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Maximally Coherent Mixed States: Complementarity Between Maximal Coherence and Mixedness

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Maximally Coherent Mixed States: Complementarity Between Maximal Coherence and Mixedness Maximally coherent mixed states: Complementarity between maximal coherence and mixedness Uttam Singh, Manabendra Nath Bera, Himadri Shekhar Dhar, and Arun Kumar Pati Harish-Chandra Research Institute, Allahabad-211019, India Quantum coherence is a key element in topical research on quantum resource theories and a primary facilitator for design and implementation of quantum technologies. However, the resourcefulness of quantum coherence is severely restricted by environmental noise, which is indicated by the loss of information in a quantum system, measured in terms of its purity. In this work, we derive the limits imposed by the mixedness of a quantum system on the amount of quantum coherence that it can possess. We obtain an analytical trade-off between the two quantities that upperbound the maximum quantum coherence for fixed mixedness in a system. This gives rise to a class of quantum states, “maximally coherent mixed states,” whose coherence cannot be increased further under any purity-preserving operation. For the above class of states, quantum coherence and mixedness satisfy a complementarity relation, which is crucial to understand the interplay between a resource and noise in open quantum systems. I. INTRODUCTION resource theory of purity [35, 36], mixedness can be obtained as a complementary quantity to global information. Since, noise tends to increase the mixedness of a quantum system, it Recent developments in modern science have shown emerges as an intuitive parameter to understand decoherence. that quantum coherence plays an important role in low- A natural question that arises is how does important physical temperature physics starting from the formulation of the basic quantities in quantum information theory, such as entangle- laws of thermodynamics to work extraction [1–11]. Further- ment [37], fare against mixedness of quantum systems? An more, it is a useful figure of merit in investigating nanoscale interesting direction is to obtain the maximum amount of en- systems [12, 13] and understanding efficient energy transfer tanglement for a given mixedness, which leads to the notion in complex biological systems [14–18]. In recent years, re- of maximally entangled mixed states [38–42]. The amount of searchers have attempted to develop a framework to formal- entanglement in such states cannot be increased further under ize the theory of quantum coherence within the realms of any global unitary operation. Also, the form of the maximally quantum information and quantum resource theories [19–30]. entangled mixed states depends on the measures employed Within this context, there are two pertinent theoretical frame- to quantify entanglement and mixedness in the system [41]. works that attempt to characterize coherence as a resource. Such states have also been investigated in Gaussian quantum The first is based on the resource theory of asymmetry rela- systems [43–45]. tive to phase shifts, where operations are restricted to phase In our work, we investigate the limits imposed by mixed- insensitive operations and symmetric states are free resources ness of a quantum system on the amount of quantum co- [19, 20, 22, 26]. The above theoretical structure has been used herence present in the system. Since we consider quantum in several resource based formulations of quantum thermo- systems where the missing phase-reference frame is appar- dynamics [4,5]. The second formalism is based on a well- ently lacking, the formalism based on the resource theory of defined set of allowed incoherent operations and a set of freely asymmetry [22] becomes over-restrictive [46]. Hence, in the available incoherent states [23]. In this framework, quantum present work, we use the theoretical approach based on the set coherence is a well-defined resource, which can be quantified of incoherent operations and states [23], to characterize and in terms of functions or coherence monotones that satisfy cer- quantify coherence. We derive an analytical trade-off between tain characteristic conditions. Some of the better known mea- the two quantities that allows us to upperbound the maximum sures of quantum coherence are those based on l1 norm and coherence in a given mixed quantum state and vice versa. Us- relative entropy [23], and skew information [25]. Incidentally, ing the l1 norm of coherence [23] as a measure of quantum a recent work proves that all measures of entanglement can be coherence and normalized linear entropy [34] as a measure of artfully used to define a family of valid measures of quantum arXiv:1503.06303v2 [quant-ph] 28 May 2015 mixedness, we prove that for a general d-dimensional quan- coherence [31]. Moreover, the latter formalism has been re- tum system the sum of the (scaled) squared coherence and the cently used to address a fundamental issue of wave-particle mixedness is always less than or equal to unity. This allows duality [32], thus, enabling coherence to be a valid indicator us to derive a class of quantum states, viz. “maximally coher- of the wave nature of quantum systems. ent mixed states” (MCMSs), that have maximal coherence, Another significant aspect in the dynamics of quantum sys- up to incoherent unitaries, for a fixed mixedness. These states tems is the role of environmental noise and the unavoidable are parametrized mixtures of a d-dimensional pure maximally phenomenon of decoherence. It is known that decoherence is coherent state and maximally mixed state. Interestingly, for detrimental to the amount of information contained in a quan- different values of mixedness the analytical form of MCMS tum state, as measured by its purity. To effectively charac- remains unchanged and, unlike maximally entangled mixed terize the role of decoherence in erasing information [33] one states, is not dependent on the choice of the measure of co- needs to quantify the purity or its complementary property, herence and mixedness, as observed for l1 norm, relative en- the mixedness of the state. A faithful measure of mixedness tropy, and geometric measures of coherence. The obtained is the normalized linear entropy [34]. From the perspective of analytical results, show an important trade-off between a rel- 2 evant quantum resource and noise in open quantum systems Measures that satisfy the above conditions, include l1 norm and a complementary behavior between coherence and mixed- and relative entropy of coherence [23] and the skew infor- ness in the class of MCMSs, which may be crucial from the mation [25]. Generic monotones of quantum coherence can perspective of quantum resource theories and thermodynam- also be derived using entanglement monotones that satisfy the ics. Significantly, since the mixedness of a quantum system above conditions [31]. In this work, we shall mainly be fo- can be experimentally measured using quantum interferomet- cused on the l1 norm of coherence. For a quantum state ρ and ric setups [47, 48], without resorting to complicated state to- the reference basis fjiig, the l1 norm of coherence is given by mography, our results provide a mathematical framework to experimentally determine the maximal coherence in a quan- X C ρ jρ j; tum state. l1 ( ) = i j (1) The paper is organized as follows. In Sec.II, we briefly dis- i, j cuss the quantification of coherence and mixedness. In Sec. III, we theorize the trade-off between coherence and mixed- where ρi j = hijρj ji. Another measure of coherence is the rela- ness in d-dimensional systems. In Sec.IV, we define a class tive entropy of coherence, which is given by Cr(ρ) = S (ρd) − of maximally coherent mixed states that satisfy a comple- S (ρ); where S (ρ) = −Tr(ρ ln ρ), is the von Neumann entropy P mentarity relation between coherence and mixedness. In Sec. and ρd = ihijρjiijiihij. Moreover, a geometric measure of co- V, we investigate the allowed set of transformations within herence had also been speculated [23, 29, 31] and was, only classes of fixed coherence or mixedness. We conclude with a recently, shown to be a full coherence monotone [31]. The discussion of the main results in Sec.VI. geometric measure is given by Cg(ρ) = 1 − maxσ2I F(ρ, σ), where I is the set of all incoherent states and F(ρ, σ) = q p p 2 II. QUANTIFYING COHERENCE AND MIXEDNESS Tr[ σρ σ] is the fidelity of the states ρ and σ. It is important to note that quantum coherence, by definition, is In this section we present a brief overview of the concepts not invariant under general unitary operation but does remain of quantum coherence and mixedness of quantum systems. To unchanged under incoherent unitaries. Furthermore, the max- Pd−1 imally coherent pure state is defined by j i = p1 jii, for characterize the coherence in a quantum system, we follow the d d i=0 theoretical approach developed in Ref. [23]. All mathematical which Cl1 (j di h dj) = d − 1 and Cr(j di h dj) = ln d. formulations and results that are subsequently presented and discussed are valid within the framework of the above theory of quantum coherence. B. Mixedness A. Quantum coherence For every quantum state, the ubiquitous interaction with en- Quantum coherence, an essential feature of quantum me- vironment or decoherence affects its purity. Noise introduces chanics arising from the superposition principle, is inherently mixedness in the quantum system leading to loss of informa- a basis dependent quantity. Therefore, any quantitative mea- tion, and hence, its characterization is an important task in sure of it must depend on a reference basis. The framework, quantum information protocols. The mixedness, which rep- to quantify coherence in the context of quantum information resents nothing but the disorder in the system, can be quan- theory, is based on the characterization of a set of incoher- tified in terms of entropic functionals, such as linear and von I ent states, denoted by I and incoherent operations Λ [23].
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