Realizing quantum advantage without entanglement in single-photon states

Quantum discord is the difference between the total mutual information shared by two qubits, I, and their locally accessible (classical) mutual information J [1, 5]. In light of the optical application that we propose in section 3, we designate the two qubits by the labels s and p, which we use below to distinguish photon spin (polarization) from interferometric path, but which can be taken to describe any two-qubit system. For a system described by the density matrix ${\rho }_{{sp}}$ in a composed Hilbert space ${{ \mathcal H }}_{s}\otimes {{ \mathcal H }}_{p}$, the total mutual information between the two qubits is

$$\begin{eqnarray}&&I({\rho }_{{sp}})=S({\rho }_{s})+S({\rho }_{p})-S({\rho }_{{sp}}),\end{eqnarray} \tag{ 1 }$$

where S denotes the von Neumann entropy and ${\rho }_{i}={\mathrm{Tr}}_{j}({\rho }_{{sp}})$ is the partial density matrix of the subsystem i, with $i\ne j=s,p$ [1]. After a projective measurement [28] on the first qubit ${{\rm{\Pi }}}_{\pm }=| \pm \rangle \langle \pm |$, the conditional state of p is given by ${\rho }_{p| \pm }={\mathrm{Tr}}_{s}({{\rm{\Pi }}}_{\pm }\otimes {{\mathbb{1}}}_{p}{\rho }_{{sp}})/{p}_{p| \pm }$ with probability ${p}_{p| \pm }\,=\,\mathrm{Tr}({{\rm{\Pi }}}_{\pm }\otimes {{\mathbb{1}}}_{p})$, where ${{\mathbb{1}}}_{p}$ is the identity for qubit p. The classical mutual information, $J({\rho }_{p| s})$, is the amount by which the uncertainty of system p is reduced after measuring s [5], and is

$$\begin{eqnarray}&&J({\rho }_{p| s})=S({\rho }_{p})-\mathop{\inf }\limits_{{{\rm{\Pi }}}_{\pm }}\displaystyle \sum _{\pm }S({\rho }_{p| \pm }),\end{eqnarray} \tag{ 2 }$$

where the average conditional entropy, ${\sum }_{\pm }S({\rho }_{p| \pm })={p}_{p| +}S({\rho }_{p| +})+{p}_{p| -}S({\rho }_{p| -})$, is minimized over all possible projective measurements. Finally, quantum discord [5], $D({\rho }_{p| s})$, corresponds to the shared information between s and p, that cannot be obtained by measuring s,

$$\begin{eqnarray}&&D({\rho }_{p| s})=S({\rho }_{p})-S({\rho }_{{sp}})+\mathop{\inf }\limits_{{{\rm{\Pi }}}_{\pm }}\displaystyle \sum _{\pm }S({\rho }_{p| \pm }).\end{eqnarray} \tag{ 3 }$$

The analogous expressions for $J({\rho }_{s| p})$ and $D({\rho }_{s| p})$ can be obtained by interchanging the roles of the qubits when computing the average conditional entropy.

In this work we restrict ourselves to a particular class of states that are generated by an incoherent superposition of maximal entangled states [1]. However, the same analysis can be performed for any symmetric two-qubit X-state. In the computational basis these states take the form

$$\begin{eqnarray}{\rho }_{{sp}}=\left(\begin{array}{cccc}a & 0 & 0 & {w}^{* }\\ 0 & b & {z}^{* } & 0\\ 0 & z & b & 0\\ w & 0 & 0 & a\end{array}\right),\end{eqnarray} \tag{ 4 }$$

with $b=1/2-a$. We show below that without loss of generality, the coherences $w\in [0,a]$ and $z\in [0,b]$ can be taken to be real and positive. The concurrence $C({\rho }_{{sp}})$ and entanglement of formation $E({\rho }_{{sp}})$ are given by the Wooters formulae [2]:

$$\begin{eqnarray}&&C({\rho }_{{sp}})=2\max [0,w-b,z-a],\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&E(C({\rho }_{{sp}}))=h((1+\sqrt{1-{C}^{2}})/2),\end{eqnarray} \tag{ 6 }$$

where $h(x)=-x{\mathrm{log}}_{2}x-(1-x){\mathrm{log}}_{2}(1-x)$. Since entanglement is a monotonic function of concurrence they have the same minima and maxima, which are 0 and 1 respectively.

For such states, which are described by density matrices of the form shown in equation (4), the conditional states after a measurement of one qubit (i.e. a local measurement) are independent of the qubit that was measured, ${\rho }_{s| p}={\rho }_{p| s}$. Because of this, the average conditional entropy is also symmetric and therefore $J({\rho }_{s| p})=J({\rho }_{p| s})=J({\rho }_{{sp}})$ and $D({\rho }_{s| p})=D({\rho }_{p| s})=D({\rho }_{{sp}})$. Following the criteria introduced by Maldonado-Trapp et al [21], the discord for this family of states is given by

$$\begin{eqnarray}D({\rho }_{{sp}})=\left\{\begin{array}{ccc}1-S({\rho }_{{sp}})+{S}_{{\sigma }_{Z}}({\rho }_{p| s}) & {\rm{if}} & 0\leqslant u\leqslant v,\\ 1-S({\rho }_{{sp}})+{S}_{{\sigma }_{X}}({\rho }_{p| s}) & {\rm{if}} & v\leqslant u\leqslant 1,\end{array}\right.\end{eqnarray} \tag{ 7 }$$

where $u=2(w+z)$ and $v=| 4a-1|$. The quantities ${S}_{{\sigma }_{Z}}({\rho }_{p| s})$ and ${S}_{{\sigma }_{X}}({\rho }_{p| s})$ correspond to the minimal average conditional entropies, when the optimal measurements are along ${\sigma }_{Z}$ and ${\sigma }_{X}$ respectively, where ${\sigma }_{i}$ are the Pauli matrices [1]. Explicitly, these entropies are given by

$$\begin{eqnarray}&&{S}_{{\sigma }_{Z}}({\rho }_{p| s})=-\displaystyle \frac{1-v}{2}{\mathrm{log}}_{2}\displaystyle \frac{1-v}{2}-\displaystyle \frac{1+v}{2}{\mathrm{log}}_{2}\displaystyle \frac{1+v}{2},\end{eqnarray} \tag{ 8 }$$

and

$$\begin{eqnarray}&&{S}_{{\sigma }_{X}}({\rho }_{p| A})=-\displaystyle \frac{1-u}{2}{\mathrm{log}}_{2}\displaystyle \frac{1-u}{2}-\displaystyle \frac{1+u}{2}{\mathrm{log}}_{2}\displaystyle \frac{1+u}{2}.\end{eqnarray} \tag{ 9 }$$

When u = v the average conditional entropy does not depend on the measurement. Note that if ${wz}\lt 0$ the optimal measurement ${\sigma }_{X}$ must be replaced by ${\sigma }_{Y}$ [21]. Although the optimal measurement changes, the value of the average conditional entropy remains the same. Since correlations do not change under local operations, the case ${wz}\lt 0$ can be avoided by considering a rotation over the two qubits of the form $\exp ({\rm{i}}{\phi }_{s}{\sigma }_{Z}/2)\otimes \exp ({\rm{i}}{\phi }_{p}{\sigma }_{Z}/2)$. Thus, as noted above, we can always choose ${\phi }_{s}$ and ${\phi }_{p}$ so that w and z are real and positive.

We now consider a scenario in which two independent parties, Alice and Bob, have access to a source of discordant ($D({\rho }_{{sp}})\gt 0$) two-qubit symmetric X-states that are fully known to both of them. Alice and Bob engage in a sequence of encoding/decoding transactions, each of which starts with the same X-state. Alice generates a random variable K that takes values $k=({b}_{1},{b}_{2})$ with a probability distribution p_k, where b₁ and b₂ are random classical bits [1, 27]. She encodes K in the qubit s and then challenges Bob to estimate K by measuring the encoded state.

In figure 1 we show a quantum circuit that illustrates the protocol described above. For each transaction, Alice applies a local unitary operation ${U}_{k}={\sigma }_{X}^{{b}_{1}}{\sigma }_{Z}^{{b}_{2}}$ to qubit s and sends the state to Bob. The four unitary operations that she can apply are ${U}_{1}={{\mathbb{1}}}_{s}$, ${U}_{2}={\sigma }_{Z}$, ${U}_{3}={\sigma }_{X}$ and ${U}_{4}={\sigma }_{X}{\sigma }_{Z}$, where ${{\mathbb{1}}}_{s}$ is the identity operator for the qubit s. The ensemble received by Bob is thus described by the density matrix

$$\begin{eqnarray}&&{\tilde{\rho }}_{{sp}}=\displaystyle \sum _{k=1}^{4}{p}_{k}{\rho }_{k}=\displaystyle \sum _{k=1}^{4}{p}_{k}{U}_{k}\otimes {{\mathbb{1}}}_{p}{\rho }_{{AB}}{U}_{k}^{\dagger }\otimes {{\mathbb{1}}}_{p}.\end{eqnarray} \tag{ 10 }$$

By performing a decoding protocol after each transaction, Bob constructs a new random variable ${K}^{* }$. Bob then estimates the value of k that was sent by Alice, and records his estimate as ${k}^{* }=({b}_{1}^{* },{b}_{2}^{* })\in {K}^{* }$. We will see that the accuracy of Bob's estimation is determined by his access to two-qubit operations.

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The maximal accessible information [1] that Bob can obtain about K corresponds to the Holevo information which for an ensemble $\epsilon =\{{p}_{k},{\rho }_{k}\}$ is given by

$$\begin{eqnarray}&&{I}_{q}=S({\tilde{\rho }}_{{sp}})-S({\rho }_{{sp}}).\end{eqnarray} \tag{ 11 }$$

Its minimum and maximum values are given by zero and two bits respectively. When ${I}_{q}=2$ the bits b₁ and b₂ can be determined deterministically, i.e. Bob can recover K with certainty. An example of this case is superdense coding [1, 29].

We note that when the protocol is applied to a symmetric X-state, the average state after the encoding ${\tilde{\rho }}_{{sp}}$ is also symmetric. Considering this, if Bob makes a projective measurement ${{\rm{\Pi }}}_{\pm }$ on qubit s, the accessible information that he can obtain from the qubit p is equivalent to the information that he can obtain from s given a measurement on p, which is

$$\begin{eqnarray}&&{I}_{c}=\mathop{\sup }\limits_{{{\rm{\Pi }}}_{\pm }}\displaystyle \sum _{\pm }\left(S({\tilde{\rho }}_{p| \pm })-\displaystyle \sum _{k}{p}_{k}S({\rho }_{p| \pm }^{k})\right)\end{eqnarray} \tag{ 12 }$$

where ${\tilde{\rho }}_{p| \pm }={\mathrm{Tr}}_{s}({{\rm{\Pi }}}_{\pm }\otimes {{\mathbb{1}}}_{p}{\tilde{\rho }}_{{sp}})/{p}_{p| \pm }$ and ${\rho }_{p| \pm }^{k}={\mathrm{Tr}}_{s}({U}_{k}^{\dagger }{{\rm{\Pi }}}_{\pm }{U}_{k}\otimes {{\mathbb{1}}}_{p}{\rho }_{{sp}})/{p}_{p| \pm }$ are the conditional states for qubit p. As for quantum discord, this quantity is non-trivial to compute since it requires the optimization of conditional entropies over all possible projective measurements.

Quantum advantage, ${\rm{\Delta }}I$, is defined as the difference between I_q and I_c, ${\rm{\Delta }}I={I}_{q}-{I}_{c}$ [15, 16]. It corresponds to the extra information that Bob can gain by performing two-qubit operations prior to making local measurements of each qubit. When the random variable K is encoded in ${\rho }_{{sp}}$, decoherence is induced in the system and therefore the correlations between s and p are modified. The discord consumption [15] is defined as the difference between quantum discord before and after the encoding, ${\rm{\Delta }}D({\rho }_{{sp}})=D({\rho }_{{sp}})-D({\tilde{\rho }}_{{sp}})$. Gu et al [15] showed that the quantum advantage of an encoding protocol and the discord consumption are related by the following inequality:

$$\begin{eqnarray}&&{\rm{\Delta }}D({\rho }_{{sp}})-J({\tilde{\rho }}_{{sp}})\leqslant {\rm{\Delta }}I\leqslant {\rm{\Delta }}D({\rho }_{{sp}}),\end{eqnarray} \tag{ 13 }$$

where $J({\tilde{\rho }}_{{sp}})$ is the classical mutual information between s and p after the protocol. Optimal encoding is the encoding that maximizes the quantum advantage [15]. It corresponds to the one in which the total mutual information is consumed, i.e. $I({\tilde{\rho }}_{{sp}})=D({\tilde{\rho }}_{{sp}})=0$. In this case the quantum advantage is equal to the initial amount of quantum discord, ${\rm{\Delta }}I=D({\rho }_{{sp}})$.

Since the von Neumann entropy is invariant under unitary local operations, we apply to the density matrix of the experimental X-states the following rotational operation, ${{\rm{e}}}^{{\rm{i}}\tfrac{\pi }{4}{\sigma }_{Z}}\otimes {I}_{p}\rho {{\rm{e}}}^{-{\rm{i}}\tfrac{\pi }{4}{\sigma }_{Z}}\otimes {I}_{p}$, such that the density matrix elements of the rotated state are real and positive. The correlations present in the original, ρ, and rotated state are the same and they can be computed by using the formulas discussed in section 2.1. In particular, concurrence is given by $C(\rho )=\max [0,{\kappa }_{h}R-T,{\kappa }_{v}T-R]$ and it vanishes when ${\kappa }_{h}R\leqslant T$ and ${\kappa }_{v}T\leqslant R$, or when R = T. The analytical expression for quantum discord is given by equation (7), with $u={\kappa }_{h}R+{\kappa }_{v}T$ and $v=| 2R-1|$. By using the second derivative test [27] we found that, with respect to variations of ${\kappa }_{h}$ and ${\kappa }_{v}$, quantum discord is maximized in two cases: ${\kappa }_{v}=1$ and ${\kappa }_{h}=0;$ and ${\kappa }_{v}=0$ and ${\kappa }_{h}=1$.

Figures 3(a) and (b) show the behavior of the pre-encoding concurrence and discord, $C(\rho )$ and $D(\rho )$, as a function of R and ${\kappa }_{h}$ when ${\kappa }_{v}=0$. When ${\kappa }_{h}=0$, the corresponding dependence on R and ${\kappa }_{v}$ can be found by reflecting these figures about the line $R=1/2$. From figure 3(a) we note concurrence increases monotonically with ${\kappa }_{h}$ and decreases monotonically with R. Concurrence is zero at the left region of the dashed line, $(1+{\kappa }_{h})R=1$. It reaches its maximum value, C = 1, when the photon is completely reflected onto path $| 0\rangle$, R = 1, and there is no time delay in the auxiliary path, ${\kappa }_{h}=1$. This maximum corresponds to the Bell state indicated by the red point, $| {\psi }_{+}\rangle =(| h0\rangle +| v1\rangle )/\sqrt{2}$ [1]. From figure 3(b) we note that quantum discord also increases monotonically with ${\kappa }_{h}$. Discord only vanishes when the photon is completely transmitted, R = 0, or when the coherence time of the photon is much less than the time delay, ${\kappa }_{h}=0$. It also reaches its maximum value for Bell states, D = 1, at R = 1 and ${\kappa }_{h}=1$. Besides the global maximum, discord also has local maxima along the black boundaries. The highest value that discord reaches without entanglement is represented by the red dot and its value is $D=1/3$ at ${\kappa }_{h}=1$ and $R=1/3$. In figure 3(c) the gray and white regions show where the optimal measurements are ${\sigma }_{Z}$ and ${\sigma }_{X}$ respectively. These are the measurements of polarization that minimize the average conditional entropy and therefore that least disturb the system but allow us to obtain more information about the system. The conditional entropy is independent of the measurement for the states along the black lines. The states along the black solid line are the Werner states [1, 20] which can be written as ${\rho }_{W}=(1-R{\kappa }_{h}){{\mathbb{1}}}_{{sp}}/4+R{\kappa }_{h}| {\psi }_{+}\rangle \langle {\psi }_{+}|$. We defined the Werner-like states to be the states along the black dashed line, which can be written as $(1+R{\kappa }_{h}){{\mathbb{1}}}_{{sp}}/4-R{\kappa }_{h}| {\psi }_{+}\rangle \langle {\psi }_{+}|$.

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In order to maximize quantum correlations we consider the pre-encoding state as the one that satisfies ${\kappa }_{h}=1$ and ${\kappa }_{v}=0$, this is

$$\begin{eqnarray}{\rho }_{{sp}}=\displaystyle \frac{1}{2}\left(\begin{array}{cccc}R & 0 & 0 & R\\ 0 & T & 0 & 0\\ 0 & 0 & T & 0\\ R & 0 & 0 & R\end{array}\right).\end{eqnarray} \tag{ 16 }$$

After each transaction of the bit pair $({b}_{1},{b}_{2})$ according to the encoding protocol described in section (2.2), the density matrix of the post-encoding state becomes

$$\begin{eqnarray}{\rho }_{k}=\displaystyle \frac{1}{4}\left(\begin{array}{cccc}1-{(-1)}^{{b}_{1}}(T-R) & 0 & 0 & {(-1)}^{{b}_{2}}(1+{(-1)}^{{b}_{1}})R\\ 0 & 1+{(-1)}^{{b}_{1}}(T-R) & {(-1)}^{{b}_{2}}(1-{(-1)}^{{b}_{1}})R & 0\\ 0 & {(-1)}^{{b}_{2}}(1-{(-1)}^{{b}_{1}})R & 1+{(-1)}^{{b}_{1}}(T-R) & 0\\ {(-1)}^{{b}_{2}}(1+{(-1)}^{{b}_{1}})R & 0 & 0 & 1-{(-1)}^{{b}_{1}}(T-R)\end{array}\right).\end{eqnarray} \tag{ 17 }$$

From equation (17) we immediately note that b₂ only appears in the non-diagonal terms. Without joint measurements Bob cannot determine the value of b₂, thus he will not be able to estimate K with certainty.

After averaging out a series of transactions, on average Bob receives the state

$$\begin{eqnarray}&&\begin{array}{l}{\tilde{\rho }}_{{sp}}=\displaystyle \frac{1}{2}\\ \,\,\,\times \,\left(\begin{array}{cccc}({p}_{3}+{p}_{4})T+({p}_{1}+{p}_{2})R & \,0 & \ 0 & \ ({p}_{1}-{p}_{2})R\\ 0 & \ ({p}_{3}+{p}_{4})R+({p}_{1}+{p}_{2})T & \,({p}_{3}-{p}_{4})R & \,0\\ 0 & \,({p}_{3}-{p}_{4})R & \,({p}_{3}+{p}_{4})R+({p}_{1}+{p}_{2})T & \,0\\ ({p}_{1}-{p}_{2})R & \,0 & \,0 & \,({p}_{3}+{p}_{4})T+({p}_{1}+{p}_{2})R\end{array}\right),\end{array}\end{eqnarray} \tag{ 18 }$$

where p₁, p₂, p₃, and p₄ are the probabilities described in section 2.2. Without loss of generality we consider only positive coherences ${p}_{1}\geqslant {p}_{2}$ and ${p}_{3}\geqslant {p}_{4}$. For this ensemble, concurrence is given by

$$\begin{eqnarray}&&C({\tilde{\rho }}_{{sp}})=\max [0,(2{p}_{1}-1)R-({p}_{1}+{p}_{2})T,(2{p}_{3}-1)R-({p}_{3}+{p}_{4})T].\end{eqnarray} \tag{ 19 }$$

Quantum discord is given by equation (7), with $u=(2({p}_{1}+{p}_{3})-1)R$ and $v=| ({p}_{3}+{p}_{4})(1-2R)+R|$.

As noted in section 2.2, an optimal encoding is the one that consumes all correlations, $I({\tilde{\rho }}_{{sp}})=0$ [15]. This is satisfied when the final average state is $\tfrac{1}{4}{{\mathbb{1}}}_{s}\otimes {{\mathbb{1}}}_{p}$; in other words, it is the encoding that introduces the greatest amount of decoherence to the system. From equation (18) we note that an optimal encoding must satisfy one of the following conditions: ${p}_{1}={p}_{2}={p}_{3}={p}_{4}=1/4$, or R = T. Thus, optimal encoding is that which yields a maximally-mixed encoded state.

Aside from the noted relationship [15] between initial quantum discord and the quantum advantage in an optimal encoding, ${\rm{\Delta }}I=D({\rho }_{{sp}})$, we found that in this case there is a direct relationship between the initial amount of information and the quantities I_c and I_q. The total accessible information, I_q, can be conceived as the amount of mutual information that has been removed from the initial state, ${I}_{q}=I({\rho }_{{sp}})-I({\tilde{\rho }}_{{sp}})$. Since for optimal encodings the total mutual information becomes zero we can state that these are the encodings that extract all the information from the system and therefore ${I}_{q}=I({\rho }_{{sp}})$. In addition to this, from equation (11) we note that I_q is also related to the randomness that has been introduced to the system. For example, when only one of the events k has preference, say ${p}_{2}=1$, no randomness is introduced and I_q is zero, while in an optimal encoding all the events have the same probability, ${p}_{k}=1/4$, and no more randomness can be introduced. Since for optimal encodings ${\rm{\Delta }}I=D({\rho }_{{sp}})$ and ${I}_{q}=I({\rho }_{{sp}})$ it is clear that the locally accessible information coincides with the initial classical information, ${I}_{c}=J({\rho }_{{sp}})$.

Although the optimal encoding is the one that maximizes the quantum advantage it is also interesting to study more general encodings. In particular, we study a group of encodings that we call 'quasi-optimal'. All quantum discord is consumed in quasi-optimal encodings, $D({\tilde{\rho }}_{{sp}})=0$, but classical mutual information is not necessarily consumed, $I({\tilde{\rho }}_{{sp}})=J({\tilde{\rho }}_{{sp}})$. Both quantum discord and classical mutual information are consumed only for a truly optimal encoding. Of the cases discussed above, all encodings with ${p}_{1}={p}_{2}$ and ${p}_{3}={p}_{4}$ are quasi-optimal, including the special case ${p}_{1}={p}_{2}={p}_{3}={p}_{4}$. In this group, the behavior of quantum correlations and advantage can be described by one variable. We choose the variable to be p₁ where ${p}_{1}\in [0,1/2]$.

Figure 4(a) shows consumed discord ${\rm{\Delta }}D$ (red) and quantum advantage ${\rm{\Delta }}I$ (blue) for the quasi-optimal encoding as a function of R and p₁. Overall, quantum advantage is always less than discord consumption, in agreement with inequality (13). In (a) and (b) the gray point denotes the Bell state $| {\psi }_{+}\rangle$, while the white and black points denote the maximum value of quantum advantage ${\rm{\Delta }}I$ when there is no entanglement for ${p}_{1}=0.25$ and ${p}_{1}=0.5$ respectively. Since quasi-optimal encoding consumes all discord, the consumed discord ${\rm{\Delta }}D$ equals the quantum discord of the pre-encoding state and is independent of p₁. From figure 4(a) we note that quantum advantage (blue) is maximal for optimal encodings, ${p}_{1}=0.25$, and it decreases symmetrically around ${p}_{1}=0.25$. Away from ${p}_{1}=0.25$ (optimal encoding), quantum advantage's behavior is drastically different. Instead of increasing monotonically with R, quantum advantage peaks at a certain R and then decreases as R increases. In the most extreme cases, where ${p}_{1}=0$ (encoding by only applying ${\sigma }_{x}$ and ${\sigma }_{y}$ and ${p}_{1}=0.5$ (encoding by only applying ${\sigma }_{z}$), quantum advantage becomes zero for R = 1, even though R = 1 corresponds to the Bell state $| {\psi }_{+}\rangle$ where both discord and entanglement is maximal. Such a difference is also evident in figure 4(b).

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Figure 4(b) shows the behavior of concurrence of the pre-encoding state, $C({\rho }_{{AB}})$, with quantum advantage, ${\rm{\Delta }}I$ as a function of R, for the optimal encoding, ${p}_{1}=0.25$ (black line), and for a particular non-optimal encoding, ${p}_{1}=0.5$ (black dashed line), which corresponds to encoding only in ${\sigma }_{z}$. For $R\leqslant 0.5$, the pre-encoding state ${\rho }_{{AB}}$ is not entangled (zero concurrence). In this regime, the maximum advantage for using the optimal encoding is 1/3 bits at $R=1/3$ (white dot), while for the non-optimal encoding it is ${\rm{\Delta }}I=3(2-{\mathrm{log}}_{2}3)/4\approx 0.311\,\mathrm{bit}$ at $R=0.5$ (black dot). The difference between optimal and non-optimal encoding in this region is small. The maximal quantum advantage between optimal and non-optimal is as low as approximately 0.022 bits. For $R\geqslant 0.5$, the pre-encoding state is entangled and is maximally entangled for R = 1 (the Bell state $| {\psi }_{+}\rangle$). In this regime, quantum advantage differs significantly between ${p}_{1}=0.25$ (optimal) and ${p}_{1}=0.5$ (non-optimal). For the non-optimal encoding ${\rm{\Delta }}I$ decreases until it reaches zero bits at R = 1, and for the optimal one ${\rm{\Delta }}I$ increases until it reaches 1 bit at R = 1 (gray dot).