We also present numerical result suggesting that this interesting situation might also happen for 2x2 entangled states.

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