Minimal Port-based Teleportation

There are two types of port-based teleportation (PBT) protocols: deterministic -- when the state always arrives to the receiver but is imperfectly transmitted and probabilistic -- when the state reaches the receiver intact with high probability. We introduce the minimal set of requirements that define a feasible PBT protocol and construct a simple PBT protocol that satisfies these requirements: it teleports an unknown state of a qubit with success probability $p_{succ}=1-\frac{N+2}{2^{N+1}}$ and fidelity $1-O(\frac{1}{N})$ with the resource state consisting of $N$ maximally entangled states. This protocol is not reducible from either the deterministic or probabilistic PBT protocol. We define the corresponding efficient superdense coding protocols which transmit more classical bits with fewer maximally entangled states. Furthermore, we introduce rigorous methods for comparing and converting between different PBT protocols.


I. INTRODUCTION
Quantum teleportation, introduced at the dawn of quantum information processing era, remains one of the pillars of quantum information transfer [1]. The concept of sending an unknown state of the quantum system without physically transferring the medium found its use in an impressive range of applications [2]. In recent years, Ishizaka and Hiroshima introduced new teleportation protocols with properties that were previously believed to be unattainable. These protocols go under a name of port-based teleportation (PBT) [3] and they possess a counterintuitive property that appears to be at odds with non-signalling principle of quantum mechanics, namely, that the teleported state requires no correction and is readily available for use after the sender performs a measurement and sends classical communication. This sparked a series of studies that investigate fundamental capabilities of different PBT protocols and their corresponding resource requirements such as the type of measurements and the amount of entanglement needed for the protocol to function [4][5][6][7][8][9].
PBT protocols found wide-ranging applications in cryptography and instantaneous non-local computation [10], they were instrumental in establishing a link between interaction complexity and entanglement in non-local computation and holography [11], they established a fundamental link between quantum communication complexity advantage and a violation of a Bell inequality [12], fundamental limitations for quantum channels discrimination by designing adaptive protocols called PBT stretching [13] and others [14][15][16].
Finding optimal protocols that work with quantum states of arbitrary dimensions necessitates the use of representation theory. The complexity of mathematical formalism precludes us from developing satisfying physical intuition about PBT protocols -in stark contrast to the elegance of the first teleportation protocol introduced in 1993 [1]. Moreover, the latter requires maximally-entangled states to operate most efficiently, whereas there exist flavours of port-based teleportation protocols which achieve the best-in-class performance using entangled states which are very different from maximally entangled states.
The underlying motivation for this work comes from the seminal work of R. Werner [17] who considered a set of remarkably distinct problems which can be reduced to performing (the first 1993-style) quantum teleportation. The emergence of distinctly different teleportation protocols raised a number of fascinating questions about the fundamental building blocks of quantum information: what other forms of transferring quantum information from one subsystem to another exist? PBT protocols offer a distinct alternative to the first teleportation scheme. We thus were motivated by a series of questions: Are there many other distinct protocols which result in quantum state transfer akin to the ordinary teleportation and PBT-like protocols? Can one treat the latter as a single class of protocols or, perhaps, there is a number of fundamentally different protocols within PBT? By now it has become clear that one can define numerous of PBT protocols, each different to another in that there was no straightforward way to provide a black-box reduction between them. Our work aims to address the above by providing two conceptual insights. First, we endeavour to derive PBT from 'minimal' requirements. In the spirit of seminal research direction where one constructs quantum theories from fewest possible axioms [18,19], we apply the same reasoning to the construction of teleportation protocols: what is the minimal number of requirements that yield a viable construction of the port-based teleportation? We introduce the first such 'minimal' set of requirements a corresponding author, email: studzinski.m.g@gmail.com arXiv:2111.05499v2 [quant-ph] 17 Jul 2023 that yield viable teleportation protocols together with the associated superdense coding schemes. Formalising the requirements immediately led to a construction of a new 'lean' or so-called minimal protocol. What's noteworthy, is that this protocol provably does not reduce to interpolation between the existing protocols and which has the exponentially improved scaling of probability of success. Second, given two PBT protocols is there a systematic way of interconverting them without having to consider the fine-grained details of the implementation? An affirmative answer to this question would provide a black-box transformation of trade-offs between imperfect but guaranteed and perfect but probabilistic information transfer LOCC protocols, which is interesting in its own right.
In our work, address both of these points: first, we outline the minimal set of assumptions that yield a working port-based teleportation protocol. We introduce a new protocol that cannot be reduced to any known protocols and that satisfies this set of minimal requirements. This protocol is exponentially more efficient than any known probabilistic PBT (pPBT), hence it cannot be obtained from any such protocol. At the same time, it uses special 'denoised' measurement operators that provably cannot result in deterministic PBT (dPBT).
Second, we develop methods for comparing port-based teleportation protocols in terms of their resource states which enable one to estimate their entanglement per port and distinctness. We further introduce methods for the conversion between the protocols and determine the conditions when it is possible to turn one type of PBT into another. We conclude by introducing a new family of superdense coding protocols which send more classical information using less entanglement compared to previously known protocol [20].
In Section II, we provide self-contained operational and mathematical preliminaries. This is followed by a minimal set of assumptions that a PBT protocol must satisfy in Section III together with a new, simplified PBT protocol. Remarkably, this remarkable simplicity does not come at a cost of reduced performance. To give a flavour of our results, in Table I we collect known results on the performance of various qubits PBT schemes compared with the minimal PBT.
Teleportation protocol Entanglement fidelity F Average success probability p succ In Section IV we show how to convert different types of PBT protocols. The non-trivial regime that needs to be rigorously investigated is transforming pPBT into dPBT.
In Section V we study PBT protocols from a different viewpoint, by investigating the properties of their respective resource states. We introduce partial ordering on the PBT resource states by considering the fidelity between resource states. We find that even in the case of two similarly performing protocols, their underlying resource states are drastically different. This ordering enables us to find the most 'frugal' teleportation protocol: the one which achieves the highest performance with substantially lower entanglement requirements.
Lastly, in Section VI we turn to superdense coding protocols, which are dual to any teleportation protocol. While ordinary superdense coding protocols are well-understood in the context of original teleportation, very little is known about their dual-PBT versions, with only one known example in [20]. We show how to take an arbitrary dPBT protocol with an established lower bound on fidelity and compute the corresponding performance of the superdense coding protocol. In particular, we find that there exist superdense coding protocols that are capable of transmitting the same amount of classical information as in [20], but using significantly less entanglement. The appendix contains the necessary source code to calculate all the quantities throughout the paper.

A. Operational preliminaries of PBT
The aim of any teleportation protocol is to transfer a given unknown quantum state ψ C from one party (Alice) to another (Bob). This is achieved by first distributing a resource state ρ AB , the sender holding subsystem labelled A and the receiver holding B. Then Alice performs a joint measurement from a set {M i } i on ψ C ⊗ ρ AB and records the outcome i ∈ N 0 . The latter is then communicated to Bob using a classical channel. After Bob receives i, he performs one of two actions (depending on the particulars of the protocol): he either discards all the subsystems except for the one that is identified by i or he aborts the protocol (if i encodes the error flag, for example when i = 0). In the first case, the teleported state of ψ C is located in the subsystem identified by i.
In what follows we present a short summary of the PBT schemes. All the results contained here have been proven elsewhere and for more details, we encourage readers to see the following papers where the formalism of PBT has been developed [3,6,[21][22][23].
We start with introducing resources used by parties wishing to apply the PBT scheme. Namely, parties share N copies of d-dimensional maximally entangled states, each of them called port: is a global operation applied by Alice to increase the efficiency of the protocol [21][22][23]. In the case when O A = 1 A , we deal with N maximally entangled state and call the total state |Ψ AB = |Ψ + AB as a non-optimised resource state and the whole protocol is called non-optimal PBT. For optimal schemes, the explicit forms of O A are presented in Appendix A for the reader's convenience, as well as in cited above literature. Further we refer to the state |Ψ AB = (O A ⊗ 1 B )|Ψ + AB as a optimised resource state and the whole protocol is called optimal PBT. Counterintuitively, due to the fact that O A = 1 A and O A is not unitary, the resource state in optimal schemes yields superior performance while not being maximally entangled.
Presently, there are two flavours of the PBT protocols: • Deterministic protocol (dPBT): An unknown quantum state ψ C is always transmitted to the receiver but the transmission is imperfect. The teleportation channel N C→ B is of the following form: where by Tr AC denotes partial trace over all systems AC but B. The states σ A i B are called signal states: where P + is projector on maximally entangled state between systems A i and B. To assess the quality of the protocol, one can evaluate entanglement fidelity F(N C→ B ) of teleportation channel N C→ B when teleporting a subsystem C of maximally entangled state P + CD , and computing overlap with the state after perfect transmission P +

BD
: where 1 D denotes identity channel leaving system D untouched. For an arbitrary dimension d the fidelity F(N C→ B ) has been evaluated explicitly using methods coming from group representation theory [22][23][24]. Due to the recent result presented in [25], we know that measurements in the form of square-root measurements (SRM) are optimal in both PBT versions (non-and optimal PBT). The optimal measurements in the nonoptimal case are: The operator ρ −1 is restricted to the support of ρ, so to ensure summation of all POVMs to identity 1 AC on the whole space (C d ) ⊗N+1 , we add to every Π AC However, this extra term does not change the entanglement fidelity F(N C→ B ) of the channel N C→ B , see also [3,21,22] for the detailed explanation.
• Probabilistic protocol (pPBT): From [21] we know that in the probabilistic scheme, the process of teleportation sometimes fails, however when it succeeds the state is faithfully teleported to Bob with fidelity F(N C→ B ) = 1.
In this scheme Alice has access to N + 1 POVMs {M AC 0 , M AC 1 , . . . , M AC N } with measurement M AC 0 corresponding to the failure of teleportation procedure. An additional POVM M AC 0 makes the teleportation channel N C→ B trace non-preserving. The efficiency of the protocol is described by the average probability of success p succ of the scheme equals to [21,22]: where In the same manner, we define a probability of failure p f ail corresponding to POVM M AC 0 averaged over all input states which of course satisfies relation p f ail = 1 − p succ . The requirement of perfect transmission F(N ) = 1 constraints form of allowed measurements accessible for Alice, namely can be of the following form [21,22]: where A i denotes all states A 1 A 2 · · · A N but A i . The optimal form of the operators Θ A i in both versions of the pPBT protocols (optimal and non-optimal) was computed for qubits and qudits in [21,22].
In both cases (deterministic and probabilistic) we used O A to denote the optimising operation. However, when the operators differ we will further write O A for dPBT and O A for pPBT. Such a distinction will be very helpful in Section V where we calculate overlaps between the resource states. In the case of pPBT protocol we also use the notion of non-optimised and optimised resource state as it was done above for dPBT scheme.

B. Mathematical preliminaries of PBT
We will now review basic representation-theoretic preliminaries. We briefly describe here the irreducible representation formalism for the permutation group S(n) with its algebra C[S(n)] [26,27] and collect the main results regarding the algebra of partially transposed permutation operators A n (d) [6,8].
A permutational representation is a map V : S(n) → Hom((C d ) ⊗n ) of the symmetric group S(n), where n = N + 1, in the space H ≡ (C d ) ⊗n defined as ∀π ∈ S(n) V(π).|e i 1 ⊗ |e i 2 ⊗ · · · ⊗ |e i n := |e i where the set {|e i } d i=1 is an orthonormal basis of the space C d , and d stands for the dimension. We drop here the lower index in every i, since it labels only position of the basis in tensor product (C d ) ⊗n . In other words, the operators V(π) just permute basis vectors according to the given permutation π ∈ S(n). The representation V extends to the representation of the group algebra C[S(n)] := span C {V(σ) : σ ∈ S(n)} ⊂ Hom((C d ) ⊗n ). All irreducible representations (irreps) of the symmetric group S(n) are labeled by so-called partitions. A partition α of a natural number n, which we denote as α n, is a sequence of positive numbers α = (α 1 , α 2 , . . . , α r ), such that α 1 ≥ α 2 ≥ · · · ≥ α r and ∑ r i=1 α i = n. Every partition can be visualised as a Young frame which is a collection of boxes arranged in left-justified rows. For illustration please see panel I in Figure 1. It means that for every fixed number n, the number of Young frames determines the number of nonequivalent irreps of S(n) in an abstract decomposition. If one fixes the representation space to H ≡ (C d ) ⊗n then when we decompose S(n) into irreps we take only Young frames α whose height h(α) is at most d. Let us take now α (n − 1) and µ n. By writing µ ∈ α we understand Young frames µ obtained from α by adding a single box. On the opposite, α ∈ µ denotes Young frames α obtained from µ by removing a single box -see panel II in Figure 1.
Recall the celebrated Schur-Weyl duality [26,27], which states that the diagonal action of the unitary group U (d) of invertible complex matrices and of the symmetric group S(n) on (C d ) ⊗n commute: FIG. 1. Panel I depicts five possible Young frames for n = 4 corresponding to all possible abstract irreducible representations of the group S(4). Its representation space is (C d ) ⊗4 and the only irreps that appear are those for which the height of corresponding Young frames is no larger than d. In particular, when one considers qubits (d = 2) we have only three admissible frames: (4), (3, 1), (2, 2). Panel II presents all possible Young frames µ 4 satisfying relation µ ∈ α for α = (2, 1). Green squares depict boxes that are added to the initial frame α. On the right, we present all possible Young frames α 3 satisfying relation α ∈ µ for µ = (2, 1, 1). Boxes subtracted from the initial Young frame µ = (2, 1, 1) are shown in red.
where σ ∈ S(n) and U ∈ U (d). In particular, it means that there exists a basis producing the decomposition of V(π) and U ⊗n into irreps simultaneously, and decomposition of the tensor product space (C d ) ⊗n as: In the above expression the symmetric group S(n) acts non-trivially on the space S α and the unitary group U (d) acts non-trivially on the space U α , labelled by the same partitions α. For a given irrep α of S(n), the space U α is a multiplicity space of dimension m α (multiplicity of irrep α), while the space S α is a representation space of dimension d α (dimension of irrep α). With every subspace U α ⊗ S α we associate the Young projector: where χ α (σ −1 ) is the irreducible character associated to the irrep indexed by α. To denote a matrix representation of an irrep of σ ∈ S(n) indexed by a frame α we will write ϕ α (σ). Equation (10) gives a way to describe irreducible representations -the minimal non-trivial blocks commuting with a diagonal action of the unitary group U (d), i.e. action of the form U ⊗n , for some natural n, and U ∈ U (d). Such reduction, in addition to displaying the interior structure of the operators allows one to specialize the analysis on the whole space (which is typically very large and complex) to every small block separately thus significantly reducing the complexity of the problem (this is especially helpful when problems involve semidefinite programming). When working with the PBT we need to introduce different type of symmetries, however, still motivated by the Schur-Weyl duality discussed above. To describe them we first briefly discuss the properties of states and measurements used in all variants of PBT described in Subsection II A.
Recall that a bipartite maximally entangled state is U ⊗ U invariant [28], where the bar denotes complex conjugation of an element U of the unitary group U (d). It means that all signal states σ A B i from (3) satisfy the following commutation rule: where U acts on B, and U ⊗N acts on systems A = A 1 · · · A N . Additionally, from the construction of the signal states σ A B i it follows that they are covariant with respect to the elements from the group S(N), acting on first N systems: In particular, choosing an arbitrary state from the set, say σ A B N all the others can be generated by acting on it with an element from the coset S(N)/S(N − 1), whose elements in the representation V are of the form V[(i, N − 1)], for i = 1, . . . , N − 1. The same kind of covariance with respect to S(N) and U ⊗N ⊗ U also holds for all measurements Finally, since the operator ρ from (5) is a sum over all possible signal states it also exhibits symmetries described in (12) and in addition, it is invariant with respect to the action of elements from the group S(N).
Satisfying the relation (12) by operators describing all variants of PBT protocols means that they belong to the algebra of partially transposed permutation operators, where the partial transposition is taken with respect to the last n−th system [8,29,30]: where for simplicity denotes the partial transposition under consideration. The algebra A n (d) is no longer a group algebra since for example, one has . This is an analog of the relation from (9) between the permutation group S(n) and unitary group U (d). This means we should expect analogous decomposition of the space (C d ) ⊗n to (10), when studying objects from A n (d). Due to the results contained in [22] we know that a similar decomposition exists, with the corresponding version of Young projectors from (11) on irreducible spaces. We denote these projectors as F µ (α), and they are labelled by two types of Young diagrams α (n − 2) and µ (n − 1), such that µ ∈ α. In particular, it was shown in [22] that the operator ρ from (5) decomposes in terms of irreducible projectors F µ (α) as with non-zero eigenvalues λ µ (α) of the form The above decomposition allows for easy calculation of the inversion ρ −1 necessary for having explicit form of the measurements from (5). We also introduce notation γ µ * (α), meaning that for a given diagram α (n − 2) we choose such µ ∈ α for which γ µ (α) from (16) is maximal, i.e.

III. PORT-BASED TELEPORTATION WITH MINIMAL REQUIREMENTS
To introduce the minimal set of requirements that defines a PBT protocol, consider the following sequence of steps outlined in the box below.
AB between sender A and receiver B; a state ψ C to be teleported; a set • Alice sends the index i to Bob by classical channel; B i ), p) that describes the performance of a teleported state. The first parameter characterises the quality of teleported state and the second -the success of the teleportation. Please notice that the parameter Q does not depend on the input state ψ C . In the general situation, one could take k ≥ n + 1 measurements, but since only first n of them give contribution to F and p the rest can be treated as a one big measurement labelled by index i = n + 1.

Definition 1.
A PBT protocol P from the above algorithm from the box is convergent if Q(P ) → (1, 1) as n → ∞.

A. Minimal PBT protocol (mPBT)
In what follows, we introduce the convergent PBT protocol that while structurally similar to both pPBT and dPBT. However, while sharing many of the common building blocks with the d(p)PBT this 'minimal' PBT protocol cannot be derived from either of them. The performance of our protocol interpolates between figures of meritthe fidelity and probability of success -between deterministic and probabilistic scheme respectively. Our protocol cannot be improved to get perfect fidelity or the probability of success with finite amount of resources (quantified as a number number of shared ports N). However, this protocol maintains exponentially faster convergence to unity, compared to the optimal pPBT. To construct the measurement operators for mPBT, we exploit the SRM that are used in dPBT, but crucially we omit an extra error term (1/N)∆ defined below expression (5): It is clear that the POVM elements do not sum up to identity on the whole space (C d ) ⊗n , where n = N + 1. To fix that and thus recover their proper probabilistic interpretation we must add an additional operator denoted here M AC 0 . This additional POVM element, in this variant, equals exactly to ∆. While operationally, it corresponds the failure of the teleportation process, we show in Appendix A that it bears no resemblance to any known M 0 (failure) operators which were used for probabilistic teleportation schemes. Note that we drop the constraint from equation (7), so the state is not teleported faithfully in this probabilistic scheme (see later discussion in this section).
The probability of success and fidelity of our protocol are given below: The probability of success p succ in mPBT, with the non-optimised resource state, when one uses square-root measurements {Π AC i } N i=1 from (18) has the form: where m α , d µ denote multiplicities and dimension of irreps of S(N − 1) and S(N) respectively in the Schur-Weyl duality. The success probability for qubits p succ has the form: Proof. To prove expression for the probability of success p succ we use equation (6) with O A = 1 A and explicit form of measurements from (18): (21) where 1 supp(ρ) is the identity operator on the support of ρ from (18). From paper [22] we know that 1 supp(ρ) has decomposition into sum of projectors F α (µ) on irreducible spaces of the algebra A n (d): since Tr F α (µ) = d µ m α . This leads us to the first statement from the theorem. To prove the second part note that in the qubit case all Young diagrams are up to two rows, and they are always of the form µ = (N − l, l), where 1 ≤ l ≤ N 2 − 1 . It is possible to derive closed-form expressions for dimensions and multiplicities of the corresponding irreps. From [31], we know that for any partition of the shape µ = (N − l, l) the following expression for the corresponding of irrep dimension and multiplicity: However, in expression (22) we have to consider two types of irreps, irreps α N − 1 and irreps µ N, for which we have the relation µ ∈ α. The trace in (22) can be written as where α ∈ µ denotes partitions α N − 1 obtained from partition µ N by removing a single box. We have two types of partitions α. The first type is when N − 1 = 2k, for N 2 − 1 , then the corresponding Young tableaux has two equal rows. In this case, we can add a box only to the first row. In the second type of partitions, when N − 1 = 2k + 1. In this case, we can add a single box to the first or second row. Using this observation and expressions (23), we can write terms d µ m α for Young tableaux satisfying relation α ∈ µ: These two expressions can be combined to get one closed expression of the form Finally, to show expression (20) we use the following chain of equalities: Now, changing range of the sum and subtracting proper terms, we arrive to where in the third line we use identities ∑ N l=0 ( N l ) = 2 N , ∑ N l=0 l( N l ) = N2 N−1 , and ∑ N l=0 l 2 ( N l ) = N(N + 1)2 N−2 . This finishes the proof.
In the right panel of figure 2 we present actual values for p succ obtained from Theorem 2 for qubits and we compare it with optimal values for the pPBT introduced in [21]. Additionally, in Figure 3 we plot efficiency characteristic for higher dimensions (d = 4) by exploiting group theoretical expression from (19) and compare it with optimal values for the pPBT for d ≥ 2 derived in [22].  Figure 2. In this case, the mPBT scheme performed by SRM measurements from Theorem 2 we outperform the optimal qudit pPBT, for which the probability of success is p succ = 1 − d 2 −1 N+d 2 −1 [22]. When one considers resulting fidelities, we see that the mPBT scheme is more efficient up to ∼ 30 ports when one compares it with the non-optimal dPBT with SRM measurements.
The next theorem shows the best achievable performance in this setting: Alice optimises both measurements and the resource state using an operation O A is given as: where v µ ≥ 0 are entries of an eignevector corresponding to a maximal eigenvalue of the teleportation matrix M F used for computation of entanglement fidelity in optimal PBT [23].
Theorem 3. The probability of success in mPBT, with the optimised resource state is given by where m α , m µ denote multiplicities of irreps of S(N − 1) and S(N) respectively in the Schur-Weyl duality.
Proof. To prove expression for the probability of success p succ we use equation (6) with O A from (30) and explicit form of measurements presented in (18): Since we know that In the second equality we use Lemma 10 from [22] stating that Tr n F µ (α) = m α m µ P µ , where P µ is a Young projector on irrep labelled by µ N. Plugging the explicit form of O A given in (30) into the above, using orthogonality property for Young projectors P ν P µ = δ νµ P µ , and taking into account Observation 11 from Appendix A we find: since Tr(P µ ) = d µ m µ . This finishes the proof.
We thus evaluated the probability of success p succ by exploiting two types of measurements used by Alice for qudits. Unlike the qubit case of Theorem 2, it does not lend itself to a nice closed form.
We now turn to entanglement fidelity of the above protocols. The entanglement fidelity, where one teleports a half of the maximally entangled state, is given by where , and A i denotes all systems but i−th. We expect to teleport the state faithfully, so we must have F(N C→ B ) = 1. Note that the term 1 is the entanglement fidelity F det of the respective protocol in deterministic scheme, since we have that Tr(M AC 0 σ AC i ) = 0. This observation allows us to express entanglement fidelity in the minimal PBT scheme (see equation (4)) as: Expressions for F det in (non-)optimal PBT are known and presented respectively in Theorem 12 in [22] and Proposition 32 in [23]. For the self-consistence we provide these expressions in Appendix A, please see expressions (A10) and (A12) respectively, with corresponding qubit forms in (A11) and (A13). In the case of qubits the probability of success in mPBT is given through expression (20) from Theorem 2 and fidelity F det is given through (A11), so (36) reads as: In the left panel of figure 2 we compare the actual fidelity of mPBT from (36) with the fidelity of the teleported state in the non-optimal PBT scheme (37), performed by SRM measurements. Figure 4 describes the known landscape of admissible protocols and their interrelation when parties exploit maximally entangled resource state. A similar plot can be make for the optimised port-based teleportation schemes, the only thing which would change is the scaling in the number of ports N, but the general shape stays the same.

IV. CONVERTING PBT PROTOCOLS
Different types of port-based teleportation require individual mathematical analysis and in this section we discuss the possibility of converting probabilistic version of PBT into the deterministic one. First, we prove a general theorem giving an explicit relation between entanglement fidelity and probability of success in such conversion. Next, we illustrate the theorem by presenting expressions for the entanglement fidelity of teleported state in the case of non-optimal pPBT and its optimal version, where explicit form of Alice's measurements is known. We also argue that the reverse conversion, i.e. converting dPBT to probabilistic one is possible only under certain constraints.
Theorem 4. Every pPBT scheme, with N ports each of dimension d, and the probability of success p succ , can be turned into deterministic with explicit entanglement fidelity of the form: where M AC 0 is measurements corresponding to the failure of probabilistic scheme, operator ρ is defined through expression (18).
Proof. Every pPBT scheme requires a set of N + 1 measurements {M AC 0 , M AC 1 , . . . , M AC N }, where the effect M AC 0 corresponds to the failure of the teleportation process. Additionally, to get F = 1 in probabilistic scheme, we require that all the measurements for 1 ≤ i ≤ N satisfy the relation (7). To design corresponding deterministic scheme, where teleported state appears on one through N ports on the Bob's side we perform the mapping: Then the teleportation channel N (ψ C ) is of the form Now, if ψ C is a half of maximally entangled state P + CD , the entanglement fidelity F(N ) of the channel N is because ρ = ∑ i σ AC i . Since we start from POVMs for probabilistic scheme, they must satisfy relation (7), which implies: Plugging the above into (41) and using (6) we get the result.
The conversion of a mPBT to dPBT protocol can be also viewed differently. Namely, every probabilistic PBT protocol leads to a deterministic one with the fidelity F = F succ p succ + F f ail (1 − p succ ), where F succ = 1, and F f ail = 1/d 2 is entanglement fidelity when one fails with the transmission process. Similar approach of turning every pPBT into dPBT shortly discussed in the paper [24]. In [24] authors consider a type of conversion when whenever Alice fails in probabilistic scheme she sends to Bob a random port index. However, our approach is a little bit different from the one presented above, since in Theorem 4 we do not have anymore possibility of the failure -see construction of the corresponding POVMs in eq. (39). Our construction divides POVM M AC 0 into N parts and add them to every POVM M AC i getting pure deterministic scheme. Summarising, Theorem 4 except the direct formula for the entanglement fidelity of such protocol gives also an algorithm for its construction by the explanation how to construct respective measurements.
From expression (38) we see that the resulting fidelity depends on two factors -probability of success in probabilistic scheme and overlap between POVM M AC 0 corresponding to failure and the state ρ from (5). In the general case it is impossible to say much about the overlap, since the operator M AC 0 depends heavily on an architecture of a given probabilistic scheme. However, when we stick with a particular probabilistic scheme we can explicitly evaluate the entanglement fidelity F in Theorem 4. In what follows we illustrate how Theorem 4 works in practice for known pPBT schemes. We shall consider both, non-optimal and optimal pPBT for an arbitrary number of ports and their arbitrary dimension, which have been analysed in [22]. Our calculations are based on the latter paper. A short summary of the existing results on pPBT together with some group theoretic methods developed in papers [22,23] is provided in Section A.

Lemma 5.
The entanglement fidelity F given in Theorem 4 for the non-optimised resource state reads: where the numbers γ ν (α), γ µ * (α) are given by (16) and (17)  Proof. The first term p succ in expression (38) is known, see Theorem 3 in [22], and equals to It remains to evaluate the term Tr(M AC 0 ρ). Using the summation rule for POVMs where 1 AC is an identity operator on (C d ) ⊗(N+1) , The last equality follows from the fact that every operator σ AC i in the sum ρ = ∑ i σ AC i is normalised, and fact that the trace and ρ are invariant under action of elements from the coset S(N)/S(N − 1). Using explicit form of POVM M AC N given in (A1) in the Appendix A, and decomposition of the state ρ into irreducible projectors of the algebra A n (d) given in (15) we get: The second equality follows from Theorem 1 and Fact 13 in the paper [22]. The third equality follows from the orthogonality property for Young projectors, saying that P β P α = δ βα P α . The fourth equality follows from the observation that only operator V acts non-trivially on last n−th system, so it can be traced out with respect to it, giving us the identity operator acting on first N systems. The fifth equality follows from the fact that only Young projector P ν acts non-trivially on N−th system, so we can apply Corollary 10 from [5], and compute the partial trace. Using orthogonality property for Young projectors an fact that Tr P α = d α m α the result follows.
It is apparent from Lemma the final result is hard to analyse analytically. However, every quantity from (43) can be computed numerically. Next, we prove Lemma 5 for the optimal pPBT, i.e. when Alice optimises simultaneously over measurements and resource state, and show that the final expression for the entanglement fidelity is in very elegant and compact form. Lemma 6. The entanglement fidelity F given through Theorem 4 for the optimised resource state reads: where p succ denotes the probability of success in the optimal pPBT.
Proof. The main idea of the proof follows the proof of Lemma 5. However, here in equation (45) The first term in the above expression is Tr(ρ) = N. We now compute the second term in (49) using explicit form of POVM M AC N from (A5): All the steps except the last equality in (50) follow the proof of Lemma 5. First, we use the observation that ∑ α∈µ d α = d µ . Then we apply Proposition 25 from [22]. Finally, combining expressions Tr(ρ) = N and (50) we conclude that Tr(M AC 0 ρ) = 0. Next, using expression for probability of success p succ in this variant of pPBT, which is p succ = 1 − d 2 −1 N+d 2 −1 due to Theorem 4 in [22] we get the result.
The above result implies that we cannot derive a deterministic scheme from probabilistic variant with entanglement fidelity scaling better than 1 − O(1/N) in the number of ports N. It follows from the fact that the probability of success in optimal pPBT scales as 1 − O(1/N), see [22]. This is due to the fact that in every probabilistic scheme we have to add an additional constraint: to ensure that the teleported state is transmitted faithfully, we have to ensure that all measurements corresponding to the probability of success satisfy (7). It means that designing, say, the optimal probabilistic scheme one does not optimise over all possible space of POVMs, but over their proper subset. This restriction is one of the reasons responsible for different scaling in (non-)optimal probabilistic and deterministic protocols respectively. This implies that one cannot turn an arbitrary deterministic protocol into probabilistic one, with better scaling than 1 − O(1/N), since measurements of such protocol do not satisfy requirement (7). In particular, we cannot turn optimal dPBT discussed in [21,23] into probabilistic scheme with F = 1 for finite N. The only way for such conversion to be feasible is when one defaults to 'interpolated' protocols, described in Section III A, where neither entanglement fidelity F nor probability of success p succ equal to one for the finite number of ports N.
The conversion between PBT protocols is illustrated below on Figure . In general, there may exist other measurements that yield PBT protocols that satisfy requirements of Section III. Every mPBT protocol from the non-shaded region can be converted into the protocol in the nonshaded dPBT by adding ∆/N to each of the POVM elements, where ∆ = 1 − ∑ N i=1 Π i (arrow 1). Any known pPBT protocol can be converted to a (shaded) subset of known dPBT protocols by taking all the POVM elements M 1 , . . . M N that correspond to successful teleportation (omitting M 0 ) and transform each of them as follows: M i = M i + M 0 /N. This operation is invertible (arrow 3). Lastly, any mPBT protocol from the shaded area can be converted to shaded subset of dPBT protocols.

V. PBT RESOURCE STATE COMPARISON
Despite the fact that all PBT protocols share operational similarities, their underlying resource states are rather different. In this section we discuss properties of the resource states in all PBT protocols by evaluating their closeness. We show that by starting from maximally entangled states, the the optimisation operator O A applied by Alice has a large effect on their mutual distances. To quantify the distance, we will use the square-root fidelity √ F [32]. See Appendix C for details. Let us denote the resource states in optimal probabilistic and optimal deterministic scheme by |Ψ AB , | Ψ AB respectively. We distinguish optimization operators on Alice's side from Eq. (52) by writing O A , O A respectively. Lemma 7. The square fidelity √ F between the resource states in optimal pPBT |Ψ AB and optimal dPBT | Ψ AB , each with N ports of dimension d equals to: √ where m µ , m ν are multiplicities of irreps of S(N) in the Schur-Weyl duality. The numbers v µ ≥ 0 are entries of an eignevector corresponding to a maximal eigenvalue of the teleportation matrix M F .
Proof. The general form of the optimal resource state in pPBT reads: where O A is taken from (A5), with normalisation constraint Tr(O † A O A ) = d N . The same holds for the optimal dPBT, but instead of O A we use O A given in (A9). The square-root fidelity √ F between the states |Ψ AB , | Ψ AB : Plugging the explicit forms of the operators O A , O A from (A9) and (A5) respectively we arrive to Taking into account orthogonality relation for Young projectors P µ P ν = P µ δ µν with trace property Tr P µ = d µ m µ and plugging the explicit form of the function g(N) from (A6) completes the proof.
The numbers v µ ≥ 0 in expression (56) are not known to admit closed form, since one does not know analytical expressions for eigenvectors of the teleportation matrix for d > 2 and d < N (see [23]). However, when d = 2 the matrix M F is tri-diagonal and we know analytical expressions for its eigenvalues and eigenvectors due to [33]. In what follows, we will invoke results from [21], where the form of the operators O A , O A was evaluated in terms of the spin angular momentum of the N−spin system. In this representation the operators are proportional to identity on subspaces with fixed quantum number j, which runs from j min = 0(1/2) when N is even (odd): for known positive numbers γ(j), ζ(j) which we describe later. The above form of the optimal Alice's operations allows us to formulate the qubit version of Lemma 7: Lemma 8. The square fidelity √ F between the resource states |Ψ AB and | Ψ AB in optimal qubit pPBT and optimal dPBT respectively, each with N ports equals to: where j runs from j min = 0(1/2) when N is even (odd). The fidelity √ F |Ψ AB , | Ψ AB converges to √ 6 π ≈ 0.778 when N → ∞ and takes maximal value √ F |Ψ AB , | Ψ AB = 0.894 which is attained for N = 2.
Proof. The proof is similar to that of Lemma 7 with the difference we now use qubit versions of the operators O A , O A in spin angular momentum representation given in (55). The dimension and the multiplicity d µ , m µ respectively depend on the quantum number j: This together with the trace rule Tr 1(j) = m j d j allows us to deduce that the square-root fidelity √ F is of the form Plugging the explicit form of the coefficients γ(j), ζ(j) (expression (35) and (54) in [21]) and observing that the value of the sinus is always positive in the allowed range of j, and with expression for h(N) (equation above (54) in [21]), which reads as we obtain the first part of the statement. From the above expression one can deduce numerically the convergence of the fidelity to the value 0.778 for N → ∞, and maximal value of F |Ψ AB , | Ψ AB = 0.894 attained for two ports.
The situation is entirely different when one considers fidelity between the resource states in non-optimal and optimal PBT, taking probabilistic and deterministic scheme for comparison. In both non-optimal pPBT and dPBT the resource state before the optimization has the same form of N copies of maximally entangled state. We start from the following lemma: Lemma 9. The square fidelity √ F between the resource states in non-optimal pPBT and its optimal version, each with N ports of dimension d equals to: where d µ , m µ are dimension and multiplicity of irreps of S(N) in the Schur-Weyl duality, and g(N) is given in (A6). In the qubit case the expression (61) has a form where j runs from j min = 0(1/2) when N is even (odd). The qubit fidelity from (62) converges to 0 with N → ∞.
Proof. We start from proving our statement for an arbitrary dimension d of the port. Using explicit form of O A in optimal pPBT given in (A5) w have: since Tr(P µ ) = d µ m µ . To get expression for qubits we use the second expressions from equations (55) and (59), and then the explicit form of the function h(N) in (60), together with equations for d j , m j in terms of quantum number j from (57). To prove convergence in the qubit case, note that the factor in front of the sum over j clearly goes to 0 with N → ∞. The second factor can be bounded from the above as follows For the completeness of our results, we include the corresponding lemma from [34] and giving the value of fidelity between resource states in non-and optimal dPBT. Lemma 10. The square fidelity between the resource state in non-optimal and optimal dPBT with N ports, each of dimension d is given as: where v µ are entries of an eigenvector corresponding to a maximal eigenvalue of the teleportation matrix M F , m µ , d µ denote multiplicity and dimension of irreps of S(N) in the Schur-Weyl duality, and P µ is a respective Young projector. For qubits, the fidelity between the resources states is of the form: where j min = 0(1/2) when N is even (odd).
The results of the all lemmas from this section for d = 2 and d > 2 are presented in table III and table III respectively located in Appendix D. In the same Appendix we include figure 6 and figure 7 which illustrate statements of Lemma 8, Lemma 9, and Lemma 10 for qubits.

VI. SENDING MORE BITS WITH FEWER EBITS: EFFICIENT PORT-BASED SUPERDENSE CODING
We now turn to superdense coding protocols induced by PBT schemes. In the case of ordinary teleportation, the underlying channel is given by an identity channel, and sending a single qubit would result in 2 bits of classical information.
A number of varying PBT protocols gives rise to an equal number of superdense coding schemes. Suppose that Alice performs a measurement in dPBT, then the unnormalised post-measurement states χ DB i , for 1 ≤ i ≤ N read: the second line is obtained by applying Lemma 12 from Appendix B twice. We introduce normalised postmeasurement state χ DB i since Tr Π BD i = d N+1 N due to Theorem 5 in [34]. To estimate the performance of the superdense protocols, we introduce where terms F i|k represent fidelities between the post-measurement state χ DB i and maximally entangled state P + DB k . These fidelities are of the form In the last equality we used the definition of states σ BD where B i denotes all systems B but i. Consider two cases: i = k and i = k. In the first case: The above follows from the fact that Tr Π BD k σ BD k does not depend on index 1 ≤ k ≤ N and F = 1 , which is entanglement fidelity in dPBT. We see that F k|k → 1 with N → ∞, since the same holds for entanglement fidelity F in dPBT. We turn to computing F i|k for i = k. Form the relation ∑ i q i|k = 1, where the coefficients are defined through (69), we write In PBT schemes p i = 1/N, since all the ports are equally probable, and F i|k = F for all i = k due to the covariance property, so We see that F → 1 d 2 for N → ∞. An explicit expression for the entanglement fidelity F in PBT can be turned into an explicit expression for F. We now present a protocol that beats the performance of the only known superdense protocol [20] derived from the non-optimized dPBT protocol with fidelity [35]: The closed form for similar lower bounds is generally hard to compute, but we know that there exist PBT protocols with fidelity scaling as 1 − O(1/N 2 ) [21,24] (optimal qubit dPBT) or non-optimal ones but better factor. For convenience, Table II lists the known expressions for fidelity derived in [24]. We see that the bound (75) is weaker than those in Table II. Using the expression for mutual information from [20] and plugging in the value of fidelity from the non-optimized dPBT from table II, we get: This function outperforms I(A : B) from [20] which exploits the bound from (75). Indeed, the function from (76) achieves maximum for Moreover, for any lower bound F * on fidelity in an arbitrary dPBT the maximum amount of information that the associated superdense coding protocol can transfer is given by: While we still used F * from the non-optimized dPBT above, we already get an improvement. We expect to have a dramatic improvement in the amount of communicated information and simultaneously the reduction of entanglement consumption when one uses bound F * from optimised dPBT protocols, for example by exploiting the second bound from Table II.

VII. DISCUSSION
In this paper, we introduce a novel variant of the PBT protocol called the minimal PBT. This protocol meets the minimal set of requirements that define a feasible PBT scheme. We analyze its efficiency and show that it over-performs optimized pPBT even with the resource state in a form of N pairs of maximally entangled states. In parallel, it offers the same efficiency as the pre-existing PBT schemes when one is interested in the entanglement fidelity of the transmission. In the second part, we investigate the possibility of conversion between different types of PBT, namely, we focus on conversion between probabilistic schemes to deterministic ones, and vice versa. We present the general recipe for such conversion and we show how it applies to existing variants of pPBT and dPBT with their connection to mPBT. We also derive the efficiency of such converted schemes. In the next part of the manuscript, we discuss the application of existing knowledge on the deterministic PBT to super-dense coding schemes, and we show the possibility of transmission of more classical bits with lower consumption of shared maximally entangled pairs (ports). Finally, we present a detailed analysis and comparison of the resource states in deterministic and probabilistic PBT by considering their mutual fidelities. We show that mutual fidelity between resource states decreases with the number of ports showing that the considered states become more distant in the trace norm.
We also leave two important open questions. The first one is to explore derived expressions for the efficiency of the mPBT protocol in the asymptotic limits when the number of ports N tends to infinity, analogously as it was done in [9]. Applying similar reasoning we can rid off group theoretical parameters like dimensions and multiplicities of irreps under interest and investigate their asymptotic behavior in terms of local dimension and the number of ports. Another important topic is to investigate noise influence on the performance of all known variants of PBT protocols, including defined here the mPBT scheme. It is well known that the noise in the practical implementation of all quantum information protocols is unavoidable and only having a detailed analysis in a real-world scenario regime can tell us about the real potential of discussed in the literature schemes. This is a very important problem, especially in the context of the recent developments in the PBT area -the first model of the PBT formalism in continuous variables [14]. The study of the impact of noise is therefore most natural in this setting, especially from the point of view of possible implementations.
1. Non-optimal pPBT. In this case O A = 1 A with measurement M AC N of the form (see Section 2.5.1 in [22]): where P α is a Young projector introduced in (11), P + N,n denotes the maximally entangled state between respective systems, and γ µ * (α) := min µ∈α 1 γ µ (α) The minimisation is taken over all µ which can be obtained from given α by adding a single box, as it is described in Section II B and Figure 1 within it. The quantity γ µ (α) can be easily connected with eigenvalues λ µ (α) from (16) of the operator ρ in (15): The probability of success p succ in this variant is given by the expression (see Theorem 3 in [22]): where the minimum is taken over all Young frames µ which can be obtained from a given Young frame α N − 1 by adding a single box (see Figure 1 for the details). Quantities m α , m µ , and d µ denote multiplicities and dimension of irreducible representations of corresponding symmetric group in the Schur-Weyl duality.
2. Optimal pPBT. In this case Alice's optimising operation is non-trivial, however optimal measurements differ only by coefficients (see Section 2.5.2 in [22]): The probability of success p succ in the optimal scheme is given by the a compact expression (see Theorem 4 in [22]): This expression for d = 2 reduces to expression (56) derived in [21] p succ = 1 − 3 N + 3 .
3. Optimal dPBT. Alice's optimising operation in the optimal variant is of the form (see Proposition 32 in [23]): where v µ ≥ 0 are entries of an eignevector corresponding to the maximal eigenvalue of the teleportation matrix M F used for computation of entanglement fidelity in optimal PBT (see Section 4 in [23]).
In the case when parties exploit maximally entangled state as a resource and Alice applies SRM measurements to run the protocol, the corresponding entanglement fidelity equals to (see Theorem 12 in [22]): where d µ , m µ denote dimension and multiplicity of irreps of S(N) in the Schur-Weyl duality. In particular case of qubits we can use expression (29) from [21], which is of the form: When Alice optimises over a resource state and measurements the entanglement fidelity can be computed as: where M d F is the principal minor of dimension d of the teleportation matrix M F introduced in Section 4 of [23]. the symbol || · || ∞ denotes the infinity norm of a matrix. For qubits, the expression (A12) reduces to (see expression (41) in [21] and Section 5.3 in [23]): Overlap between states for optimal dPBT and optimal pPBT calculated for qubits by using expression (56) from Lemma 8. We see that for these two states the overlap between them saturates on the value 0.778. Right hand side: Overlap between states for non-optimal and optimal dPBT for qubits by using expression (66) from Lemma 10. The maximal value of the overlap which is F = 0.9977 is attained for N = 6. In the asymptotic limit the both states are orthogonal. This plot has been firstly obtained in [34] in the context of resource state degradation. In both figures we see completely different behaviour of the overlaps between the resource states. Overlap between states for non-optimal pPBT and optimal pPBT calculated for qubits by using expression (62) from Lemma 9. We see that for these two states the overlap between them approaches 0 even for a very small number of ports N making them orthonormal. This happens much faster than for the overlap between states for non-optimal and optimal dPBT depicted in Figure 6.