Colloquium: quantum coherence as a resource
Summary (14 min read)
Introduction
- The coherent superposition of states, in combination with the quantization of observables, represents one of the most fundamental features that mark the departure of quantummechanics from the classical realm.
- The development of a rigorous theory of quantum coherence as a physical resource has been initiated only recently.
- In a nutshell, any such theory first considers constraints that are imposed on us in a specific physical situation (e.g., the inability to perform joint quantum operations between distant laboratories due to the impossibility to transfer quantum systems from one location to the other while preserving their quantum coherence, and thus restricting us to local operations and classical communication).
- These applications include so-called “QuantumTechnologies 2.0,” such as quantum-enhancedmetrology and communication protocols, and extend farther into other fields, such as thermodynamics and even certain branches of biology.
A. Constraints, operations, and resources
- These constraints may be due to either fundamental conservation laws, such as superselection rules and energy conservation, or constraints due to the practical difficulty of executing certain operations, e.g., the restriction to local operations and classical communication (LOCC) which gives rise to the resource theory of entanglement (Plenio and Virmani, 2007; Horodecki et al., 2009).
- The states that can be generated from the maximally mixed state1 by the application of free operations in F alone, are considered to be available free of charge, forming the set I of free states.
- All the other states attain the status of a resource, whose provision carries a cost.
- For the purposes of the present exposition of the resource theory of coherence, the authors begin by adopting the latter point of view and then proceed to require additional desirable properties of their classes of free operations.
1. Incoherent states
- Coherence is naturally a basis dependent concept, which is why the authors first need to fix the preferred or reference basis in which to formulate their resource theory.
- If the reference basis for a single qubit is taken to be the computational basis fj0i; j1ig, i.e., the eigenbasis of the Pauli σz operator, then any density matrix with a nonzero off-diagonal element jϱ01j ¼ jh0jϱj1ij ≠ 0 is outside the set I of incoherent states, and hence has a resource content.
- If the Hamiltonian is nondegenerate, the corresponding set of free states is exactly the set of incoherent states described, where the incoherent basis is defined by the eigenbasis of the Hamiltonian.
- In the following, whenever the authors refer to incoherent states, they explicitly mean states of the form (1).
2. Classes of incoherent operations
- The definition of free operations for the resource theory of coherence is not unique and different choices, often motivated by suitable practical considerations, are being examined in the literature.
- The authors also mention genuinely incoherent operations (GIO) and fully incoherent operations (FIO) (de Vicente and Streltsov, 2017).
- A classification of the different frameworks of coherence, motivated by the notion of speakable and unspeakable information (Peres and Scudo, 2002; Bartlett, Rudolph, and Spekkens, 2007), was proposed by Marvian and Spekkens (2016).
1. Maximally coherent states and state transformations via incoherent operations
- The authors start by identifying a d-dimensional maximally coherent state as a state that allows for the deterministic generation of all other d-dimensional quantum states by means of the free operations.
- The authors note that not all frameworks of coherence that were discussed in Sec. II.
- One may also determine maximally coherent states under certain additional constraints, such as the degree of mixedness, which gives rise to the class of maximally coherent mixed states (Singh, Bera, Dhar, and Pati, 2015).
- A notable result in this context was provided by Chitambar and Gour 2016a, 2016b, 2017), who gave a full characterization of single-qubit state conversion via SIO, DIO, IO, or MIO.
- The optimal rate for this process can be evaluated analytically; see Sec. III.
2. States and maps
- A more complex task beyond state preparation is that of the generation of a general quantum operation from a supply of coherent states and incoherent operations.
- Just as the maximally entangled state allows for the generation of all quantum operations (Eisert et al., 2000) via LOCC, so does the maximally coherent state allow for the generation of all quantum operations via IO.
- The explicit construction for an arbitrary single-qubit unitary can be found in Baumgratz, Cramer, and Plenio (2014), and the extension to general quantum operations of arbitrary dimension was studied by Chitambar and Hsieh (2016) and Ben Dana et al. (2017).
- The corresponding construction makes use of maximally coherent states even if the targeted quantum operation may only be very slightly coherent.
- By virtue of the monotonicity of coherence quantifiers under incoherent operations, lower bounds to the amount of coherence required to implement a quantum operation can be provided (Mani and Karimipour, 2015; Ben Dana et al., 2017; Bu et al., 2017).
C. Quantum coherence in distributed scenarios
- Based on the framework of LOCC known from entanglement theory (Plenio and Virmani, 2007; Horodecki et al., 2009), one can introduce the framework of local incoherent operations and classical communication (LICC) (Chitambar and Hsieh, 2016; Streltsov, Rana, Bera, and Lewenstein, 2017).
- While LICC operations in general have a difficult mathematical structure, it is possible to introduce the more general class of separable incoherent (SI) operations which has a simple mathematical form (Streltsov, Rana, Bera, and Lewenstein, 2017): Λ½ϱAB ¼ X i ðAi ⊗ BiÞϱABðA†i ⊗ B†i Þ; ð14Þ where Ai and Bi are local incoherent operators.
- Asymmetric scenarios where only one of the parties is restricted to IO locally have also been considered (Chitambar et al., 2016; Streltsov, Rana, Bera, and Lewenstein, 2017).
- The latter task was also performed experimentally (Wu et al., 2017) and will be discussed in more detail in Sec. III.L.3.
- Here σAi are arbitrary states on the subsystem A, while jiiB are incoherent states on the subsystem B. Moreover, LQICC is a strict subset of SQI (Streltsov, Rana, Bera, and Lewenstein, 2017) and GOIB (Matera et al., 2016).
D. Connection between coherence and entanglement theory
- The resource theory of coherence exhibits several connections to the resource theory of entanglement.
- These results were generalized to quantum discord (Ma et al., 2016) and general types of nonclassicality (Killoran, Steinhoff, and Plenio, 2016); see Secs. III.
- Another relation between entanglement and coherence was provided by Streltsov, Chitambar et al. (2016), where they introduced and studied the task of incoherent quantum state merging.
- An important open question in this context is whether any of the aforementioned classes of incoherent operations Rev. Mod.
A. Postulates for coherence monotones and measures
- The first axiomatic approach to quantify coherence was presented by Åberg (2006), and an alternative framework was more recently developed by Baumgratz, Cramer, and Plenio (2014).
- At this point it is instrumental to compare conditions C2 and C3 to the corresponding conditions in entanglement theory (Vedral et al., 1997; Vedral and Plenio, 1998; Plenio and Virmani, 2007; Horodecki et al., 2009).
- The authors remark that the terminology adopted here differs from the one used in some recent literature, which is mainly based on Baumgratz, Cramer, and Plenio (2014).
- With the more stringent approach presented here, inspired from entanglement theory, the authors aim to distinguish two important coherence quantifiers: distillable coherence (which is equal to the relative entropy of coherence) and coherence cost (which is equal to the coherence of formation).
B. Distillable coherence and coherence cost
- The distillable coherence is the optimal number of maximally coherent single-qubit states jΨ2iwhich can be obtained per copy of a given state ϱ via incoherent operations in the asymptotic limit.
- For pure states this result was independently found by Yuan et al. (2015).
- Condition C5 also follows from Eq. (27) and the definition of coherence of formation; see Sec. III.D. Moreover, the coherence cost is additive, i.e., condition C6 is also satisfied (Winter and Yang, 2016).
- In general, the distillable coherence cannot be larger than the coherence cost CdðϱÞ ≤ CcðϱÞ: ð28Þ.
- For pure states this inequality becomes an equality, which implies that the resource theory of coherence is reversible for pure states.
C. Distance-based quantifiers of coherence
- A general distance-based coherence quantifier is defined as (Baumgratz, Cramer, and Plenio, 2014) 10A similar situation occurs in the case of entanglement, for which a fully reversible resource theory can also be constructed if one considers the largest set of operations preserving separability (Brandão and Plenio, 2008, 2010), which is a strict superset of separable operations.
- Monotonicity C2 is also fulfilled for any set of operations discussed in Sec. II.
- A.2 if the distance is contractive (Baumgratz, Cramer, and Plenio, 2014), DðΛ½ϱ ;Λ½σ Þ ≤ Dðϱ; σÞ ð30Þ for any quantum operation Λ. Moreover, any distance-based coherence quantifier fulfills convexity C4 whenever the corresponding distance is jointly convex (Baumgratz, Cramer, and Plenio, 2014), D X i piϱi; X j pjσj ≤ X i piDðϱi; σiÞ: ð31Þ.
- In the following, the authors explicitly discussed three important distance-based coherence quantifiers.
1. Relative entropy of coherence
- If the distance is chosen to be the quantum relative entropy Sðϱ∥σÞ ¼ Tr½ϱlog2ϱ − Tr½ϱlog2σ ; ð32Þ the corresponding quantifier is known as the relative entropy of coherence11 (Baumgratz, Cramer, and Plenio, 2014): CrðϱÞ ¼ min σ∈I Sðϱ∥σÞ: ð33Þ.
- The relative entropy of coherence fulfills conditions C1, C2, and C4 for any set of operations discussed in Sec. II.
- Moreover, it is equal to the distillable coherence Cd, and therefore both quantities admit the same closed expression (Gour, Marvian, and Spekkens, 2009; Baumgratz, Cramer, and Plenio, 2014; Winter and Yang, 2016) CrðϱÞ ¼ CdðϱÞ ¼ SðΔ½ϱ Þ − SðϱÞ; ð34Þ where Δ½ϱ is the dephasing operation defined in Eq. (5).
- As further shown by Singh, Bera, Misra, and Pati (2015), the relative entropy of coherence can also be interpreted as the minimal amount of noise required for fully decohering the state ϱ.
2. Coherence quantifiers based on matrix norms
- The authors now consider coherence quantifiers based on matrix norms, i.e., such that the corresponding distance has the form Dðϱ; σÞ ¼ kϱ − σk with some matrix norm k·k.
- Note first that any such distance is jointly convex, i.e., fulfills Eq. (31), as long as the corresponding norm k·k fulfills the triangle inequality and absolute homogeneity12 (Baumgratz, Cramer, and Plenio, 2014).
- Furthermore, the quantity on the right-hand side of Eq. (34) was independently proposed as a coherence quantifier by Herbut (2005) under the name coherence information.
- Since the trace norm is contractive under quantum operations, the corresponding coherence quantifier C1 satisfies the conditions C1, C2, and C4 for any set of operations discussed in Sec. II.
- Similar results were obtained for general X states: also in this case C1 is equivalent to Cl1 , and condition C3 is fulfilled for the set IO (Rana, Parashar, and Lewenstein, 2016).
3. Geometric coherence
- The geometric coherence fulfills conditions C1, C2, and C4 for any set of operations discussed in Sec. II.
- A.2. For pure states, the geometric coherence takes the form CgðjψiÞ ¼ 1 −maxijhijψij2, which means that Cg does not meet conditions C5 and C6.
- Another related quantity was introduced by Baumgratz, Cramer, and Plenio (2014) and studied by Shao et al. (2015) where it was called fidelity of coherence: CFðϱÞ ¼ 1 −maxσ∈I ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fðϱ; σÞp .
- While CF fulfills conditions C1, C2, and C4 for any set of operations discussed in Sec. II.
D. Convex roof quantifiers of coherence
- Provided that a quantifier of coherence has been defined for all pure states, it can be extended to mixed states via the standard convex roof construction (Yuan et al., 2015): CðϱÞ ¼ inf fpi;jψ iig X i piCðjψ iiÞ; ð41Þ where the infimum is taken over all pure state decompositions of ϱ ¼.
- Thus, 13Yuan et al. (2015) called this quantity intrinsic randomness.
E. Coherence monotones from entanglement
- The supremum is performed over all bipartite incoherent operations Λi ∈ IO, i.e., such that the corresponding Kraus operators map the product basis jkijli onto itself.
- In particular, CE fulfills conditions C1–C4 for the set IO whenever E fulfills the corresponding conditions in entanglement theory (Streltsov et al., 2015).
- For distillable entanglement, relative entropy of entanglement, and geometric entanglement, the supremum in Eq. (47) is achieved when Λi is the generalized CNOT operation, i.e., the optimal incoherent operation is the unitary UCNOTjiij0i ¼ jiijii: ð49Þ.
- In both cases, the bound is saturated by the generalized CNOT operation; see Eq. (49).
- These results also imply that a state ϱS can be used to create entanglement via incoherent operations if and only if ϱS has nonzero coherence (Streltsov et al., 2015).
F. Robustness of coherence
- Another coherence monotone was introduced by Napoli et al. (2016) and Piani et al. (2016) and termed robustness of coherence.
- A similar quantity was studied earlier in entanglement theory under the name robustness of entanglement (Vidal and Tarrach, 1999; Steiner, 2003).
- The latter result also implies that RC does not comply with C5 and C6.
- Interestingly, for any state ϱ there exists a witnessW saturating this inequality.
- The robustness of coherence is moreover a figure of merit in the task of quantum phase discrimination; see Sec. V.D for its definition and detailed discussion.
G. Coherence quantifiers from interferometric visibility
- Clearly, quantum coherence is required for the observation of interference patterns, e.g., in the double slit experiment.
- Recently, this idea was formalized by von Prillwitz, Rudnicki, and Mintert (2015), where they studied the problem to determine coherence properties from interference patterns.
- Bagan et al. (2016) derived two exact complementarity relations between quantifiers of coherence and path information in a multipath interferometer, using, respectively, the l1 norm and the relative entropy of coherence.
- Then by placing an output detector which implements a measurement described by a positiveoperator valued measure M ¼ ðMωÞ, one observes outcomes ω sampled from the Born probability pMjϱðωjφ⃗Þ ¼ Tr½ϱðφ⃗ÞMω , which constitute the interference pattern.
- Various examples of visibility functionals and the corresponding coherence monotones were further discussed by Biswas, García-Díaz, and Winter (2017), showing, in particular, that the robustness of coherence presented in Sec. III.
H. Coherence of assistance
- This quantity is dual to the coherence of formation defined in Eq. (42) as the minimum over all decompositions.
- For this reason the authors get Cað1=2Þ ¼ 1, i.e., the coherence of assistance is maximal for the maximally mixed state.
- On the one hand, this means that the coherence of assistance is not a coherence monotone, as it indeed violates condition C1.
- Moreover, for all single-qubit states ϱ the coherence of assistance is n-copy additive and thus can be written as CaðϱÞ ¼ SðΔ½ϱ Þ. Further, there is a close relation between the coherence of assistance and the entanglement of assistance Rev. Mod.
I. Coherence and quantum correlations beyond entanglement
- The states of Eq. (58) are also known as classical quantum (Piani, Horodecki, and Horodecki, 2008), and the corresponding set will be denoted by CQ.
- This notion can be straightforwardly extended to more than two parties (Piani et al., 2011).
- They showed that the creation of general quantum correlations from a state ϱ is bounded above by its coherence.
J. Coherence in continuous variable systems
- The resource theory framework for quantum coherence adopted in this Colloquium assumes a finite-dimensional Hilbert space.
- Some of the previously listed quantifiers of coherence have also been studied in continuous variable systems and specifically in bosonic modes of the radiation field.
- These systems are characterized by an infinitedimensional Hilbert space, spanned by the Fock basis fjnig∞n¼0 of eigenstates of the particle number operator a†a (Braunstein and van Loock, 2005).
- Similarly to what was done by Eisert, Simon, and Plenio (2002) in entanglement theory, Y.-R. Zhang et al. (2016) imposed a finite mean energy constraint ha†ai≡ n̄ < ∞ to address the quantification of coherence in such systems with respect to the Fock reference basis.
1. Asymmetry monotones
- This framework is based on the notion of TIO, which were already introduced in Sec. II.
- Thus, in this case the relative entropy of asymmetry.
2. Quantifying superpositions
- A very general approach to quantify coherence was presented by Åberg (2006) within the framework of quantum superposition.
- Pk all have rank 1, the total operation Π corresponds to the total dephasing Δ. However, in general the projectors Pk can have rank larger than 1.
- Åberg (2006) also proposed a set of conditions a faithful quantifier of superposition ought to satisfy and showed that the relative entropy of superposition fulfills these conditions.
- The relative entropy of superposition is a special case of the relative entropy of asymmetry presented in Sec. III.K.1 and admits the following expression (Åberg, 2006): SrðϱÞ ¼ min Π½σ ¼σ Sðϱ∥σÞ ¼ Sðϱ∥Π½ϱ Þ: ð73Þ.
3. Coherence rank and general quantifiers of nonclassicality
- An alternative approach was taken by Killoran, Steinhoff, and Plenio (2016), Mukhopadhyay et al. (2017), Regula et al. (2017), and Theurer et al. (2017), who investigated a very general form of nonclassicality, also going beyond the framework of coherence.
- Killoran, Steinhoff, and Plenio (2016) introduced the coherence rank of a general pure state.
- Note, however, that the incoherent channels of Levi and Mintert (2014) are in general different from the IO defined by Baumgratz, Cramer, and Plenio (2014) and can be rather identified with the SIO (Winter and Yang, 2016; Yadin et al., 2016).
- The authors also note that a related framework was presented recently by Yadin and Vedral (2016) to quantify macroscopic coherence.
- The possibility to establish superpositions of unknown quantum states via universal quantum protocols was investigated by Oszmaniec et al. (2016).
4. Optical coherence and nonclassicality
- The framework of Killoran, Steinhoff, and Plenio (2016) and Theurer et al. (2017) is partly motivated by the seminal theory of optical coherence in continuous variable systems 15The coherence rank can be generalized to mixed states via a procedure similar to the convex roof described in Sec. III.D.
- The resulting quantity rCðϱÞ ¼ inffpi;jψ iigmaxirCðjψ iiÞ is called the coherence number of ϱ (Regula et al., 2017).
- If the set of classical states is identified with the convex hull of Glauber-Sudarshan coherent states, as in the theory of optical coherence, then the corresponding CO includes so-called passive operations, i.e., operations preserving the mean energy ha†ai, which can be implemented by linear optical elements such as beam splitters and phase shifters.
- In particular, Asbóth, Calsamiglia, and Ritsch (2005) proposed to quantify optical nonclassicality for a single-mode state ϱ in terms of the maximum two-mode entanglement that can be generated from ϱ using linear optics, auxiliary classical states, and ideal photodetectors.
- Furthermore, Vogel and Sperling (2014) independently defined a notion analogous to the coherence rank rC of Eq. (74) for optical nonclassicality, i.e., with fjciig being a subset of (linearly independent) Glauber-Sudarshan coherent states.
1. General distance-based coherence quantifiers
- S, the aforementioned quantities are related via the following inequality (Yao et al., 2015): CDðϱÞ ≥ QDðϱÞ ≥ EDðϱÞ: ð75Þ.
- These results can be straightforwardly generalized to more than two parties (Yao et al., 2015).
2. Quantum-incoherent relative entropy
- Br ðϱABÞ ¼ min σAB∈QI SðϱAB∥σABÞ; ð76Þ where the minimum is taken over the set of quantumincoherent statesQI defined in Eq. (15).
- As further discussed by Chitambar et al. (2016), the quantum-incoherent relative entropy admits the following closed expression: CAjBr ðϱABÞ ¼ SðΔB½ϱAB Þ − SðϱABÞ; ð77Þ whereΔB denotes a dephasing operation on subsystem B only.
3. Distillable coherence of collaboration
- The distillable coherence of collaboration was introduced and studied by Chitambar et al. (2016) as the figure of merit for the task of assisted coherence distillation.
- The distillable coherence of collaboration is the highest achievable rate for this procedure (Chitambar et al., 2016): CAjBLQICCðϱÞ¼ sup n.
- Here the infimum is taken over all LQICC operations Λ, and τ ¼ jΨ2ihΨ2jB is the maximally coherent single-qubit state on Bob’s subsystem.
- Streltsov, Rana, Bera, and Lewenstein (2017) extended this framework to other sets of operations, such as LICC, SI, and SQI; see Sec. II.C for their definitions.
- The experiment used polarization-entangled photon pairs for creating pure entangled states and also mixed Werner states.
4. Recoverable coherence
- It is defined in the same way as the distillable coherence of collaboration in Eq. (78), but with the set of LQICC operations replaced by GOIB; see Sec. II.C for their definition.
- Following the analogy to distillable coherence of collaboration, the authors denote the recoverable coherence by CAjBGOIB.
- As shown by Matera et al. (2016), the recoverable coherence is additive, convex, monotonic on average under GOIB operations, and upper bounded by the quantum-incoherent relative entropy.
- Notably the recoverable coherence has an operational interpretation, as it is directly related to the precision of estimating the trace of a unitary via the deterministic quantum computation with one qubit (DQC1) quantum algorithm (Matera et al., 2016); see also Sec. V.B for a more general discussion on the role of coherence in quantum algorithms.
- Additionally, minimizing the recoverable coherence over all local bases leads to an alternative quantifier of discord (Matera et al., 2016).
5. Uncertainty relations and monogamy of coherence
- Uncertainty relations for quantum coherence, both for a single party and for multipartite settings, have been studied by Peng et al. (2016) and Singh, Pati, and Bera (2016).
- If coherence is defined with respect to two different bases fjiig and fjaig, the corresponding relative entropies of coherence Cir and Car fulfill the following uncertainty relation (Singh, Pati, and Bera, 2016): CirðϱÞ þ Car ðϱÞ ≥ −2 log2 max i;a jhijaij −.
- The discussion on monogamy of quantum coherence is also inspired by results from entanglement theory (Coffman, Kundu, and Wootters, 2000; Horodecki et al., 2009).
- Further results on monogamy of coherence were also presented by Radhakrishnan et al. (2016).
IV. DYNAMICS OF QUANTUM COHERENCE
- Quantum coherence is typically recognized as a fragile feature: the vanishing of coherence in open quantum systems exposed to environmental noise, commonly referred to as decoherence (Breuer and Petruccione, 2002; Zurek, 2003; Schlosshauer, 2005), is perhaps the most distinctive manifestation of the quantum-to-classical transition observed at their macroscopic scales.
- Numerous efforts have been invested into devising feasible control schemes to preserve coherence in open quantum systems, with notable examples including dynamical decoupling (Viola, Knill, and Lloyd, 1999), quantum feedback control (Rabitz et al., 2000), and error correcting codes (Shor, 1995).
- In this section the authors review more recent work concerning the dynamical evolution of coherence quantifiers (defined in Sec. III) subject to relevant Markovian or non-Markovian evolutions.
- Coherence effects in biological systems and their potential functional role are discussed in Sec. V.
- Here the authors also discuss generic properties of coherence in mixed quantum states, the cohering (and decohering) power of quantum channels, and the role played by coherence quantifiers in defining speed limits for closed and open quantum evolutions.
A. Freezing of coherence
- One of the most interesting phenomena observed in the dynamics of coherence is the possibility for its freezing, that is, complete time invariance without any external control, in the presence of particular initial states and noisy evolutions.
- Cr is strictly decreasing (Bromley, Cianciaruso, and Adesso, 2015).
- Specifically, all coherence monotones (respecting, in particular, property C2 for the set SIO) are frozen for an initial state subject to a strictly incoherent channel if and only if the relative entropy of coherence is frozen for such initial state (Yu, Zhang, Liu, and Tong, 2016).
- Liu et al. (2016) theoretically explored freezing of coherence for a system of two-level atoms interacting with the vacuum fluctuations of an electromagnetic field bath.
B. Coherence in non-Markovian evolutions
- Some attention has been devoted to the study of coherence in non-Markovian dynamics.
- Addis et al. (2014) studied the phenomenon of coherence trapping in the presence of nonMarkovian dephasing.
- Addis et al. (2014) showed that the specifics of coherence trapping depend on the environmental spectrum: its low-frequency band determines the presence or absence of information backflow, while its high-frequency band determines the maximum coherence trapped in the stationary state.
- The dynamics of the l1 norm of coherence for two qubits globally interacting with a harmonic oscillator bath was investigated by Bhattacharya, Banerjee, and Pati (2016), finding that non-Markovianity slows down the coherence decay.
- A proposal to witness non-Markovianity of incoherent evolutions via a temporary increase of coherence quantifiers was discussed by Chanda and Bhattacharya (2016), inspired by more general approaches to witness and measure nonMarkovianity based on revivals of distinguishability, entanglement, or other informational quantifiers (Rivas, Huelga, and Plenio, 2014; Breuer et al., 2016).
C. Cohering power of quantum channels and evolutions
- When considering unitary channels U and adopting the l1 norm, García-Díaz, Egloff, and Plenio (2016) and Bu et al. (2017) proved that PCðUÞ ¼ ~PCðUÞ in the case of a single qubit, but PCðUÞ < ~PCðUÞ strictly in any dimension larger than 2.
- Finally, the authors mention that similar studies have been done in entanglement theory (Zanardi, Zalka, and Faoro, 2000; Linden, Smolin, and Winter, 2009).
D. Average coherence of random states and typicality
- While some dynamical properties of coherence may be dependent on specific channels and initial states, it is also interesting to study typical traits of coherence quantifiers on randomly sampled pure or mixed states.
- Note that generic random states, exhibiting the typical features of coherence summarized in the following, can be in fact generated by a dynamical model of a quantized deterministic chaotic system, such as a quantum kicked top (Puchała, Pawela, and Życzkowski, 2016).
- This was proven rigorously resorting to Lévy’s lemma, hence showing that states with coherence bounded away from its average value occur with exponentially small probability.
1. Average relative entropy of coherence
- The exact average of the relative entropy of coherence for pure d-dimensional states jψi ∈.
- This shows that random pure states have 17More precisely, this would quantify the asymmetry or symmetry power rather than the cohering or decohering power.
- Typicality of the relative entropy of coherence for random mixed states was investigated by Puchała, Pawela, and Życzkowski (2016), Zhang (2017), and Zhang, Singh, and Pati (2017).
- Considering the probability measure induced by partial tracing, that is, corresponding to random mixed states ϱ ¼ TrCd0 jψihψ j, with jψi ∈.
- The concentration of measure phenomenon for Cr was then proven by Puchała, Pawela, and Życzkowski (2016) and Zhang, Singh, and Pati (2017).
2. Average l1 norm of coherence
- Concerning the l1 norm of coherence, Singh, Zhang, and Pati (2016) derived a bound to the average ECl1ðjψiÞ for pure Haar-distributed d-dimensional states jψi, exploiting a relation between the l1 norm of coherence and the so-called classical purity (Cheng and Hall, 2015).
- This shows that, for asymptotically large d, the l1 norm of coherence of random pure states scales linearly with d and stays smaller than the maximal value (d − 1) by a factor π=4, while the l1 norm of coherence of random mixed states scales only with the square root of d.
3. Average recoverable coherence
- Miatto et al. (2015) considered a qubit interacting with a ddimensional environment, of which an a-dimensional subset is considered accessible, while the remaining k-dimensional subset (with d ¼ ak) is unaccessible.
- While for d ≫ 1 such an interaction leads to decoherence of the principal qubit, its coherence can be partially recovered by quantum erasure, which entails measuring (part of) the environment in an appropriate basis to erase the information stored in it about the system, hence restoring coherence of the latter.
- Cd of the system plus environment composite and studied the average recoverable l1 norm of coherence ECl1ðϱÞ of the marginal state ϱ of the principal qubit, following an optimal measurement on the accessible a-dimensional subset of the environment.
- By virtue of typicality, this means that, regardless of how a high-dimensional environment is partitioned, suitably measuring half of it generically suffices to project a qubit immersed in such environment onto a nearmaximally coherent state, a fact reminiscent of the quantum Darwinism approach to decoherence (Zurek, 2009; Brandão, Piani, and Horodecki, 2015).
E. Quantum speed limits
- In the dynamics of a closed or open quantum system, quantum speed limits dictate the ultimate bounds imposed by quantum mechanics on the minimal evolution time between two distinguishable states of the system.
- Interestingly, for pure states, even the inverses of the Mandelstam-Tamm and Margolus-Levitin quantities appearing on the right-hand side of Eq. (99) are themselves asymmetry monotones, bounding the asymmetry monotone given by 1=τ⊥ðjψiÞ.
- Marvian, Spekkens, and Zanardi (2016) then derived new Mandelstam-Tamm–type quantum speed limits for unitary dynamics based on various measures of distinguishability, including a bound featuring the Wigner-Yanase skew information with respect to H (obtained when D is set to the relative Rényi entropy of order 1=2), which was also independently obtained by Mondal, Datta, and Sazim (2016).
- Λ½ϱ under a (generally open) dynamics Λ. Quantum speed limits then ensue from a simple geometric observation, namely, that the geodesic connecting ϱ to ϱτ, whose length can be indicated byLgðϱ; ϱτÞ, is the path of shortest length among all physical evolutions between the given initial and final states Lgðϱ; ϱτÞ ≤ lgΛðϱ; ϱτÞ∀Λ.
- The authors finally mention that looser speed limits involving the skew information were also recently presented by Pires, Céleri, and Soares-Pinto (2015) and Mondal, Datta, and Sazim (2016).
V. APPLICATIONS OF QUANTUM COHERENCE
- Ranging from quantum information processing to quantum sensing and metrology, thermodynamics, and biology.the authors.
- Particular emphasis is given to those settings in which a specific coherence monotone introduced in Sec. III acquires an operational interpretation, hence resulting in novel insights stemming from the characterization of quantum coherence as a resource.
A. Quantum thermodynamics
- Recently, the role of coherence in quantum thermodynamics was discussed by several authors.
- The authors review the main concepts in the following, being in large part based on the resource theory of quantum thermodynamics (Brandão et al., 2013; Gour et al., 2015; Goold et al., 2016) defined by the framework of thermal operations (Janzing et al., 2000).
1. Thermal operations
- The importance of thermal operations arises from the fact that they are consistent with the first and second laws of thermodynamics (Lostaglio et al., 2015).
- First, they are TIO with respect to the Hamiltonian HS, i.e., Λth½e−iHStϱSeiHSt ¼ e−iHStΛth½ϱS eiHSt: ð104Þ.
- Here preservation of the thermal state is the only requirement on the quantum operation.
- Interestingly, Faist, Oppenheim, and Renner (2015) showed that these maps are strictly more powerful than thermal operations.
2. State transformations via thermal operations
- Several recent works studied necessary and sufficient conditions for two states ϱ and σ to be interconvertible via thermal operations.
- More general conditions which allow for the addition of ancillas and catalytic conversions (Jonathan and Plenio, 1999), known as the second laws of quantum thermodynamics, were presented by Brandão et al. (2015).
- As discussed by Lostaglio, Jennings, and Rudolph (2017), it is important to distinguish between two cases: namely, whether the state γE is incoherent, or has nonzero coherence, with respect to the eigenbasis of HE.
- And apart from that the process can also create coherence in the system.
- In particular, they investigated the creation of coherence for a quantum system (with respect to the eigenbasis of its Hamiltonian H) via unitary operations from a thermal state and also explored the energy cost for such coherence creation.
3. Work extraction and quantum thermal machines
- Here hWiðϱSÞ denotes the amount of work that can be extracted from the state ϱS, and Π½ϱS ¼ Pi Tr½ΠiϱS Πi, where Πi are projectors onto the eigenspaces of HS.
- Further results on the role of coherence for work extraction have also been presented by 20Note that thermomajorization is also related to the mixing distance studied by Ruch, Schranner, and Seligman (1978); see Egloff et al. (2015), where a relation between majorization and optimal guaranteed work extraction up to a risk of failure was investigated.
- Moreover, it was shown by Vacanti, Elouard, and Auffeves (2015) that work is typically required for keeping coherent states out of thermal equilibrium.
- Quantum coherence was also shown to be useful for transient cooling in absorption refrigerators (Mitchison et al., 2015).
B. Quantum algorithms
- The role of coherence in quantum algorithms was discussed by Hillery (2016), with particular focus on the Deutsch-Jozsa algorithm (Deutsch and Jozsa, 1992).
- Another important quantum algorithm is known as the DQC1 (Knill and Laflamme, 1998).
- The role of quantum discord for DQC1 was later questioned by Dakić, Vedral, and Brukner (2010), who showed that certain nontrivial instances do not involve any quantum correlations.
- This issue was further discussed by Datta and Shaji (2011).
- The latter work indicates that coherence is indeed a suitable figure of merit for this protocol.
C. Quantum metrology
- As the bound in Eq. (108) is asymptotically achievable for n ≫ 1 by means of a suitable optimal measurement, the quantum Fisher information directly quantifies the optimal precision of the estimation procedure and is thus regarded as the main figure of merit in quantum metrology (Paris, 2009; Giovannetti, Lloyd, and Maccone, 2011).
- Quantum coherence in the form of asymmetry is the primary resource behind the power of quantum metrology.
- In the absence of noise, the Heisenberg scaling can be equivalently achieved using n entangled probes in parallel, each subject to one instance of Uφ (Huelga et al., 1997; Giovannetti, Lloyd, and Maccone, 2006).
D. Quantum channel discrimination
- Consider a d-dimensional probe and a set of unitary channels fUφg generated by H ¼ P d−1 i¼0 ijiihij, where fjiig sets the reference incoherent basis for the probe system and φ can take any of the d values f2πk=dgd−1k¼0 with uniform probability 1=d.
- This provides a direct operational interpretation for the robustness of coherence RC in quantum discrimination tasks.
- Such an interpretation can be extended to more general channel discrimination scenarios (i.e., with nonuniform prior probabilities and including nonunitary incoherent channels) and carries over to the robustness of asymmetry with respect to arbitrary groups (Napoli et al., 2016; Piani et al., 2016).
E. Witnessing quantum correlations
- Recently, several authors tried to find Bell-type inequalities for various coherence quantifiers.
- They found a Bell-type bound for this quantity for all product states ϱA ⊗ ϱB and showed that the bound is violated for maximally entangled states and a certain choice of observables X and Y.
- In a similar spirit, the interplay between coherence and quantum steering was investigated by Mondal and Mukhopadhyay (2015) and Mondal, Pramanik, and Pati (2017), where steering inequalities for various coherence quantifiers were found, and by X. Hu and Fan (2016) and Hu et al. (2016), where the maximal coherence of steered states was investigated.
- As further shown by Girolami and Yadin (2017), detection of coherence can also be used to witness multipartite entanglement.
- In particular, an experimentally accessible lower bound on the quantum Fisher information (which does not require full state tomography) can serve as a witness for multipartite entanglement, as explicitly demonstrated for mixtures of Greenberger-Horne-Zeilinger states (Girolami and Yadin, 2017).
F. Quantum biology and transport phenomena
- Transport is fundamental to a wide range of phenomena in the natural sciences and it has long been appreciated that coherence can play an important role for transport, e.g., in the solid state (Deveaud-Plédran, Quattropani, and Schwendimann, 2009; Li et al., 2012).
- It is now recognized that typically both coherent and noise dynamics are required to achieve optimal performance.
- A range of mechanisms to support this claim and understand its origin qualitatively [see Huelga and Plenio (2013) for an overview] has been identified.
- In the studies of the impact of coherence on transport dynamics, formal approaches using coherence and asymmetry quantifiers based on the Wigner-Yanase skew information were used (Vatasescu, 2015, 2016), but the connection to function has remained tenuous so far.
- Another question of interest in this context concerns that of the distinction between classical and quantum coherence (O’Reilly and Olaya-Castro, 2014) and dynamics (Wilde, McCracken, and Mizel, 2009; Li et al., 2012) in biological systems, most notably photosynthetic units.
G. Quantum phase transitions
- Coherence and asymmetry quantifiers have been employed to detect and characterize quantum phase transitions, i.e., changes in the ground state of many-body systems occurring at or near zero temperature and driven purely by quantum fluctuations.
- Karpat, Çakmak, and Fanchini (2014) and Çakmak, Karpat, and Fanchini (2015) showed that single-spin coherence reliably identifies the second-order quantum phase transition in the thermal ground state of the anisotropic spin-1=2 XY chain in a transverse magnetic field.
- In particular, the singlespin skew information with respect to the Pauli spin-x operator σx, as well as its experimentally friendly lower bound which can be measured without state tomography (Girolami, 2014), exhibits a divergence in its derivative at the critical point, even at relatively high temperatures.
- Malvezzi et al. (2016) extended the previous analysis to ground states of spin-1 Heisenberg chains.
- Focusing on the one-dimensional XXZ model, they found that no coherence and asymmetry quantifier (encompassing skew information, relative entropy, and l1 norm) is able to detect the triple point Rev. Mod.
VI. CONCLUSIONS
- Quantum thermodynamics, and quantum biology, as well as physics more widely.the authors.
- The set F allows one to create any incoherent state from any other state.
- A.2. In place of the first condition, the authors give the corresponding literature reference, where suitable motivations can be found for each set.
- The authors further reviewed in Sec. III the current progress on quantifying coherence and related manifestations of nonclassicality in compliance with the underlying framework of resource theories, in particular, highlighting interconnections between different measures and, where possible, their relations to entanglement measures.
- Most of these advances are still at a very early stage, and the operational value of coherence still needs to be pinpointed clearly in many contexts.
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Cites background from "Colloquium: quantum coherence as a ..."
...…in parallel schemes and coherence (namely, superposition in the eigenbasis of the generator) (Baum- gratz et al., 2014; Marvian and Spekkens, 2016; Streltsov et al., 2017) in sequential schemes further extends to certain schemes of quantum metrology in the presence of noise, namely when the…...
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Frequently Asked Questions (8)
Q2. What is the preferred basis for a multipartite incoherence?
In the case of more than one party, the preferred basis with respect to which coherence is studied will be constructed as the tensor product of the corresponding local reference basis states for each subsystem.
Q3. What is the role of coherence in biological transport?
Transport is fundamental to a wide range of phenomena in the natural sciences and it has long been appreciated that coherence can play an important role for transport, e.g., in the solid state (Deveaud-Plédran, Quattropani, and Schwendimann, 2009; Li et al., 2012).
Q4. Why do thermal operations have such a great importance?
The importance of thermal operations arises from the fact that they are consistent with the first and second laws of thermodynamics (Lostaglio et al., 2015).
Q5. What is the construction for a arbitrary single-qubit unitary?
The explicit construction for an arbitrary single-qubit unitary can be found in Baumgratz, Cramer, and Plenio (2014), and the extension to general quantum operations of arbitrary dimension was studied by Chitambar and Hsieh (2016) and Ben Dana et al. (2017).
Q6. What is the known classical procedure for estimating the trace of a unitary matrix?
This quantum algorithm provides an exponential speedup over the best known classical procedure for estimating the trace of a unitary matrix (given as a sequence of two-qubit gates).
Q7. What is the limiting case of the same resource states identified by Baumgratz, Cram?
This demonstrates that continuous variable states exhibiting optical nonclassicality can be seen essentially as the limiting case of the same resource states identified by Baumgratz, Cramer, and Plenio (2014), when the incoherent basis is chosen as the set of Glauber-Sudarshan coherent states.
Q8. What is the meaning of the term separable operations?
General quantum operations of the form in Eq. (14), but without the incoherence restriction, have been studied extensively in entanglement theory, where they are called separable operations (Vedral and Plenio, 1998).