The second law of thermodynamics places a limitation into which states a system can evolve into. For systems in contact with a heat bath, it can be combined with the law of energy conservation, and ...it says that a system can only evolve into another if the free energy goes down. Recently, it's been shown that there are actually many second laws, and that it is only for large macroscopic systems that they all become equivalent to the ordinary one. These additional second laws also hold for quantum systems, and are, in fact, often more relevant in this regime. They place a restriction on how the probabilities of energy levels can evolve. Here, we consider additional restrictions on how the coherences between energy levels can evolve. Coherences can only go down, and we provide a set of restrictions which limit the extent to which they can be maintained. We find that coherences over energy levels must decay at rates that are suitably adapted to the transition rates between energy levels. We show that the limitations are matched in the case of a single qubit, in which case we obtain the full characterization of state-to-state transformations. For higher dimensions, we conjecture that more severe constraints exist. We also introduce a new class of thermodynamical operations which allow for greater manipulation of coherences and study its power with respect to a class of operations known as thermal operations.
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Port-based teleportation (PBT), introduced in 2008, is a type of quantum teleportation protocol which transmits the state to the receiver without requiring any corrections on the receiver's side. ...Evaluating the performance of PBT was computationally intractable and previous attempts succeeded only with small systems. We study PBT protocols and fully characterize their performance for arbitrary dimensions and number of ports. We develop new mathematical tools to study the symmetries of the measurement operators that arise in these protocols and belong to the algebra of partially transposed permutation operators. First, we develop the representation theory of the mentioned algebra which provides an elegant way of understanding the properties of subsystems of a large system with general symmetries. In particular, we introduce the theory of the partially reduced irreducible representations which we use to obtain a simpler representation of the algebra of partially transposed permutation operators and thus explicitly determine the properties of any port-based teleportation scheme for fixed dimension in polynomial time.
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One of the crucial steps in building a scalable quantum computer is to identify the noise sources which lead to errors in the process of quantum evolution. Different implementations come with ...multiple hardware-dependent sources of noise and decoherence making the problem of their detection manyfoldly more complex. We develop a randomized benchmarking algorithm which uses Weyl unitaries to efficiently identify and learn a mixture of error models which occur during the computation. We provide an efficiently computable estimate of the overhead required to compute expectation values on outputs of the noisy circuit relying only on the locality of the interactions and no further assumptions on the circuit structure. The overhead decreases with the noise rate and this enables us to compute analytic noise bounds that imply efficient classical simulability. We apply our methods to ansatz circuits that appear in the variational quantum eigensolver and establish an upper bound on classical simulation complexity as a function of noise, identifying regimes when they become classically efficiently simulatable.
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Herein we continue the study of the representation theory of the algebra of permutation operators acting on the n-fold tensor product space, partially transposed on the last subsystem. We develop the ...concept of partially reduced irreducible representations, which allows us to significantly simplify previously proved theorems and, most importantly, derive new results for irreducible representations of the mentioned algebra. In our analysis we are able to reduce the complexity of the central expressions by getting rid of sums over all permutations from the symmetric group, obtaining equations which are much more handy in practical applications. We also find relatively simple matrix representations for the generators of the underlying algebra. The obtained simplifications and developments are applied to derive the characteristics of a deterministic port-based teleportation scheme written purely in terms of irreducible representations of the studied algebra. We solve an eigenproblem for the generators of the algebra, which is the first step towards a hybrid port-based teleportation scheme and gives us new proofs of the asymptotic behaviour of teleportation fidelity. We also show a connection between the density operator characterising port-based teleportation and a particular matrix composed of an irreducible representation of the symmetric group, which encodes properties of the investigated algebra.
Let <inline-formula> <tex-math notation="LaTeX">U_{d} </tex-math></inline-formula> be a unitary operator representing an arbitrary <inline-formula> <tex-math notation="LaTeX">d ...</tex-math></inline-formula>-dimensional unitary quantum operation. This work presents optimal quantum circuits for transforming a number <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> of calls of <inline-formula> <tex-math notation="LaTeX">U_{d} </tex-math></inline-formula> into its complex conjugate <inline-formula> <tex-math notation="LaTeX">\overline {U_{d}} </tex-math></inline-formula>. Our circuits admit a parallel implementation and are proven to be optimal for any <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula> with an average fidelity of <inline-formula> <tex-math notation="LaTeX">\left \langle{ {F}}\right \rangle =\frac {k+1}{d(d-k)} </tex-math></inline-formula>. Optimality is shown for average fidelity, robustness to noise, and other standard figures of merit. This extends previous works which considered the scenario of a single call (<inline-formula> <tex-math notation="LaTeX">k=1 </tex-math></inline-formula>) of the operation <inline-formula> <tex-math notation="LaTeX">U_{d} </tex-math></inline-formula>, and the special case of <inline-formula> <tex-math notation="LaTeX">k=d-1 </tex-math></inline-formula> calls. We then show that our results encompass optimal transformations from <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> calls of <inline-formula> <tex-math notation="LaTeX">U_{d} </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">f(U_{d}) </tex-math></inline-formula> for any arbitrary homomorphism <inline-formula> <tex-math notation="LaTeX">f </tex-math></inline-formula> from the group of <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula>-dimensional unitary operators to itself, since complex conjugation is the only non-trivial automorphism on the group of unitary operators. Finally, we apply our optimal complex conjugation implementation to design a probabilistic circuit for reversing arbitrary quantum evolutions.
We analyse the problem of transmitting a number of unknown quantum states or one composite system in one go. We derive a lower bound on the performance of such process, measured in the entanglement ...fidelity. The obtained bound is effectively computable and outperforms the explicit values of the entanglement fidelity calculated for the pre-existing variants of the port-based protocols, allowing for teleportation of a much larger amount of quantum information. The comparison with the exact formulas and similar analysis for the probabilistic scheme is also discussed. In particular, we present the closed-form expressions for the entanglement fidelity and for the probability of success in the probabilistic scheme in the qubit case in the picture of the spin angular momentum.
In this paper we present a new method for entanglement witnesses construction. We show that to construct such an object we can deal with maps which are not positive on the whole domain, but only on a ...certain sub-domain. In our approach a crucial role is played by such maps which are surjective between sets of rank projectors and the set of rank one projectors acting in the d dimensional space. We argue that our method can be used to check whether a given observable is an entanglement witness. In the second part of this paper we show that an inverse reduction map satisfies this requirement and using it we can obtain a bunch of new entanglement witnesses.
Irreducible representations (irreps) of a finite group G are equivalent if there exists a similarity transformation between them. In this paper, we describe an explicit algorithm for constructing ...this transformation between a pair of equivalent irreps, assuming that we are given an algorithm for computing the matrix elements of these irreps. Along the way, we derive a generalization of the classical orthogonality relations for matrix elements of irreps of finite groups. We give an explicit form of such unitary matrices for the important case of conjugated Young-Yamanouchi representations, when our group G is the symmetric group S(N).
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