Quantifying quantumness of channels without entanglement

A bipartite quantum state $\tilde{\rho}_{A_{0},B_{0}}$ shared between Alice and Bob is entangled if the corresponding density operator is not separable, namely

where $p(j)\in\mathcal{P}(\mathcal{J})$ is a probability distribution, and $\tilde{\sigma}_{{A_{0}}}(j)\in\mathcal{D}(H_{A_{0}})$ $(\tilde{\eta}_{{B_{0}}}(j)\in\mathcal{D}(H_{B_{0}}))$ is a local density operator. In general, the EB channel is defined by sending Alice’s subsystem into a quantum channel $\mathcal{E}\in O(H_{A_{0}},H_{A_{1}})$, such that the entanglement is broken for arbitrary entangled states. We can explicitly formulate the EB channel as

Here, the superscript EB is used to denote the channel $\mathcal{E}$ to be EB, and the set of separable states is denoted by $\mathcal{F}_{\text{SEP}}$.

Entanglement-breaking channels are equivalent to measure-and-prepare channels, namely

where $M_{j}$ is a positive-operator valued measurement (POVM) element satisfying $M_{j}>0\leavevmode\nobreak\ \forall\leavevmode\nobreak\ j$, and $\sum_{j}M_{j}=\openone$ with classical outcomes $j$. Here, with some slight abuse of notation, by $t_{0}$ ($t_{1}$) we denote a time indicating that the system is in the Hilbert space $\mathcal{H}_{t_{0}}$ ($\mathcal{H}_{t_{1}}$) before (after) the operation of the quantum channel. The physical interpretation of the EB channel can be explained as follows: one measures the original system $\tilde{\rho}_{{t_{0}}}\in\mathcal{D}(H_{t_{0}})$ at $t_{0}$, after that, based on the outcome $j$, the corresponding state $\tilde{\sigma}_{{t_{1}}}(j)\in\mathcal{D}(H_{t_{1}})$ is prepared at $t_{1}$. Obviously, when we send one of the entangled pairs into a measure-and-prepare channel, the system becomes separable since one has locally prepared another quantum state without any direct interaction with the other party.

It has been shown that a non-EB channel is a criterion for a functional quantum memory because one would like a quantum memory to, at the very least, preserve the entanglement of a state. In the framework of the resource theory of quantum memory [34, 31], one can introduce a set of quantum-memory free operations transforming any EB channel to another EB channel (see Appendix. A for more details). With these quantum-memory free operations, we can recall that the robustness of a quantum memory is the minimal noise $\alpha$ mixing with the input quantum memory, such that the whole memory is lost, namely

Here, $\mathcal{E}^{\prime}$ is an arbitrary quantum channel and $\mathcal{F}_{\text{EB}}$ is the set of EB channels. It has been shown that the robustness of a quantum memory is a monotonic function after applying a quantum-memory free operation [31].

Before introducing the notion of the NLB channel, let us briefly recall the definition of Bell nonlocality. A spatially separated state is Bell-local when local measurements with finite inputs $x\in\mathcal{X}$ ($y\in\mathcal{Y}$) and outcomes $a\in\mathcal{A}$ ($b\in\mathcal{B}$) generate a correlation $p(a,b|x,y)=\text{Tr}\left[(M_{a|x}\otimes M_{b|y})\tilde{\rho}_{A,B}\right]$ which admits a local-hidden variable (LHV) model [12, 13], namely,

which the correlations are pre-determined by a hidden variable $j$. Here, we denote a set of correlations admitting a LHV model as $\mathcal{F}_{\text{LHV}}$. Since quantum correlations are a strict superset of $\mathcal{F}_{\text{LHV}}$ forming a convex set, one can distinguish the local correlation from the quantum ones by testing the famous Bell inequalities given by the parameters $\beta^{x,y}_{a,b}$ [58, 12, 13], namely

where $\delta^{\beta}$ is the local bound for a given Bell inequality.

Analogous to the EB channels, a NLB channel is the channel under which a correlation always satisfies a LHV model for arbitrary measurements and states, namely [43]

Similar to the definition of the robustness of a quantum memory, we now consider a noise $\alpha$ mixing with a given channel $\mathcal{E}$, namely

This noisy channel always generates a correlation satisfying a LHV model for arbitrary measurements and states. We denote the minimal value $\alpha$ as the robustness of the non-NLB channel $\text{R}_{\text{n-NLB}}(\mathcal{E})$. In Appendix A, we show that the robustness of a non-NLB channel is a monotonic function under a non-NLB free operation.

NLB channels have some other important known properties, which we summarize here. Unlike the situation with EB channels, the input of a maximally entangled state is not sufficient for verifying if the channel is NLB [43, 59]. In particular, because of this, we denote the case with the input being the maximally entangled state $|\Phi\rangle=\sum_{i}(1/\sqrt{d})|i\rangle\otimes|i\rangle$ as NLB channels for the maximally entangled state. In Ref. [43], the authors considered a particular NLB channel [the Clauser-Horne-Shimony-Holt (CHSH) NLB channel], i.e., channels that break CHSH-nonlocality for any state and measurements. Here, we say that a system satisfies CHSH nonlocality if it violates the CHSH inequality, namely:

where $2$ is the local bound for the CHSH inequality [58], $E(x_{i},y_{j})\equiv p(a=b|x_{i},y_{j})-p(a\neq b|x_{i},y_{j})$ is the expectation value of $a\cdot b$, for $a\in\mathcal{A}$, $b\in\mathcal{B}$, $x_{i}\in\mathcal{X}$, and $y_{j}\in\mathcal{Y}$, with $\mathcal{X}=\mathcal{Y}=\{1,2\}$, and $\mathcal{A}=\mathcal{B}=\{\pm 1\}$. The quantum bound of the CHSH inequality is given by $2\sqrt{2}$. We note that the CHSH-NLB channels for the maximally entangled state have the property [43]: if the channel is unital, then it is also CHSH-NLB.

Now we introduce our first main result: the concept of “steerability-breaking channel” which breaks any quantum information tasks using quantum steerability as a resource. We then investigate the properties of the SB channel by showing that the channel is SB if and only if it breaks the quantum steerability of the maximally entangled state. The above property is useful not only for experimental-friendly certifications of SB, but also for the theoretical analysis of SB channels. For instance, (1) we derive that a SB channel is equivalent to the incompatible-breaking channel [45], (2) we propose the robustness as a quantification of a non-SB channel, and (3) we discuss the hierarchy relationship between breaking channels.

Quantum steering refers to the ability of remotely projecting Bob’s quantum states by Alice’s collection of measurements $\{M_{a|x}\}$ with finite inputs $x$ and outcomes $a$ [60, 18]. A set of measurements gives rise to a collection of quantum states, termed as an assemblage,

where $\tilde{\rho}_{A,B}$ is a bipartite state shared by Alice and Bob. We say that an assemblage is unsteerable when it admits a local-hidden-state (LHS) model, namely

Otherwise, the assemblage is steerable. The physical interpretation of a LHS model is that an assemblage can be pre-determined by a hidden variable $j$, which simultaneously distributes over the statistics $p(a|x,j)$ and the states $\tilde{\rho}_{B}(j)\in D(H_{B})$. The set of all assemblages admitting a LHS model is denoted as $\mathcal{F}_{\text{LHS}}$. Violation of a LHS model simultaneously implies that (i) the shared state is entangled and (ii) Alice’s measurement violates incompatibility [61, 62, 63, 64]. It has been shown that quantum steering is a central resource for one-sided device-independent quantum information tasks including: metrology [65], quantum advantages on the subchannel-discrimination problems [66, 67], key distribution [7], and random number generation [68, 69, 70].

In analogy to the EB and NLB channels, we propose a SB channel as a channel which breaks the steerability for any collection of measurements $\{M_{a|x}\}$ acting on the state sent through the channel $\tilde{\rho}^{\prime}_{{A_{1}},{B_{0}}}$. More specifically, the assemblage after a SB channel can always be expressed by a LHS model, namely

We denote the set of all SB channels as $\mathcal{F}_{\text{SB}}$. Moreover, we define $|\mathcal{X}|$-SB channels which break the steerability with a finite input $|\mathcal{X}|$. For instance, if the finite index set is $\mathcal{X}=\{1,2,3\}$, we can define the $3$-SB channel. Here, we show that SB channels have the following properties:

Proof.—We present a proof of this theorem in Appendix B. We now use the above result to simplify the definition of the SB channel.

The definition of the SB channel is similar to that of the incompatibility-breaking channel, which maps an incompatible measurement $\{M_{a|x}\}$ to a jointly measurable one in the Heisenberg picture [45, 71]. More specifically, a set of measurements after incompatibility-breaking channel can always expressed as

where $\sum_{j}p(a|x,j)M_{j}$ is a joint measurable model with an intrinsic POVM $\{M_{j}\}$ and postprocessing $p(a|x,j)$. Here, $\mathcal{E}^{\dagger}$ is the dual map of the quantum channel which is CP and unital. The set of the joint measurements is denoted by $\mathcal{F}_{\text{JM}}$. Intuitively, the incompatibility-breaking channels form a proper subset of SB channels because a joint measurement cannot generate a steerable assemblage [62, 63, 64]. With Theorem 1, we provide a stronger connection between two breaking channels by the following theorem:

Proof.—We present the proof in Appendix B. We recall that there is a one-to-one relation between an unsteerable assemblage and a joint measurement [64, 62, 63]. Our result provides a similar but not identical analog in terms of breaking channels. This result can be naturally extended in a quantitative manner (as shown below).

To quantify the degree of a non-SB channel, we consider, again, a noisy channel consisting of a noise $\alpha$ and the input non-SB channel $\mathcal{E}$ [cf. Eq. (8)]. The standard robustness of the non-SB channel is defined as the minimal value of the noise $\alpha$, such that the whole channel is SB for any measurement set $\{M_{a|x}^{\text{T}}\}$ and any entangled state, namely

where T denotes the transposition. Inserting the maximally entangled state $|\Phi\rangle=\sum_{i}(1/\sqrt{d})|i\rangle\otimes|i\rangle$ (Theorem 1) into Eq. (14) , we can simplify the standard robustness of the non-SB channel in the Heisenberg picture to arrive at

Here, we use the fact that $(\mathbf{X}\otimes\mathbf{Y})|\Phi\rangle\langle\Phi|=\mathbf{X}\mathbf{Y}^{\text{T}}\otimes\openone|\Phi\rangle\langle\Phi|$, where $\mathbf{X}(\mathbf{Y})$ is any operator, and the subscript in $\text{R}_{\text{n-IB}}$ is used to denote the non-incompatibility-breaking channels. One can see that Eq. (15) is the same as the robustness of the non-incompatiability-breaking channel proposed in Ref. [38]. In Appendix A, we further show that the robustness of a non-SB channel is a monotonic function under the most general non-SB free operation.

It has been shown that the sets of all incompatibility-breaking channels (also SB channels) and NLB channels are both supersets of the set of all EB channels [43, 45]. To complete this hierarchy relationship between breaking channels, we show the following:

Proof.—We present the proof of Theorem 3 in Appendix D, and we illustrate this theorem for the qubit depolarizing channel

in Fig. 2. The experimental demonstration of this theorem is also presented in Sec. IV. We note that $\mathcal{F}_{\text{NLB}_{\text{CHSH}}}$ denotes the set of CHSH-NLB channels.

We first briefly recall the definition of a pseudo-density operator (PDO), which has primarily been used in the study of the causality of quantum theory [49]. We then show that the temporal nonseparability extracted from a PDO can be used to measure the quality of a quantum memory. Afterwards, we show that the PDO formalism is operationally equivalent to the formalism of temporal semi-quantum games in the task of verifying non-EB channels [34].

To reconstruct the state of a quantum system from trusted measurements performed at different times, one can generalize the concept of a standard density operator in the time domain [49] (see also Fig. 3). Without loss of generality, the events, or observations, collated at different times can be connected by quantum channels to an input state. In what follows, we only consider a two-event PDO with a maximally mixed input. The information of a channel is contained in the PDO, namely [75]

where $P_{\mathcal{\openone}}=\text{SWAP}/d$ is the PDO of the identity channel with the swap operator defined by $\text{SWAP}|ij\rangle=|ji\rangle$. For the qubit case, we have $\text{SWAP}=1/2\sum_{i=0}^{3}\sigma_{i}\otimes\sigma_{i}$. Here, $\{\sigma_{i}\}_{i=0,1,2,3}=\{\openone,\hat{X},\hat{Y},\hat{Z}\}$ is a set containing the identity and Pauli operators. A PDO is: (1) Hermitian and (2) unit trace but not necessarily positive semidefinite (the latter is a necessary condition for a standard density operator). Similar to the standard density operator, a PDO is called temporally separable when it admits

where $p(j)\in\mathcal{P}(\mathcal{J})$, $\tilde{\omega}_{{t_{0}}}(j)\in D(H_{t_{0}})$, and $\tilde{\theta}_{{t_{1}}}(j)\in D(H_{t_{1}})$. In Eq. (20), there exist definite states described by $\tilde{\omega}_{{t_{0}}}(j)$ and $\tilde{\theta}_{{t_{1}}}(j)$ at each moment of time, $t_{0}$ and $t_{1}$.

It has been shown that the PDO is identical to the partial transpose of the CJ state of the channel $\mathcal{E}$ [76, 77],

where PT is the partial transposition with respect to the first subsystem. Therefore, a separable PDO implies that the CJ state is also separable and, thus, the corresponding channel must be EB [32, 74]. With the above properties, we have the following:

Recall that one can still use the PDO to distinguish the temporal and spatial correlations, while the partial transposition of the PDO cannot [49]. To quantify the degree of a quantum memory with a PDO, we can use again the idea of robustness-based measure. Namely, the robustness of a quantum memory is equal to the minimum ratio of a noisy operator one has to mix with the partial transpose of the PDO before the mixture becomes separable. In fact, it can be easily shown that such a measure is the same as the measure derived from the channel formalism in Eq. (4) [31].

Finally, it is useful to compare the PDO and the temporal semi-quantum scenario, which certifies all non-EB channels with minimal assumptions in the sense that only state preparation devices are trusted [34]. In detail, the temporal semi-quantum scenario is constructed by an unknown joint measurement, a given quantum channel $\mathcal{E}$, and a set of trusted states. In the beginning, one chooses a quantum state from the set as an input to the quantum channel. A joint measurement is performed on the output state and a state chosen from the same set. One can certify an arbitrary non-EB channel if the set is tomographically complete. From the PDO perspective, we now consider a set of trusted measurements $\{M_{a|x}\}$ at $t_{0}$ acting on the PDO, which is used for generating a set of states $\tilde{\rho}_{\mathcal{E}}(a|x)\in\mathcal{D}(H_{t_{1}})=\text{Tr}_{t_{0}}\left[M_{a|x}\otimes\openone P_{\mathcal{E}}\right]$, up to renormalization, at $t_{1}$ [78]. This is the so-called “normalized” temporal assemblage, and we will formally introduce it later. The joint measurement is then performed on a characterized quantum input $\{\tilde{\tau}_{y}\}\in\mathcal{D}(H_{t^{\prime}_{1}})$ and the normalized temporal assemblage. The above steps are exactly the same as the procedure used in temporal semi-quantum games, and we have

Proof.—We present a detailed comparison in Appendix C. We note that, although in Ref. [50] the authors have already shown the relationship between the PDO formalism and the temporal semi-quantum scenario, we provide a clearer physical interpretation in the proof by using the property found in Ref. [78].

Here, we first recall the concept of channel steering from a PDO perspective. In this way, we can show the relationship between channel steering and non-SB channels in a quantitative manner. This will allow us to measure all non-SB channels in the temporal domain. We do not consider the more general case of channel steering used in Ref. [37] because there it is employed to discuss the coherent properties of an extended channel, which is beyond the scope of this work.

In the PDO formalism, the measurements performed at two moments of time are assumed to be characterized. Now, we replace the characterized measurement at time $t_{1}$ with an uncharacterized one with finite inputs $y\in\mathcal{Y}$ and outcomes $b\in\mathcal{B}$ (see also Fig. 3). In short, the measurements at times $t_{0}$ and $t_{1}$ are trusted and untrusted, respectively. We note that the above scenario is channel steering [37] with only classical outputs (measurement results) at time $t_{1}$. To put it another way, the measurements at $t_{1}$ on the PDO generate a set of evolved states by

where $d$ is the dimension of the PDO. Note that the resulting state can be seen as the evolution of the measurement in the Heisenbeg picture. Since Eq. (22) is a valid assemblage, one can test whether Eq. (22) admits a hidden-state (HS) model, i.e.,

The above formula suggests that once the HS model is satisfied, the corresponding measurement set $\{\mathcal{E}^{\dagger}(M_{b|y})\}$ is jointly measurable (see also Appendix B). By Theorem 2, if the dual channel breaks the incompatibility of an arbitrary measurement set $\{M_{b|y}\}$, the channel is SB. Therefore, one can use channel steering to certify all SB channels. We also note that if one inserts an EB channel into Eq. (22), the assemblage also satisfies the HS model. However, due to Theorem 3, not all EB channels can be witnessed with channel steering [36].

We now define the robustness of channel steering, $\text{R}_{\text{CS}}$, as the minimal ratio $\alpha$ of noise mixed with the dual map of the underlying channel such that the evolved assemblage admits a HS model, or, equivalently, the evolved measurement assemblage admits a jointly measurable model, i.e.,

If we now test all incompatible measurements $\{M_{b|y}\}$ in the robustness of channel steering, the minimal one corresponds to the robustness of the non-SB channel in Eq. (15) and we arrive at:

Here, we introduce the last two temporal scenarios: (1) the temporal analogy of quantum steering (temporal steering) and (2) non-macrorealism in the form of the temporal Bell inequality. Both scenarios can be derived from the assumptions of macrorealism (in the sense that the system is assumed to have well defined pre-existing properties (realism) and can be measured without disturbance (non-invasiveness) [53]). Beyond quantifying non-Markovianity [79, 80, 81] and connections to the security of quantum key distribution [82] , we show that temporal steerability can be used to quantify the unital non-SB channels. We also establish for the first time a link between two previously close but not directly related concepts: Bell nonlocality and non-macrorealism. In particular, we show that the value of non-macrorealism provides a lower bound on the magnitude of the non-NLB channel. In other words, if the Leggett-Garg inequality (LGI) can be violated, there exists a corresponding violation of the Bell scenario with the same measurement sets and channel.

In the temporal steering scenario, we consider a set of uncharacterized measurements with finite inputs $x\in\mathcal{X}$ and outcomes $a\in\mathcal{A}$ at $t_{0}$, while measurements at $t_{1}$ are assumed to be fully characterized (see Fig. 3). With the above setting, one can obtain a set of subnormalized quantum states, termed a temporal assemblage [46], namely

We say that a temporal assemblage is temporally unsteerable when it admits an HS model:

[c.f. Eq. (11)]. In general, an HS model in the temporal domain pre-assigns ontic probability and states in an asymmetric way, such that the system is well defined. For brevity, whenever there is no ambiguity we denote the assemblage $\{\rho_{\mathcal{E}}(a|x)\}$ as $\vec{\rho}_{\mathcal{E}}$. It is convenient to quantify the degree of temporal steerability by the robustness of temporal steerability [83, 84], which again refers to the minimum noise $\alpha$ mixed with a given assemblage before the mixture admits an HS model:

where $\vec{\sigma}$ is an arbitrary assemblage.

From Eq. (25) and the definition of the PDO, one immediately sees that the temporal assemblage $\vec{\rho}_{\mathcal{E}}$ can be formulated as

Because now the quantum channel is acting on the measurement, in what follows we only consider the unital quantum channel denoted as $\mathcal{E}_{\text{unital}}$. It is easy to see that the set of quantum channels is a superset of the unital quantum channels. Obviously, if $\text{R}_{\text{TS}}(\vec{\rho}_{\mathcal{E_{\text{unital}}}})=0$ when considering an arbitrary measurement set $\{M_{a|x}\}$, then the channel $\mathcal{E}_{\text{unital}}$ is a unital SB channel. With Theorem 1, we can certify all unital SB channels with temporal steering. Furthermore, $\text{R}_{\text{TS}}(\vec{\rho}_{\mathcal{E}_{\text{unital}}})$ provides a lower bound on the $\text{R}_{\text{n-SB}}(\mathcal{E}_{\text{unital}})$ in Eq. (15). If the entire measurement set $\{M_{a|x}\}$ is considered, we have $\text{R}_{\text{TS}}(\vec{\rho}_{\mathcal{E}_{\text{unital}}})=\text{R}_{\text{n-SB}}(\mathcal{E}_{\text{unital}})$. We arrive at:

Let us now turn our attention to the temporal Bell scenario (see Fig. 3), which is equivalent to the LGI scenario. The measurements at both times $t_{0}$ and $t_{1}$ are viewed as black boxes with finite inputs $x\in\mathcal{X}$ and $y\in\mathcal{Y}$, respectively. The index $a\in\mathcal{A}$ ($b\in\mathcal{B}$) is used to denote the measurement outcome at $t_{0}$ ($t_{1}$) [48, 47, 53]. The system is assumed to obey the aforementioned macrorealism, which implies that a temporal correlation can be described by a hidden-variable (HV) model, namely

One can see that the hidden parameter $j$ causally determines the well-defined probability distributions $p(a|x,j)$ and $p(b|y,j)$ at times $t_{0}$ and $t_{1}$, respectively. The violation of the temporal Bell inequality, which has the same form of Eq. (6) but with the correlation obtained from temporally separated measurements, reveals a quantum correlation effect. This quantum correlation is interpreted as non-macrorealism. This is because the temporal Bell inequality is a special kind of LGI, which is compatible with all macrorealistic physical theories.

In quantum theory, the observed correlation can be described by

One can note that certifying the non-macrorealistic probability distribution is mathematically identical to witnessing the locality of the CJ state in Eq. (30). If the observed correlation admits an HV model, the channel breaks nonlocality for the maximally entangled states by definition. Now, we define the robustness of non-macrorealism as

where $\mathcal{F}_{\text{HV}}$ denotes the set of temporal correlations admitting an HV model. We can see that $\text{R}_{\text{n-MR}}$ provides a lower bound for $\text{R}_{\text{n-NLB}}$. Therefore, we have:

Since other works [43, 59] focused on CHSH-NLB channels, we briefly discuss the particular case of the temporal CHSH inequality. Since the unital channel is CHSH-NLB, when the channel breaks the CHSH nonlocality for the maximally entangled states, the temporal CHSH inequality can be used to certify all unital CHSH-NLB channels. Finally, we emphasize that due to the hierarchy relation of temporal quantum correlations [78, 51], temporal separability implies macrorealism, but not vice versa. Thus, the concept of non-macrorealism can be used to witness non-EB channels. A similar argument for certifying non-SB channels can also be applied for testing non-macrorealism.